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QUICK REFRESHER GUIDE For

Computer Science & Information Technology

By

www.thegateacademy.com

Quick Refresher Guide

Contents

CONTENTS Topic #1.Mathemathics 1.1Linear Algebra 1.2Probability and Distribution 1.3Numerical Methods 1.4Calculus

#2.Discrete Mathematics and Graph Theory 2.1Mathematical Logic 2.2Combinatorics 2.3Sets and Relations 2.4Graph Theory

#3.Data Structures and Algorithms 3.1Data Structure and Algorithm Analysis 3.2Elements of Discrete Mathematics for Data Structure 3.3Abstract Data Type (ADT) 3.4Stacks 3.5Queue 3.6Trees 3.7 Height Balanced Trees (AVL Trees, B and B+) 3.8Hashing 3.9Graph Algorithms 3.10Sorting Algorithms

#4.

Operating System 4.1Introduction to Operating System 4.2Process Management 4.3Threads 4.4CPU Scheduling 4.5Deadlocks 4.6Memory Management & Virtual Memory 4.7File System 4.8I/O Systems 4.9Protection and Security

#5.Data Base Management Systems 5.1ER Diagrams 5.2Functional Dependencies & Normalization 5.3Relational Algebra & Relational Calculus

Page No. 1 – 30 1–8 9 – 14 15 – 19 20 – 30

31 – 70 31 – 36 37 – 42 43 – 56 57 – 70

71 – 105 71 – 75 76 – 77 78 79 80 81 - 85 86 – 94 95 – 97 98 – 100 101 – 105

106 – 157 106 – 108 109 – 123 124 – 125 126 – 128 129 – 133 134 – 144 145 – 149 150 – 153 154 – 157

158 – 193 158 – 163 164 – 170 171 – 174

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5.4SQL 5.5Transactions and Concurrency Control 5.6File Structures (Sequential files, indexing B & B+ trees)

#6.Theory of Computation 6.1Introudction 6.2Finite Automata 6.3Regular Expression 6.4Context free grammar 6.5Turing Machines

#7.Computer Organization 7.1Introduction of Computer Organization 7.2Memory Hierarchy 7.3Pipelining 7.4Instruction Types 7.5Addressing Modes 7.6I/O Data Transfer

#8.Digital Logic 8.1Number Systems & Code Conversions 8.2Boolean Algebra &Karnaugh Maps 8.3Logic Gates 8.4Combinational Digital Circuits 8.5Semiconductor Memory

#9. Compiler Design 9.1 Introduction 9.2Syntax Analysis 9.3Syntax Directed Translation 9.4Run Time Environment 9.5Intermediate Code Generation 9.6Code Optimization 9.7Code Generation

#10. Computer Networks 10.1 Introduction to Computer Networks 10.2Multiple Access Protocols 10.3The Data Link Layer 10.4Routing & Congestion Control 10.5TCP/IP, UDP and Stocks, IP(V4) 10.6Application Layer 10.7Network Security

Contents

175 - 180 181 – 188 189 – 193

194 – 238 194 195 – 198 199 – 208 209 – 218 219 – 238

239 – 278 239 – 244 245 –252 253 – 258 259 – 263 264 – 266 267 – 278

279 – 292 279 – 280 281 – 282 283 – 286 287 – 291 292

293 – 349 293 – 300 301 – 325 326 – 334 335 – 336 337 – 342 343 – 344 345 – 349

350 – 381 350 – 357 358 – 360 361 – 365 366 – 369 370 – 376 377 – 378 379 – 381

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#11. Software Engineering and Web Technology 11.1Introduction 11.2Process Models and Software Estimation 11.3 Validation and Verification 11.4 HTML 11.5 XML & DTDs

# Reference Book

Contents

382 – 417 382 – 389 390 – 401 402 – 407 408 – 413 414 – 417

418 – 419

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Mathematics

Part - 1: Mathematics 1.1 Linear Algebra 1.1.1

Matrix Definition: A system of “m n” numbers arranged along m rows and n columns. Conventionally, single capital letter is used to denote a matrix. Thus,

A=[

a a

a a

a a

a a a

a

a a a a

]

ith row, jth column

1.1.1.1 Types of Matrices 1.1.1.2 Row and Column Matrices  Row Matrix [ 2, 7, 8, 9] 

Column Matrix

[

]

single row ( or row vector) single column (or column vector)

1.1.1.3 Square Matrix     -

Same number of rows and columns. Order of Square matrix no. of rows or columns Principle Diagonal (or Main diagonal or Leading diagonal): The diagonal of a square matrix (from the top left to the bottom right) is called as principal diagonal. Trace of the Matrix: The sum of the diagonal elements of a square matrix. tr (λ A) = λ tr(A) [ λ is scalar] tr ( A+B) = tr (A) + tr (B) tr (AB) = tr (BA)

1.1.1.4 Rectangular Matrix Number of rows

Number of columns

1.1.1.5 Diagonal Matrix A Square matrix in which all the elements except those in leading diagonal are zero. e.g.[

]

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Page 1


Mathematics

1.1.1.6 Unit Matrix (or Identity Matrix) A Diagonal matrix in which all the leading diagonal elements are ‘ ’. e.g. =

[

]

1.1.1.7 Null Matrix (or Zero Matrix) A matrix is said to be Null Matrix if all the elements are zero. e.g.[

]

1.1.1.8 Symmetric and Skew Symmetric Matrices:  Symmetric, when a = +a for all i and j. In other words  Skew symmetric, when a = - a In other words = -A

=A

Note: All the diagonal elements of skew symmetric matrix must be zero. Symmetric Skew symmetric a h g h g [h b ] [h ] g c g

Symmetric Matrix

=A

Skew Symmetric Matrix

=-A

1.1.1.9 Triangular Matrix  A matrix is said to be “upper triangular” i all the elements below its principal diagonal are zeros.  A matrix is said to be “lower triangular” i all the elements above its principal diagonal are zeros. a a h g [ ] [ g b ] b h c c Upper Triangular Matrix Lower Triangular Matrix 1.1.1.10

Orthogonal Matrix: If A.A = , then matrix A is said to be Orthogonal matrix.

1.1.1.11

Singular Matrix: If |A| = 0, then A is called a singular matrix.

1.1.1.12

̅ ) = transpose of a conjugate of matrix A Unitary Matrix: If we define, A = (A Then the matrix is unitary if A . A =


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1.1.1.13

Mathematics

Hermitian Matrix: It is a square matrix with complex entries which is equal to its own conjugate transpose.

A = A or a = a̅̅̅ 1.1.1.14

Note: In Hermitian matrix, diagonal elements

always real

1.1.1.15 Skew Hermitian matrix: It is a square matrix with complex entries which is equal to the negative of conjugate transpose. A = A ora =

a̅̅̅

Note: In Skew-Hermitianmatrix , diagonal elements 1.1.1.16

either zero or Pure Imaginary

Idempotent Matrix If A = A, then the matrix A is called idempotent matrix.

1.1.1.17

Multiplication of Matrix by a Scalar:

Every element of the matrix gets multiplied by that scalar. Multiplication of Matrices: Two matrices can be multiplied only when number of columns of the first matrix is equal to the number of rows of the second matrix. Multiplication of (m n) [ ] and(n p) matrices results in matrix o (m p)dimension [ ] =[ ] . 1.1.1.18

Determinant:

An n order determinant is an expression associated with n

n square matrix.

If A = [a ] , Element a with ith row, jth column. For n = 2 ,

a D = det A = |a

a a |=a a

-a a

Determinant o “order n”

D = |A| = det A = ||

a a

a

a

a

a

a a

| |

a


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1.1.1.19 

Mathematics

Minors & Co-Factors:

The minor of an element in a determinant is the determinant obtained by deleting the row and the column which intersect that element. Co actor is the minor with “proper sign”. The sign is given by (-1) (where the element th th belongs to i row, j column).



1.1.1.20 Properties of Determinants: 1. A determinant remains unaltered by changing its rows into columns and columns into rows. 2. If two parallel lines of a determinant are inter-changed, the determinant retains its numerical values but changes its sign. (In a general manner, a row or a column is referred as line). 3. Determinant vanishes if two parallel lines are identical. 4. If each element of a line be multiplied by the same factor, the whole determinant is multiplied by that factor. [Note the difference with matrix]. 5. If each element of a line consists of the m terms, then determinant can be expressed as sum of the m determinants. 6. If each element of a line be added equi-multiple of the corresponding elements of one or more parallel lines, determinant is unaffected. e.g.by the operation, + p +q , determinant is unaffected. 7. Determinant of an upper triangular/ lower triangular/diagonal/scalar matrix is equal to the product of the leading diagonal elements of the matrix. 8. If A & B are square matrix of the same order, then |AB|=|BA|=|A||B|. 9. If A is non singular matrix, then |A |=| | (as a result of previous). 10. 11. 12. 13.

Determinant of a skew symmetric matrix (i.e. A =-A) of odd order is zero. If A is a unitary matrix or orthogonal matrix (i.e. A = A ) then |A|= ±1. If A is a square matrix of order n, then |k A| = |A|. | | = 1 ( is the identity matrix of order n).

1.1.1.21 Inverse of a Matrix  A = | |   

|A| must be non-zero (i.e. A must be non-singular). Inverse of a matrix, if exists, is always unique. a b d If it is a 2x2 matrix [ ] , its inverse will be [ c d c

b ] a

Important Points: 1. A = A = A, (Here A is square matrix of the same order as that of ) 2. 0 A = A 0 = 0, (Here 0 is null matrix) 3. AB = , then it is not necessarily that A or B is null matrix. Also it doesn’t mean BA = . 4. If the product of two non-zero square matrices A & B is a zero matrix, then A & B are singular matrices. 5. If A is non-singular matrix and A.B=0, then B is null matrix. 6. AB BA (in general) Commutative property does not hold 7. A(BC) = (AB)C Associative property holds 8. A(B+C) = AB AC Distributive property holds 9. AC = AD , doesn’t imply C = D [even when A ]. 10. If A, C, D be matrix, and if rank (A)= n & AC=AD, then C=D. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

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11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Mathematics

(A+B)T = A + B (AB)T = B . A (AB)-1 = B . A AA =A A= (kA)T = k.A (k is scalar, A is vector) (kA)-1 = . A (k is scalar , A is vector) (A ) = (A ) ̅̅̅̅) = (A ̅ ) (Conjugate of a transpose of matrix= Transpose of conjugate of matrix) (A If a non-singular matrix A is symmetric, then A is also symmetric. If A is a orthogonal matrix , then A and A are also orthogonal.

21. If A is a square matrix of order n then (i) |adj A|=|A| (ii) |adj (adj A)|=|A|( ) (iii) adj (adj A) =|A|

A

1.1.1.22 Elementary Transformation of a Matrix: 1. Interchange of any 2 lines 2. Multiplication of a line by a constant (e.g. k ) 3. Addition of constant multiplication of any line to the another line (e. g.

+p

)

Note:  Elementary trans ormations don’t change the ran o the matrix.  However it changes the Eigen value of the matrix. 1.1.1.23

Rank of Matrix

If we select any r rows and r columns from any matrix A,deleting all other rows and columns, then the determinant formed by these r r elements is called minor of A of order r. Definition: A matrix is said to be of rank r when, i) It has at least one non-zero minor of order r. ii) Every minor of order higher than r vanishes. Other definition: The rank is also defined as maximum number of linearly independent row vectors. Special case: Rank of Square matrix Rank = Number of non-zero row in upper triangular matrix using elementary transformation. Note: 1. 2. 3. 4.

r(A.B) min { r(A), r (B)} r(A+B) r(A) + r (B) r(A-B) r(A) - r (B) The rank of a diagonal matrix is simply the number of non-zero elements in principal diagonal. 5. A system of homogeneous equations such that the number of unknown variable exceeds the number of equations, necessarily has non-zero solutions. 6. If A is a non-singular matrix, then all the row/column vectors are independent. 7. If A is a singular matrix, then vectors of A are linearly dependent. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 5


Mathematics

8. r(A)=0 iff (if and only if) A is a null matrix. 9. If two matrices A and B have the same size and the same rank then A, B are equivalent matrices. 10. Every non-singular matrix is row matrix and it is equivalent to identity matrix. 1.1.1.24

Solution of linear System of Equations:

For the following system of equations A X = B a

a a

x x

a a

Where, A =

, [

a

a

a

]

=

, [x ]

B = [

]

A= Coefficient Matrix, C = (A, B) = Augmented Matrix r = rank (A), r = rank (C), n = Number of unknown variables (x , x , - - - x ) Consistency of a System of Equations: For Non-Homogenous Equations (A X = B) i) If r r , the equations are inconsistent i.e. there is no solution. ii) If r = r = n, the equations are consistent and there is a unique solution. iii) If r = r < n, the equations are consistent and there are infinite number of solutions. For Homogenous Equations (A X = 0) i) If r = n, the equations have only a trivial zero solution ( i.e. x = x = - - -x = 0). ii) If r < n, then (n-r) linearly independent solution (i.e. infinite non-trivial solutions). Note: Consistent means:

one or more solution (i.e. unique or infinite solution)

Inconsistent means:

No solution

Cramer’s ule Let the following two equations be there a x +a x = b ---------------------------------------(i) a x +a x = b ---------------------------------------(ii) a D = |b

a b |

b D =| b

a | a


Page 6


a D =| a

Mathematics

b | b

Solution using Cramer’s rule: x =

and x =

In the above method, it is assumed that 1. No of equations = No of unknowns 2. D 0 In general, for Non-Homogenous Equations D 0 single solution (non trivial) D = 0 infinite solution For Homogenous Equations D 0 trivial solutions ( x = x =………………………x = 0) D = 0 nontrivial solution (or infinite solution) Eigen Values & Eigen Vectors 1.1.1.25

Characteristic Equation and Eigen Values:

Characteristic equation: | A λ |= 0, The roots of this equation are called the characteristic roots /latent roots / Eigen values of the matrix A. Eigen vectors: [

]X=0

For each Eigen value λ, solving for X gives the corresponding Eigen vector. Note: For a given Eigen value, there can be different Eigen vectors, but for same Eigen vector, there can’t be di erent Eigen values. Properties of Eigen values 1. The sum of the Eigen values of a matrix is equal to the sum of its principal diagonal. 2. The product of the Eigen values of a matrix is equal to its determinant. 3. The largest Eigen values of a matrix is always greater than or equal to any of the diagonal elements of the matrix. 4. If λ is an Eigen value of orthogonal matrix, then 1/ λ is also its Eigen value. 5. If A is real, then its Eigen value is real or complex conjugate pair. 6. Matrix A and its transpose A has same characteristic root (Eigen values). 7. The Eigen values of triangular matrix are just the diagonal elements of the matrix. 8. Zero is the Eigen value of the matrix if and only if the matrix is singular. 9. Eigen values o a unitary matrix or orthogonal matrix has absolute value ‘ ’. 10. Eigen values of Hermitian or symmetric matrix are purely real. 11. Eigen values of skew Hermitian or skew symmetric matrix is zero or pure imaginary. | | 12. is an Eigen value of adj A (because adj A = |A|. A ). 13. If λ is an Eigen value of the matrix then , i) Eigen value of A is 1/λ THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

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ii) iii) iv) v)

Eigen value of A is Eigen value of kA are Eigen value of A Eigen value of (A

Mathematics

λ λ (k is scalar) are λ + k )2 are ( )

Properties of Eigen Vectors 1) Eigen vector X of matrix A is not unique. Let is Eigen vector, then C is also Eigen vector (C = scalar constant). 2) If λ , λ , λ . . . . . λ are distinct, then , . . . . . are linearly independent . 3) If two or more Eigen values are equal, it may or may not be possible to get linearly independent Eigen vectors corresponding to equal roots. 4) Two Eigen vectors are called orthogonal vectors if T∙ = 0. ( , are column vector) (Note: For a single vector to be orthogonal , A = A or, A. A = A. A =  ) 5) Eigen vectors of a symmetric matrix corresponding to different Eigen values are orthogonal. Cayley Hamilton Theorem: Every square matrix satisfies its own characteristic equation. 1.1.1.26

Vector:

Any quantity having n components is called a vector of order n. Linear Dependence of Vectors  If one vector can be written as linear combination of others, the vector is linearly dependent. Linearly Independent Vectors  If no vectors can be written as a linear combination of others, then they are linearly independent. Suppose the vectors are x x x x  

Its linear combination is λ x + λ x + λ x + λ x = 0 Ifλ , λ , λ , λ are not “all zero” they are linearly dependent. If all λ are zero they are linearly independent.


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Mathematics

1.2 Probability and Distribution 1.2.1

Probability

Event: Outcome of an experiment is called event. Mutually Exclusive Events (Disjoint Events): Two events are called mutually exclusive, if the occurrence o one excludes the occurrence o others i.e. both can’t occur simultaneously. A

B =φ, P(A

B) =0

Equally Likely Events: If one of the events cannot happen in preference to other, then such events are said to be equally likely. Odds in Favour of an Event = Where m n

no. o ways avourable to A

no. o ways not avourable to A

Odds Against the Event = Probability: P(A)= =

. .

P(A)+ P(A’)= Important points:  P(A B) Probability of happening of “at least one” event o A & B  P(A B) ) Probability of happening of “both” events o A & B  If the events are certain to happen, then the probability is unity.  If the events are impossible to happen, then the probability is zero. Addition Law ofProbability: a. For every events A, B and C not mutually exclusive P(A B C)= P(A)+ P(B)+ P(C)- P(A B)- P(B C)- P(C A)+ P(A B C) b. For the event A, B and C which are mutually exclusive P(A B C)= P(A)+ P(B)+ P(C) Independent Events: Two events are said to be independent, if the occurrence of one does not affect the occurrence of the other. If P(A B)= P(A) P(B)

Independent events A & B

Conditional Probability: If A and B are dependent events, then P( ) denotes the probability of occurrence of B when A has already occurred. This is known as conditional probability. P(B/A)=

(

) ( )


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For independent events A & B

Mathematics

P(B/A) = P(B)

Theorem of Combined Probability: If the probability of an event A happening as a result of trial is P(A). Probability of an event B happening as a result of trial after A has happened is P(B/A) then the probability of both the events A and B happening is P(A B)= P(A). P(B/A), = P(B). P(A/B),

[ P(A) 0]

[ P(B) 0]

This is also known as Multiplication Theorem. For independent events A&B P(B/A) = P(B), P(A/B )= P(A) Hence P(A B) = P(A) P(B) Important Points: If P 1. 2. 3. 4.

&P are probabilities of two independent events then P (1-P ) probability o irst event happens and second ails (i.e only irst happens) (1-P )(1-P ) probability o both event ails 1-(1-P )(1-P ) probability o at least one event occur PP probability o both event occurs

Baye’s theorem: An event A corresponds to a number of exhaustive events B ,B ,..,B . If P(B ) and P(A/B ) are given then, P( )=

( (

). ( ) ). ( )

This is also known as theorem of Inverse Probability. Random variable: Real variable associated with the outcome of a random experiment is called a random variable. 1.2.2

Distribution

Probability Density Function (PDF) or Probability Mass Function: The set of values Xi with their probabilities P constitute a probability distribution or probability density function of the variable X. If f(x) is the PDF, then f(x ) = P( = x ) , PDF has the following properties:  Probability density function is always positive i.e. f(x) (x)dx = (Continuous)  ∫ (x ) = (Discrete) 


Page 10


Mathematics

Discrete Cumulative Distribution Function (CDF) or Distribution Function The Cumulative Distribution Function F(x) of the discrete variable x is defined by, (x) = (x)= P(X x) =

P(x )=

(x )

Continuous Cumulative Distribution function (CDF) or Distribution Function: If (x) = P(X x) =∫ (x)dx, then F(x) is defined as the cumulative distribution function or simply the distribution function of the continuous variable. CDF has the following properties: ( ) (x) =f(x) 0 i) = (x) 0 ii) 1 iii) If x x then (x ) (x ) , i.e. CDF is monotone (non-decreasing function) )=0 iv) ( ( ) =1 v) (a) vi) P(a x b) =∫ (x)dx =∫ (x)dx - ∫ (x)dx = (b) Expectation [E(x)]: 1. E(X) = x (x ) (Discrete case) 2. E(X) =∫ x (x )dx (Continuous case) Properties of Expectation 1. E(constant) = constant 2. E(CX) = C . E(X) [C is constant] 3. E(AX+BY) = A E(X)+B E(Y) [A& B are constants] 4. E(XY)= E(X) E(Y/X)= E(Y) E(X/Y) E(XY) E(X) E(Y) in general But E(XY) = E(X) E(Y) , if X & Y are independent Variance (Var(X)) Var(X) =E[(x

) ]

Var(X)= (x x

) (xx ) (Discrete case)

Var(X)=∫ (xx Var(X) =E(

) f(x)dx (Continuous case)

)-[E(x)]

Properties of Variance 1. Var(constant) = 0 2. Var(Cx)= C Var(x) -Variance is non-linear [here C is constant] THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

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Mathematics

3. Var(Cx D)=C Var(x) -Variance is translational invariant [C & D are constants] 4. Var(x-k) = Var(x) [k is constant] 5. Var(ax+by) =a Var(x) +b Var(y) 2ab cov(x,y) (if not independent) [A & B are constants] = a Var(x) +b Var(y) (if independent) Covariance Cov (x,y)=E(xy)-E(x) E(y) If independent

covariance=0,

E(xy) = E(x) . E(y)

(if covariance = 0, then the events are not necessarily independent) Properties of Covariance 1. Cov(x,y) = Cov(y,x) (i.e. symmetric) 2. Cov(x,x) = Var(x) 3. |Cov(x,y)| Standard Distribution Function (Discrete r.v. case): 1. Binomial Distribution : P(r) = C p Mean = np, Variance = npq, S.D. =√np 2. Poisson Distribution: Probability of k success is P (k) = no. o success trials ,n no. o trials , P success case probability mean o the distribution For Poisson distribution: Mean = , variance = , and =np Standard Distribution Function (Continuous r.v. case): 1. Normal Distribution (Gaussian Distribution): f(x) =

√

e

(

)

Where and are the mean and standard deviation respectively  P(
P(x1< x < x2) =∫

√

e

(

)

dx = Area under the curve from x1 to x2

2. Exponential distribution : f(x) = λe , x = , x 3. Uniform distribution: f(x)= , b f(x) a = , otherwise 4. Cauchy distribution : f(x)= .( ) 5. Rayleigh distribution function : f(x) =

e

, here λ

,x

Mean:  For a set of n values of a variant X=(x , x , … . . , x ) THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

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Mathematics

The arithmetic mean,̅=   

For a grouped data if x , x , … . . , x are mid values of the class intervals having frequencies , ,….., ,then, ̅= If ̅ is mean for n data;̅ is mean for n data; then combined mean of n +n data is ̅

̅

̅= If ̅̅̅ , be mean and SD of a sample size n and m , SD of combined sample of size n +n is given by, (n

n )

D = m -m ( n)

=n

+n

be those for a sample of size n then

+n D +n D

(m , = mean, SD of combined sample)

= (n

)

(n D )

Median: When the values in a data sample are arranged in descending order or ascending order of magnitude the median is the middle term if the no. of sample is odd and is the mean of two middle terms if the number is even. Mode: It is defined as the value in the sampled data that occurs most frequently. Important Points:  Mean is best measurement ( all observations taken into consideration).  Mode is worst measurement ( only maximum frequency is taken).  In median, 50 % observation is taken.  Sum o the deviation about “mean” is zero.  Sum o the absolute deviations about “median” is minimum.  Sum o the s uare o the deviations about “mean” is minimum. Co-efficient of variation =̅

100

Correlation coefficient =(x,y) =      

( , )

-1 (x, y) 1 (x,y) = (y,x) |(x,y)| = 1 when P(x=0)=1; or P(x=ay)=1 [ for some a] If the correlation coefficient is -ve, then two events are negatively correlated. If the correlation coefficient is zero, then two events are uncorrelated. If the correlation coefficient is +ve, then two events are positively correlated.

Line of Regression: The equation of the line of regression of y on x is y The equation of the line of Regression of x on y is (x

̅̅̅̅

y= x) =

̅̅̅̅

(x

̅̅̅̅

x) (y

y)

is called the regression coefficient of y on x and is denoted by byx.


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̅̅̅̅

Mathematics

is called the regression coefficient of x on y and is denoted by bxy.

Joint Probability Distribution: If X & Y are two random variables then Joint distribution is defined as, Fxy(x,y) = P(X x ; Y y) Properties of Joint Distribution Function/ Cumulative Distribution Function: 1. ( , ) = 2. ( , ) = 3. ( , ) = { ( , ) = P( y) = 0 x 1 = 0 } (x, ) = P( ) = (x) . = (x) 4. x ( , y) = (y) 5. Joint Probability Density Function: Defined as (x, y) = Property: ∫

∫

(x, y) (x, y) dx dy

=

Note: X and Y are said to be independent random variable If fxy(x,y) = fx(x) . fy(y)


Page 14


Mathematics

1.3 Numerical Methods 1.3.1

Solution of Algebraic and Transcendental Equation / Root Finding : Consider an equation f(x) = 0

1. Bisection method This method inds the root between points “a” and “b”. If f(x) is continuous between a and b and f (a) and f (b) are of opposite sign then there is a root between a & b (Intermediate Value Theorem). First approximation to the root is x1 =

.

If f(x1) = 0, then x1 is the root of f(x) = 0, otherwise root lies between a and x1 or x1 and b. Similarly x2 and x3 . . . . . are determined.  Simplest iterative method  Bisection method always converge, but often slowly.  This method can’t be used or inding the complex roots.  Rate of convergence is linear 2. Newton RaphsonMethod (or Successive Substitution Method or Tangent Method) ( ) xn+1 = xn – (

)

This method is commonly used for its simplicity and greater speed. Here (x) is assumed to have continuous derivative ’(x). This method ails i ’(x) = . It has second order of convergence or quadratic convergence, i.e. the subsequent error at each step is proportional to the square of the error at previous step.  Sensitive to starting value, i.e. The Newton’s method converges provided the initial approximation is chosen sufficiently close to the root.  Rate of convergence is quadratic.    

3. Secant Method x =x (

)– (

)

(x )

 Convergence is not guaranteed.  If converges, convergence super linear (more rapid than linear, almost quadratic like Newton Raphson, around 1.62). 4. RegulaFalsiMethod or (Method of False Position)  Regulafalsi method always converges.  However, it converges slowly.  If converges, order of convergence is between 1 & 2 (closer to 1). THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 15


Mathematics

 It is superior to Bisection method. Given f(x) = 0 Select x0 and x1 such that f(x0) f(x1) < 0 x =x -

– (

)

(

)

, (x ) =

(

)–

(

)

(

(i.e. opposite sign)

( ) )

Check if f(x0) f(x2) <0 or f(x1) f(x2) < 0 Compute x

………

which is an approximation to the root. 1.3.2 1.

Solution of Linear System of Equations Gauss Elimination Method Here e uations are converted into “upper triangular matrix” orm, then solved by “bac substitution” method. Consider a1x + b1x + c1z = d1 a2x + b2x + c2z = d2 a3x + b3x + c3z = d3 Step 1: To eliminate x from second and third equation (we do this by subtracting suitable multiple of first equation from second and third equation) a1x + b1y + c1z = d ’ (pivotal equation, a1 pivot point.) b ’y + c ’ z = d ’ b ’y + c ’ z = d ’ Step 2: Eliminate y from third equation a1x + b1y + c1z = d b ’y + c2z = d ’ c ’’z = d ”

’

(pivotal equation, b ’ is pivot point.)

Step 3: The value of x , y and z can be found by back substitution. Note: Number of operations:N =

2.

+n -

Gauss Jordon Method  Used to find inverse of the matrix and solving linear equations.  Here back substitution is avoided by additional computations that reduce the matrix to “diagonal rom”, instead to triangular orm in Gauss elimination method.  Number of operations is more than Gauss elimination as the effort of back substitution is saved at the cost of additional computation. Step 1: Eliminate x from 2nd and 3rd THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 16


Mathematics

Step 2: Eliminate y from 1st and 3rd Step 3: Eliminate z from 1st and 2nd 3.

a31

L U Decomposition It is modification of the Gauss eliminiation method. Also Used for finding the inverse of the matrix. [A]n x n = [ L ] n x n [U] n x n a11 a12 a13 1 0 0 a21 b22 c23 L21 1 0 = b32 c33 L31 L32 1

U11 U12 U13 0 U22 U23 0 0 U31

Ax = LUX = b can be written as a)LY=b and b) UX=Y Solve for rom a) then solve or rom b). This method is nown as Doolittle’s method.  Similar methods are Crout’s method and Choles y methods. 4. Iterative Method (i) Jacobi Iteration Method a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3 If a1, b2 , c3 are large compared to other coefficients, then solving these for x, y, z respectively x = k1 – l1y – m1z y = k2 – l2x – m2z z = k3 – l3x – m3y Let us start with initial approximation x0 , y0 , z0 x1= k1 – l1y0 – m1z0 y1= k2 – l2y0 – m2z0 z1= k3 – l3y0 – m3z0 Note: No component of x(k) is used in computation unless y(k) and z(k)are computed. The process is repeated till the difference between two consecutive approximations is negligible. In generalized form: x(k+1) = k1 – l1 y(k) – m1z(k) y(k+1) = k2 – l2 x(k) – m2z(k) z(k+1) = k3 – l3 x(k) – m3y(k) (ii) Gauss-Siedel Iteration Method Modi ication o the Jacobi’s Iteration Method Start with (x0, y0, z0)= (0, 0, 0) or anything [No specific condition] THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 17


Mathematics

In first equation, put y = y0 z = z0 which will give x1 In second equation, put x = x1 and z = z0 which will give y1 In third equation, put x = x1and y = y1 which will give z1 Note: To compute any variable, use the latest available value. In generalized form: x(k+1) = k1 – l1y(k) – m1z(k) y(k+1) = k2 – l2x(k+1) – m2z(k) z(k+1) = k3 – l3x(k+1) – m3y(k+1) 1.3.3

Numerical Integration

Trapezoidal Formula:Step size h = (x)dx =

∫

h

{( irst term

last term)

(remaining terms)}

Error = Exact - approximate The error in approximating an integral using Trapezoidal rule is bounded by h

a) max | ( )|

(b

[ , ]

Simpson’s One Third Rule (Simpson’s Rule): h (x)dx = {( irst term last term) ∫

(all odd terms)

(all even terms)}

The error in approximating an integral using Simpson’s one third rule is h

(b

a) max |

( )

[ , ]

( )|

Simpson’s Three Eighth Rule: (x)dx =

∫

h ( irst term {

last term)

(all multiple o terms) } (all remaining terms)

The error in approximating an integral using Simpson’s / rule is (b

a)

max | [ , ]

( )

( )|

1.3.4 Solving Differential Equations (i) Euler method (for first order differential equation ) Given equation is y = f(x, y); y(x0) = y0 Solution is given by, Yn+1 = yn + h f(xn,yn)


Page 18


Mathematics

(ii) Runge Kutta Method Used for finding the y at a particular x without solving the 1st order differential equation = (x, y) K1 = h f(x0, y0) K2 = h f(x0 + , y0 + ) K3 = h f(x0 + , y0 + ) K4 = h f(x0 +h, y0 + k3) K = (k1 + 2k2 + 2k3 + k4) Y(x0+h) = y0 + k


Page 19


Mathematics

1.4 Calculus 1.4.1

Limit of a Function

Let y = f(x) Then lim (x)= 0<|x a|< ,| (x)

i.e, “ (x) |<

as x a” implies or any

(>0), (>0) such that whenever

Some Standard Expansions (

x) =

x

a =x a

x

(

nx x

)

a

x

e =1+x+ +

.........

log(

x) = x

+

log(

x) =

x

Sin x = x

n(n

a

)(n

)

x

.........x

.........a

......... ......... .........

Cos x = 1

+

Sinh x = x

......... .........

Cosh x = 1

+

.........

Some Important Limits lim

sinx = x

lim (

x

lim( lim lim

) =

x) =

a x e x

= log a =


Page 20


lim lim

log(

x) x

x x

Mathematics

=

a = a

a

lim log|x| = L – Hospital’s ule  When function is of limit.

or

form, differentiate numerator &denominator and then apply

Existence of Limits and Continuity: 1. f(x) is defined at a, i.e, f(a) exists. 2. If lim f(x) = lim f(x) = L ,then the lim f(x) exists and equal to L. 3.

lim

(x) = lim

(x)= f(a) then the function f(x) is said to be continuous.

Properties of Continuity If f and g are two continuous functions at a; then a. (f+g), (f.g), (f-g) are continuous at a b. is continuous at a, provided g(a) 0 c. | | or |g| is continuous at a olle’s theorem If (i) f(x) is continuous in closed interval [a,b] (ii) ’(x) exists or every value o x in open interval (a,b) (iii) f(a) = f(b) Then there exists at least one point c between (a, b) such that

( )=0

Geometrically: There exists at least one point c between (a, b) such that tangent at c is parallel to x axis

C C 2

a

C1

b


Page 21


Mathematics

Lagrange’sMean Value Theorem If (i) f(x) is continuous in the closed interval [a,b] and (ii) ’(x) exists in the open interval (a,b), then atleast one value c o x exist in (a,b) such that ( )

( )

= (c).

Geometrically, it means that at point c, tangent is parallel to the chord line.

Cauchy’sMean Value Theorem If (i) f(x) is continuous in the closed interval [a,a+h] and (ii) (x) exists in the open interval (a,a+h), then there is at least one number (0< <1) such that f(a+h) = f(a) + h f(a+ h) Let f1 and f2 be two functions: i) f1,f2 both are continuous in [a,b] ii) f1, f2 both are differentiable in (a,b) iii) f2’ 0 in (a,b) then, for a ( ) ( )

1.4.2

( ) = ( )

( ) ( )

Derivative:

’( ) =lim

(

)

( )

Provided the limit exists ’( ) is called the rate of change of f at x. Algebra of derivative:i. ( g) = g ii. ( g) = – g iii. ( . g) = . g iv. ( /g) =

.

.g

.


Page 22


Mathematics

Homogenous Function Any function f(x, y) which can be expressed in from xn ( ) is called homogenous function of order n in x and y. (Every term is of nth degree.) f(x,y) = a0xn + a1xn-1y + a2xn-2y2 f(x,y) = xn

………… an yn

( )

Euler’s Theorem on Homogenous Function If u be a homogenous function of order n in x and y then,  x +y = nu 

x

1.4.3

+ 2xy

+y

= n(n

)u

Total Derivative

u= (x,y) ,x=φ(t), y=Ψ(t) =

.

u=

+

.

x+

y

Monotonicity of a Function f(x) 1. f(x) is increasing function if for , ( ) Necessary and su icient condition, ’ (x) 2. f(x) is decreasing functionif for ,, ( ) Necessary and sufficient condition, (x) Note:

( ) ( )

is a monotonic unction on a domain ‘D’ then is one-one on D.

Maxima-Minima a) Global

b) Local

Rule for finding maxima & minima:  If maximum or minimum value of f(x) is to be found, let y = f(x)  Find dy/dx and equate it to zero and from this ind the values o x, say x is , β, …(called the critical points). 

Find

at x = ,

If

, y has a minimum value

If

,y has a maximum value


Page 23


If

Mathematics

= , proceed urther and ind at x = .

If

, y has neither maximum nor minimum value at x =

But If

= , proceed further and find

If

, y has minimum value

If

, y has maximum value

If

at x = .

= , proceed further

Note: Greatest / least value exists either at critical point or at the end point of interval. Point of Inflexion If at a point, the following conditions are met, then such point is called point of inflexion

Point of inflexion i) ii)

=

,

=0,

iii)

 Neither minima nor maxima exists

Taylor Series: (a

h)= (a)

h ’(a)

”(a)

.........

Maclaurian Series: (x) = ( )

x ’( )

( )

h

( )

Maxima &Minima (Two variables) r=

,s=

1.

= 0,

2. (i) if rt (ii) ifrt (iii) ifrt (iv) ifrt

, t= =

solve these e uations. Let the solution be (a, b), (c, d)…

s and r maximum at (a, b) s and r minimum at (a, b) s < 0 at (a, b), f(a,b) is not an extreme value i.e, f(a, b) is saddle point. s > 0 at (a, b), It is doubtful, need further investigation.


Page 24


1.4.4

Mathematics

Standard Integral Results

1. ∫ x dx =

, n

2. ∫ dx = log x 3. ∫ e dx = e 4. ∫ a dx = (prove it ) 5. 6. 7. 8. 9. 10. 11.

∫ cos x dx = sin x ∫ sin x dx = cos x ∫ sec x dx = tan x ∫ cosec x dx = cot x ∫ sec x tan x dx = sec x ∫ cosec x cot x dx = cosec x dx = sin ∫ √

12. ∫

√

dx =

sec

13. ∫ dx = sec x √ 14. ∫ cosh x dx = sinh x 15. ∫ sinh x dx = cosh x 16. ∫ sech x dx = tanh x 17. ∫ cosech x dx = coth x 18. ∫ sech x tanh x dx = sech x 19. ∫ cosec h x cot h x dx = cosech x 20. ∫ tan x dx = log sec x 21. ∫ cot x dx = log sin x 22. ∫ sec x dx = log( sec x tan x) = log tan( ⁄ 23. ∫ cosec x dx = log(cosec x cot x) = log tan

x⁄ )

24. ∫ √

dx = log(x

√x

a ) = cosh ( )

25. ∫ √

dx = log(x

√x

a ) = sinh ( )

26. ∫ √a

x dx =

27. ∫ √a

x dx = √x

a

log(x

√x

a )

28. ∫ √x

a dx = √x

a

log(x

√x

a )

√

sin

29. ∫

dx =

tan

30. ∫

dx =

log (

) where x
31. ∫

dx =

log (

) where x > a

32. ∫ sin x dx = 33. 34. 35. 36.

sin x

sin x ∫ cos x dx = ∫ tan x dx = tan x x ∫ cot x dx = cot x x ∫ ln x dx = x ln x x

37. ∫ e

sin bx dx =

(a sin bx

b cos bx )

38. ∫ e

cos bx dx =

(a cos bx

b sin bx )


Page 25


39. ∫ e [ (x)

Mathematics

(x)]dx = e (x)

Integration by parts: ∫

u v dx = u. ∫ v dx

∫(

∫ v dx)dx

I L A T E E

Selection of U & V Inverse circular (e.g. tan x)

Exponential

Logarithmic

Algebraic Trigonometric

Note: Take that function as “u” which comes first in “ILATE” 1.4.5 Rules for Definite Integral 1. ∫ (x)dx =∫ (x)dx+∫ (x)dx 2. ∫ (x)dx =∫ (a 3. ∫ (x)dx =∫

/

b

x)dx

(x)dx+∫

a
∫ (x)dx =∫ (a

x)dx

/

(a x)dx ∫ (x)dx = ∫ if f(a-x)=f(x) =0 if f(a-x)=-f(x) 4. ∫ (x)dx =2 ∫ (x)dxif f(-x) = f(x), even function =0 if f(x) = -f(x), odd function

/

(x)dx

Improper Integral Those integrals for which limit is infinite or integrand is infinite in a then it is called as improper integral. 1.4.6 Convergence:  ∫ (x)dxis said to be convergent if the value of the integral is finite. (i) (x) g(x) for all x and (ii) ∫ g(x)dx converges , then ∫  

(i) (x)

g(x) ( ) ( )

for all x and (ii) ∫ g(x)dx diverges, then ∫



If lim



diverge. is converges when p ∫



∫ e



The integral ∫



The integral ∫

= c where c 0, then both integrals ∫

dx and ∫

)

(

)

b in case of ∫ (x)dx,

(x)dx also converges (x)dx also diverges

(x)dx and ∫ g(x)dx converge or both

and diverges when p

e dx is converges for any constant p

(

x

and diverges for p

is convergent if and only ifp is convergent if and only ifp


Page 26


1.4.7

Mathematics

Vector Calculus:

Scalar Point Function: If corresponding to each point P of region R there is a corresponding scalar then (P) is said to be a scalar point function for the region R. (P)= (x,y,z) Vector Point Function: If corresponding to each point P of region R, there corresponds a vector defined by F(P) then F is called a vector point function for region R. F(P) = F(x,y,z) = f1(x,y,z) ̂ +f2(x,y,z)ĵ

3(x,y,z) ̂

Vector Differential Operator or Del Operator:

=(

ĵ

̂

)

Directional Derivative: ⃗⃗ is the resolved part of The directional derivative of f in a direction N

⃗⃗ . in directionN

⃗⃗ = | |cos .N ⃗ is a unit vector in a particular direction Where ⃗N Direction cosine: l

m

n =

Where, l =cos , m=cos β , n=cos , 1.4.8

Gradient:

The vector function as grad f.

is defined as the gradient of the scalar point function f(x,y,z) and written

grad f =

+̂

   1.4.9

=î

ĵ

is vector function If f(x,y,z)= 0 is any surface, then is a vector normal to the surface f and has a magnitude equal to rate of change of f along this normal. Directional derivative of f(x,y,z) is maximum along and magnitude o this maximum is | |. Divergence:

The divergence of a continuously differentiable vector point function F is denoted by div. F and is defined by the equation. div. F = . THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 27


Ψ̂

F=f + ĵ div.F= . = ( =  

Mathematics

+

̂

ĵ

) .( f + ĵ

Ψ̂)

+

. is scalar . = is Laplacian operator

1.4.10 Curl: The curl of a continuously differentiable vector point function F is denoted by curl F and is defined by the equation. ĵ Curl F =

̂

=|

| φ

Ψ

is vector function 1.4.11 SolenoidalVector Function If .A = 0 , then A is called as solenoidal vector function. 1.4.12 IrrotationalVectorFunction If

A =0, then A is said to be irrotational otherwise rotational.

1.4.13 DEL Applied Twice to Point Functions: 1. div grad f = 2. 3. 4. 5.

f=

+

+

---------- this is Laplace equation

curl grad f = =0 divcurl F = . =0 curl curl F = ( )= ( . )grad div F = ( . )= ( )+

F F

1.4.14 Vector Identities: f, g are scalar functions & F, G are vector functions 1. ( g) = + g )= . 2. . ( . ( )= 3. 4. ( g) = f g + g f 5. . ( )= . . 6. ( )= ) 7. ( . ) = F ( ( ) )= G.( 8. . ( ) .( ) ( )= F( 9. ) ( ) THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com

Page 28


Mathematics

Also note: 1. (f/g)= (g f – f g)/g 2. ( . )’ = ’. . ’ 3. (F )’ = ’ G + F ’ 4. (fg) = g f + 2 f. g + f

g

1.4.15 Vector product 1. Dot product of A B with C is called scalar triplet product and denoted as [ABC] Rule: For evaluating the scalar triplet product (i) Independent of position of dot and cross (ii) Dependent on the cyclic order of the vector [ABC] = A B. C = A. B C = B C. A= B.C A = C A. B = C.A B A B. C = -(B A. C) ⃗ B ⃗ = (extreme adjacent) Outer ⃗) C 2. (A = (Outer. extreme) adjacent (Outer. adjacent) extreme ⃗⃗⃗⃗ ⃗⃗⃗ ⃗ = (C ⃗ .A ⃗ )B ⃗ .B ⃗ ⃗ - (C ⃗ )A  (A B) C ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗  A (B C ) = (A . C ) B - (A . B )C ⃗ ⃗B ) ⃗C ⃗A (B ⃗ ⃗C )  (A 1.4.16 Line Integral, Surface Integral & Volume Integral 

Line integral = ∫ ( )d ( )= (x,y,z) ĵ (x,y,z) + ̂ Ψ(x,y,z) d = dx ĵ dy ̂ dz dy Ψ dz ) ∫ ( )d = ∫ ( dx



⃗⃗⃗⃗ or∫ ⃗ . ⃗N ⃗ ds, Where N is unit outward normal to Surface. Surface integral: ∫ ⃗ .ds



Volume integral : ∫

dv

If F(R ) = f(x,y,z)î +

(x,y,z)ĵ

∫ 1.4.17

dv = î∫ ∫ ∫ dxdydz

Ψ (x,y,z) ̂ and v = x y z , then

ĵ ∫ ∫ ∫ dxdydz + ̂ ∫ ∫ ∫ Ψdxdydz

reen’s Theorem

If R be a closed region in the xy plane bounded by a simple closed curve c and if P and Q are continuous unctions o x and y having continuous derivative in , then according to reen’s theorem. ∮ (P dx

dy) = ∫ ∫ (

x

P ) dxdy y


Page 29


Mathematics

1.4.18 Sto e’s theorem If F be continuously differentiable vector function in R, then ∮ . dr = ∫

.N ds

1.4.19 Gauss divergence theorem The normal surface integral of a vector point function F which is continuously differentiable over the boundary of a closed region is equal to the ∫

. N. ds = ∫ div

dv


Page 30


DMGT

Part – 2: Discrete Mathematics and Graph Theory 2.1: Mathematical Logic

Logic is a formal language and it has a syntax and semantics and a way of manipulating expressions in the language. Syntax is a description of what you are allowed to write down (i.e., what expressions are legal) Semantics give meanings to legal expressions. A language is used to describe about a set Logic usually comes with a proof system which is away of manipulating syntactic expressions which will give you new syntactic expressions The new syntactic expressions will have semantics which tell us new information about sets. In the next 2 topics we will discuss 2 forms of logic 1. Propositional logic 2. First order logic Propositional logic Sentences are usually classified as declarative, exclamatory interrogative, or imperative Proposition is a declarative sentenceto which we can assign one and only one of the truth values “true” or “false” or A statement which is either “true” or “false” Assumptions about propositions  For every proposition p, if p is not true, then it is false & vice-versa.  For every proposition p, it cannot be simultaneously “true” or “false”

 Atomic proposition: are the proposition which cannot be further divided Ex: p, q, r  2 or more propositions may be combined to form compound proposition using logical connectives. Conjunction (v), disjunction (∧), implication ⟶, by implication (⟷), negation ( ) are 5 basic connectives If p is a proposition then negation p is a compound proposition. We represent it as p. Example: p: 2+2=4 p: 2+2≠4

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 31


DMGT

Truth table for p

p

p T

F

F

T

If p and q are 2 propositions then p conjunction q (p and q) is a compound proposition denoted by p ∧ . p ∧ is true if both p and q are true otherwise it is false Truth table

p

q

p∧ q

T

T

T

T

F

F

F

T

F

F

F

F

If p and q are 2 propositions then p disjunction q (p or q) is a compound proposition denoted by p . p is false if both p and q are false otherwise it is true. If p and q are 2 propositions the p “implication” q is a compound proposition denoted by (antecedent) p ⟶ (consequent). p ⟶ is false if p is true and q is false. Otherwise it is true. Whenever p is “false” p ⟶ is true. truth table for p ⟶

truth table for p

p

q

p⟶ q

p

q

p q

T

T

T

T

T

T

T

F

F

T

F

T

F

T

T

F

T

T

F

F

T

F

F

F

{ p We also write p ⟶

p, converse , inverse or opposite p, contra positive

as if p then q or q if p

If p and q are 2 propositions then p bi implication q is a compound proposition denoted by p⟷ p ⟷ is true if both p, q should have same truth value otherwise it is false. Also denoted as p iff q Truth table



p

q

p⟷ q

T

T

T

T

F

F

F

T

F

F

F

T

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Two compound propositions are said to be equivalent if they have the same truth tables. Given below are some of the equivalence propositions. ( (p)) p (p ) p∧ (p ∧ ) p p⟶ p ( p )∧( p⟷ p), p (p ⟶ ) ∧ ( ⟶ p) A propositional function is a function whose variables are propositions A propositional function p is called tautology if the truth table of p contains all the entries as true ⟶ e p p A propositional function p is called contradiction if the truth table of p contains all the entries as false⟶ e p ∧ p A propositional function p which is neither tautology nor contradiction is called contingency (i e A statement should be “true” for atleast one case, A statement should be “false” for atleast one case) ⟶ e p ⟶ Whenever p ⟶ is a tautology, then we can replace the symbol ⟶ with i.e. p Ex. If p q, then p ⟶ is a tautology, is a tautological implication. Inference will be used to designate a set of premises accompanied by a suggested conclusion. (p ∧ p ∧ p p )⟶Q Here each p is called premise and Q is called conclusion. If (p ∧ p ∧ p p ) ⟶ Q is tautology then we say Q is logically derived from p ∧ p p i.e., from set of premises i.e., {p , p , , p i.e. whenever the premises p , p , , p are true, then Q is true. or p , p , p if conclusion ‘ ’ is obtained by rule of inference Q Otherwise it is called invalid inference Some rules of inference are: ∧ I1 ∧ I2 I3 I4 I5 P ( ) I6 I7 I8 I9 I10 I11

( ( , P,P ,

) ) ∧Q

Q Q

}

(simplification)

}

(addition)

(A false antecedent p implies P to be “true”) (A “true” conse uent is implied by any proposition p) (if Q statement P is false, p is “true”) (if a statement P is “false”, is false) (disjunctive syllogism) (modus ponens)



I12 I13 I14 I15 I16 I17

,

P

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(modus tollens) (hypothetical syllogism) (dilemma) (constructive dilemma) (destructive dilemma

, R R , R, R R , R, S R S R, S, R S V Q ( Λ ), ⇨ Q P P

(conjunctive syllogism)

EQUIVALENCES E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E25 E26 E27 E28 E29 E30

⇔ Λ ⇔ Λ } ⇔ ( Λ )ΛR ⇔ Λ( ( ) R⇔ ( Λ( R) ⇔ ( Λ ( ΛR) ⇔ ( ( Λ )⇔ ( )⇔ Λ ⇔ Λ ⇔ R ( Λ )⇔R RΛ( )⇔R R ( )⇔T RΛ( Λ )⇔F ⇔ ( )⇔ Λ ⇔ ( R)⇔( Λ ( )⇔~ ⇔( )Λ( (P Q) ⇔( Λ ) ( Λ ) ⇔P PV~P⇔T Λ ⇔F PVF⇔P PVT⇔T ΛT⇔P ΛF⇔F

(double negation) (commutative laws) ΛR) } R) ) ( ΛR) } )Λ( R)

(associative laws) (distributive laws) (De Morgan’s law)

}

(contrapositive) ) R

(

) Λ

)

(absorption laws)

(trivial laws)



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First Order Logic 1.

Let A be a given set. A propositional function (or an open sentence or condition)defined on A is an expression p(x) which has the property that p(a)is true or false for each ‘a’ in A. That is, p(x) becomes a statement (with a truth value) whenever any element a ∈ A is substituted for the variable x.

2.

The set A is called the domain of p(x), and the set Tp of all elements of A for which p(a) is true is called the truth set of p(x).

3.

Tp= {x | x ∈ A, p(x) is true} or Tp= {x | p(x)}.

4.

Frequently, when A is some set of numbers, the condition p(x) has the form of an equation or inequality involving the variable x.

5.

The symbol ∀which reads “for all” or “for every” is called the universal quantifier.

6. 7.

The expression (∀x ∈ A) p(x) or ∀x p(x) is true if and only if p(x) is true for all x in A. The symbol ∃which reads “there e ists” or “for some” or “for at least one” is called the existential quantifier.

8.

The expression (∃x ∈ A)p(x) or ∃x, p(x) is true if and only if p(a) is true for atleast one element x in A. Sentence Abbreviated Meaning ∀ x, F(x)

all true

∃x, F(x)

at least one true

~[∃x, F(x)]

none true

∀ x, [~F(x)]

all false

∃x, [~F(x)]

at least one false

~(∃x,[~F(x)])

none false

~(∀ x,[F(x)])

not all true

Statement

Negation

“all true”

∀x, F(x)

∃ , [ F( )] “ at least one false”

“at least one false”

∃x, [~F(x)]

∀ , F( )

“all false”

∀x, [~F(x)]

∃x, F(x)

“at least one true”

∃x, F(x)

∀ , [ F( )] “all false”

“all true” “at least one true”



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We see that to form the negation of a statement involving one quantifier, we need to only change thequantifier from universal to existential, or from existential to universal and negate the statements which it quantifies. i.e. (∀ , F( )) ∃x[ F( )], (∃ , F( )) ∀ [ F( )] Sentences with Multiple quantifiers: In general if P (x, y) is any predicate involving the two variables x and y, then the following possibilitiesexist: (∀x)(∀y)P (x,y)

(∀x)(∃y) P (x,y)

(∃x)(∀y)P (x,y)

(∃x)(∃y) P (x,y)

(∀y)(∀x)P (x,y)

(∃y)(∀x) P (x,y)

(∀y)(∃x)P (x,y)

(∃y)(∃x) P (x,y)

(∀ )(∀ )

(∀ )(∀ ) (∃ )(∀ )

∃∀

(∃ )(∀ )

P(x,y) (∀ )(∃ )

(∀ )(∃ ) (∃ )

(∃ )(∃ )

(∃ )(∃ )

∀∃

Rules of Inference For Quantified Propositions: “Universal Specification”: If a statement of the form ∀x, P(x) is assumed to be true then the universal quantifier can be dropped to obtain that P(c) is true for any arbitrary object c in the ∀ , ( ) universe. This rule may be represented as ( ) Universal generalization : If a statement P(c) is true for each element c of the universe, then the universal quantifier may be prefixed to obtain ∀x, P(x), In symbols, this rule is ( )

∀ , ( )

This rule holds provided we know P(c) is true for each element c in the universe. Existential specification: If ∃x, P(x) is assumed to be true, then there is an element c in the universe such that P(c) is true. This rule takes the form. ∃ , ( ) ( )



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2.2: Combinatorics Counting: Let x be a set. Let us use |x| to denote the number of elements in x. Two elementary principles act as “building blocks” for all counting problems Sum rule:  If a set x is the union of disjoint non empty subsets s . . . . . . s , then |x| = |s | + |s |+ . . . . . . |s |. 

If E1, E2, En are mutually exclusive events, and E1 can happen in e1 ways, E2 can happen in e2 ways, ,En can happen in en ways, then E1 or E2 or En can happen e1 + e2 + en ways.

Product rule:  If s , s . . . . . s are non empty subsets then the number of elements in the Cartesian product s s s . . . . . s is the product ∏ s  If events E1, E2 En can happen in (e1, e2 , en) ways respectively, then the sequence of events E1 first followed by E2 , followed by En can happen in (e1 x e2 en ) ways.

Permutations: P (n, r) = = the number of permutations of n objects taken r at a time repetitions) = n (n – 1)(n – 2)

(without any

(n +r + 1)

NOTE: 1. 2. 3. 4. 5. 6.

P (n, n) = n! i.e.., there are n! permutations of n objects There are (n - 1)! permutations of n distinct objects in a circle. A permutation of n objects taken r at a time (also called an r - permutation of n objects) is an ordered selection or arrangement of r of the objects. U (n, r) = The number of r-permutations of n objects with unlimited repetitions =nr. The number of permutations of n objects of which n1 are alike, n2 are alike nr are alike is The number of ordered partitions of a set S of type (q1, q2 q2 z) =

z).

Where |S| = n is p (n; q1,



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Combinations A combination of n – objects taken r at a time (called an r-combination of n objects) is an unordered selection of n objects. C (n, r) = The number of combinations of n – objects taken r at time (without repetition) n r (n r) NOTE 1. P (n, r) = r! C(n, r) 2. C (n, n) = 1 3. V (n, r) = The number of combinations of n distinct objects taken r at a time with unlimited repetitions. 4. V (n, r) = C ( n – 1 + r, r ) = C (n – 1 + r, n – 1) V (n, r) =The number of ways distributing r – similar balls in to n numbered boxes. V (n, r) =The number of non negative integral solutions to x1 + x2 + + n = r. Generating functions can be used to compute V (n, r). Enumeration of permutations and combinations The number of r-permutations of n-elements without repetitions = n

= P(n, r)

The number of r-combinations of n-elements without repetitions = nC = C(n, r)

Enumerating combinations and permutations with repetitions The number of r-permutations with unlimited repetitions = U(n, r) = n The number of r-combinations of n objects with unlimited repetitions = V(n, r) = (n + r –1)C Enumerating permutations with constrained repetitions: a , a . . . . . . a are t distinct objects, each a is repeating Let

+

.....

times.

= n, then

The number of n-permutations of that objects denoted by P (n ,

,

,

.....

)=



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Pigeonhole Principle: If A is the average number of pigeons per hole, then some pigeon hole contains at least ⌈A⌉ pigeons and some pigeon hole contains at most ⌊A⌋ pigeons. E ample: If n + 1 pigeons are distributed among ‘n’ pigeon holes, then some pigeon hole contains at least 2 pigeons. E ample: If 2n + 1 pigeons are distributed among ‘n’ pigeon holes, then some pigeon hole contains at least 3 pigeons. E ample: In general, if k is a positive integer and ‘kn+1’ pigeons are distributed among ‘n’ pigeon holes, then some hole contains at least ‘k + 1’ pigeons Example: In a group of 61 people at least 6 people were born in the same month. Example: If 401 letters were delivered to 50 apartments, then some apartment received at most 8 letters. Euler’s ∅ - Function: If n is a ‘+ve’ integer then ∅ (n) = The number of integers such that 1 n and such that n and are relatively prime ∅ (n) = n [(1 – (1/p1)) (1- (1/p2)) (1- (1/pk))] Where p1, p2 ,

pk are prime divisors of n.

Derangements: Among the permutations of 1, 2, , n there are some, called derangements, in which none of the n integers appears in its natural place. Dn = The number of derangements of n elements Dn = n ∑

(

)

Summation: 1. 2. 3. 4. 5.

C (n, r) C (r, k) = C (n, k) = C (n – k, r – k) = for integers n > r > k > 0 (Newtons identity) C (n, r).r = n. C (n -1, r – 1) P (n, r) = n. P (n – 1, r – 1) ascal’s identity C (n, r) = C (n - 1, r) + C (n - 1, r – 1) C (n, 0) = 1 = C (n, n) C (n, 1) = n = C (n, n – 1)

Notation: C (n, 0) = C0,

C (n, 1)= C1,

C (n, 2) = C2

C (n, r) = Cr

C (n, n) = Cn.



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Row Summation: C 0 + C1 + C2

+ Cn = 2n

Row square summation: 6. C + C + C + +C = C (2n, n) 7. C0 - C1 + C2 C3 + + (-1)nCn = 0 8. (C0 + C2 + C4 + ) = ( C 1 + C3 + C5 + 9. C1 + 2C2 +3C3 + + n Cn = n 2n-1 0 1 2 10. 2 C0 + 2 C1 + 2 C2 + + 2nCn = 3n

) = 2n-1

Diagonal Summation: 11. C (n, 0) + C (n + 1, 1) + C (n + 2, 2) + 12. C (m, 0) C (n, 0) + C (m, 1) C (n, 1) + m>n>0

+ C (n + r, r) = C (n +r + 1, r) + C (m, n) C (n, n) = C (m + n, n) for integers

Column Summation: 13. 14. 15. 16.

C (r, r) + C (r + 1, r) + + C (n, r) = C (n + 1, r + 1), for any positive integer n > r 2.1 C2 + 3.2 C2 + + n (n – 1) Cn = 2n - 1 n (n - 1) 12 C1 + 22 C2 + + n2Cn = 2n – 2 n (n + 1) 3 3 1 C1 + 2 C2 + + n3Cn = 2n – 3 n2 (n + 3)

Recurrence Relations Definition: A recurrence relation is a formula that relates for any integer n>1, the n-th term of sequence an to one or more of the terms a0, a1 an-1. Examples of recurrence relations: If Sn denotes the sum of the first n positive integers, then 1. Sn=n+Sn-1. Similarly if d is a real number, then the nth term of an arithmetic progression with common difference satisfies the relation 2. an=an-1+d. Likewise if Pn denotes the nth term of a geometric progression with common ratio r, then 3. Pn = rPn-1. We list other examples as: Fibonacci relation: The relation F = F conditionF = 0 and = F = 1

+F

is called the Fibonacci relation with the initial

The numbers generated by the Fibonacci relation with the initial condition are called Fibonacci numbers. 0, 1, 1, 2, , , , 1 ,



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Linear Recurrence Relation A relation of the form c (n)a + c (n)a c (n), c (n)

+

c (n)a

= f(n), when n,

∈

,n >

c (n)&f(n) are function of n is linear recurrence relation.

ifc (n) are constants, then Linear Recurrence Relation with constant coefficient Homogeneous f(n) = 0 Non-homogeneous f(n) ≠ 0 Degree: the number of previous terms of sequence used to define an Ex: 1. f = f 2. 3. 4. 5.

a a a a

+ f

=a =a =a = a

(degree z linear)

(degree 5 linear) + a ( non-linear) + 2a ( linear homogeneous degree 2 ) + 11a + a (linear homogeneous degree 3)

Solution Of Recurrence Relations Homogenous linear recurrence relations with coefficients: In general, to solve the equation y

+a y

+a y

+

+a y = 0

Where a’s are constants: i) ii) iii)

Write the equation in the symbolic form (E + a E Write down the auxillary equation i.eE + a E + write the solution as follows:

Roots for A.E 1.

,

,

2.

,

,

3.

,

4. i

+ +a )y =0 +a = 0 and solve it for E

Solution i.e.C.F ,

(real and distinct roots) ,

,

c ( ) +c ( ) +c ( ) +

(2 real and e ual roots) (c + c n)( ) + c ( ) +

( real and e ual roots)

+i , (a pair of imaginary roots)

(c + c n + c n )( ) + r (c cos n + c sin n ) where r= √(

+

) and = tan (

)



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Non Homogeneous Linear Recurrence Relations With Constant Coefficients. Rules for Finding The Particular Solution: Consider the equation y

+a y

+

+ a y = f(n)

Which in symbolic form is Where ∅(E) =E + a E

+ +a .

Then the particular integral is given by P.I =[1/∅(E)]f(n) Case (i): when f(n)=a P.l =[1/∅ (E) ]a , Put E=a. = [1/∅(a) ]a ,provided ∅(a) ≠ 0 If ∅(a) =0, then for equation (i) (E-a)y = a , (ii) (E

I = (1/E-a ) a = na

a) y = a ,

(iii) ((E

I=(

a) y = a ,

a =

)

I=(

)

a =

(

) (

a )(

)

a

and so on. Case(ii): When f(n)=np P.I. =

( )

np =

(

)

nP.

(1) Expand [ (1 + )]-1 in ascending powers of by the Binomial theorem, as far as the term in P. (2) Express np in the factorial form and operate on it with each term of the expansion. Case(iii): When f(n) = an F(n), F(n) being a polynomial of finite degree in n. P.I. =

( )

an F(n)=an.

(

)

F(n)

Now F(n) being a polynomial in n, proceed as in case ii.

Divide And Conquer Relations Relations of the form a = c a

+ f(n) where c and d are constants.

Usually these relations can be solved by substituting a power of d for n.



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2.3: Sets and Relations Set: Set is a unordered collection of well defined objects (called the elements of the set). A, B, C,

denotes the set

a, b, c,

denotes the elements

∈

is an element of the set “A” is not an element of the set “A”

Set builder notations:A={

( )

( )property that element of A should have ex: 0 =

∈

is odd}

0 = 1, , , 7, Sub Set: Let A and B be any two sets. If every element of A is an element of B, then A is called a subset of B Symbolically this relation is denoted by A⊆ B Note: For any set A, A⊆A and ∅ ⊆ A i e , for any set A,∅ and A are called trivial sub sets Some examples of set are N = the set of natural numbers or positive integers: 1, 2, 3, . . . Z = the set of all integers: . . . , 2, 1, 0, 1, 2, . . . Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers Observe that N ⊆ ⊆ ⊆R ⊆C. Proper Sub Set: If A⊆ B and A ≠ B, then A is called a proper sub set of A This relationship is denoted by A⊂B Equal Sets: Two sets A and B are equal [A = B] iff A⊆ B and B⊆ A Sets are equal if they have same elements ∀ ( ) Equivalent set: of two sets A and B are equivalent iff

=

denoted by A ~ B

Empty Set: A set with no elements is called the empty set and it is denoted by ∅ Or { } [ { ≠ ] Universal Set: The set of all objects under consideration or discussion, and is denoted by U. Singleton: A set with only one element is called singleton. Ex { Set Difference:

or {0 , {1

A B=



Set Union:

A∪B=

Set Intersection:

A

Set Complement:

Ac =

[A∪

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=(

)∪(

)∪(

)]

B= A

Disjoint Sets: Two sets A and B are said to be disjoint sets if A

B=∅

Symmetric Difference (Boolean Sum): A

B = (A – B) ∪ (B – A) = (A ∪ B) – (A

B)

A⊕B= Venn diagram: It is a pictorial representation of sets in which sets are represented by enclosed areas in the plane. The universal set U is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle.

U B

A

(d)

∪

(e)

(f) A

B

Cardinality of a Set: The number of elements in a set is called the cardinality of that set. It is denoted by |A|. ∪

=

+ ∪

∪ ∪

= ∪

+ =

=

+

+

iff A & B are disjoint sets +

+

+



Equivalent set: Two sets A and B are equivalent iff

=

DMGT

denoted by A ~ B

Ex: A = { 1, 2 } P(A) = {{ } ,{1}, {2},{1,2}} Power Set: Let A be any set. Then the set of all subsets of A is called the power set of A. It is denoted by p (A) Note: If a set A contains n elements, then its power set P(A) contains 2n elements. The set operations satisfy the following laws:

Table 3.1 Laws of the algebra of sets Idempotent Laws:

(1a)

Associative Laws:

(2a)( ∪ ) ∪

Commutative Laws: (3a) Distributive Laws:

Identity Laws Involution Laws: Complement Laws: DeMorgan’s Laws:

∪

∪

=

=

=

∪( ∪ )

∪

(4a) ∪( ( ∪ )

)=( ∪ )

(1b)

=

(2b) (

)

(3b)

=

=

( ∪ )=(

(4b) ( )

(5a)

∪∅=

(5b)

=

(6a)

∪

(6b)

∅=∅

(7) (

) =

(8a)

∪

(9a)

=

=

=∅

(

(8b)

)

)∪

=∅

(9b) ∅ =

(10a) ( ∪ ) =

(10b) (

) =

∪

Each of the above laws has the corresponding dual. Cartesian product: Cartesian product of two sets A and B, written as A × B and is defined as = ( , )

∈

= ( , , ) Ex: {1, 2}

∈ ∈

,

∈ , ∈

{a, b} = {(1, a), (1, b), (2, a), (2, b)}



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Note: 1. 2. 3. 4. 5. 6. 7.

In general, ≠ If = then either = or ( If = , = and = then ( ∪ )=( )∪( ) ( )=( ) ( ) ( )=( ) ( ) ( ) ( )=( ) ( )

∅) =

and

=

Relation: A (binary) relation R from A to B is a sub set of A × B. [If A= B, we say R is a relation on A (R ⊆ A

)].

Domain and Range: Let R be a relation from A to B. Then, Domain of R = {x x ∈ A and (x, y) ∈ R for some y ∈ B}. Range of R = {y y ∈ B and (x, y) ∈ R for some x ∈ A}. Clearly, dom R ⊆ A and ran R ⊆ B. Moreover, domain of R is the set of first co-ordinates in R and its range is set of all second coordinates in R. We some times write, (x, y) ∈ R as x R y which reads 'x relates y'. Inverse Relation: Let R be a relation from a set A to B. The inverse of R, denoted by R -1 is the relation from B to A which consists of those ordered pairs, which when reversed belongs to R i.e. R-1 = { (b, a)| (a, b) ∈ R} Equivalence Relation: Let R be relation on A. R is said to be an equivalence relation if the following conditions are satisfied. 1) 2) 3)

xRx ∀ x ∈ A ( R is reflexive) If x Ry then y Rx (R is symmetric) If x R y and y R z then xRz( R is transitive)

Reflexive Relation:- A relation R on a set A is said to be reflexive if ( , ) ∈

∀

∈

Symmetric Relation :-A relation R on a set A is called symmetric if ( , ) ∈ , then ( , )) ∈ ∀ , ∈ , i.e. ∀ ∀ (( , ) ∈ ( , )∈ ) Transitive Relation :-A relation R on a set a is transitive if whenever ( , ) ∈ then ( , )∈ ∀ , , ∈ ( , ) ∈ ( , ) ∈ ) ∀ ∀ ∀ (( , ) ∈ ( ) Note: there are 2 different reflexive relation possible on a set of n element.

and ( , ) ∈



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Anti-Symmetric Relation: A relation R on a set A is anti symmetric relation if aRb and bRa a = b ; i.e. ∀

∀

(( , ) ∈

i.e. whenever a R b and b R a then a = b ( , )∈

)

a=b

Note: 1) The properties of being symmetric and being anti symmetric are not negatives of each other. 2) For the sets A and B, A × B is called universal relation from A to B and ∅ is called the empty relation from A to B. 3) A relation can be symmetric an antisymmetric both or cannot have both Irreflexive relation: A relation R on a set A is irreflexive if (x , x)

R for all x ∈ A.

Note: Any relation which is not reflexive is not necessarily irreflexive and vice versa . Asymmetric relation: ∀ x, y if (x ,y) ∈ R then (y ,x) R Diagonal Relation: Let A be any set. The diagonal relation consists of all ordered pairs (a,b) such that a = b ie = (a, a) I a ∈ A} Composition of Relations: 9.

Let A, B and C be sets, and let R be a relation from A to B and let S be a relation from B tC. That is, R isa subset of A× B and S is a subset of B × C. Then R and S give rise to a relation from A to C denoted by R◦S and defined by: a(R◦S)c if for some b ∈B we have aRband bSc.

10.

Equivalently, R ◦S = {(a, c) | there exists b ∈B for which (a, b) ∈R and (b, c) ∈S}.

Partial Ordering Relation:a relation R on a set is called a partial order if R is reflexive, antisymmetric and transitive. Poset:A set S together with a partial order R is called a partially ordered set or poset. Complementary Relation: R is a relation from A to B then Rc = {(a, b) ∈ (A

) ( , )

}

Note: i) A relation R* is the transitive ( symmetric, reflexive) closure of R , if R* is the smallest relation containing R which is transitive ( symmetric, reflexive). ii) R ∪ R-1 is the symmetric closure of R and R∪ A is the reflexive closure of R. Partition: A partition of S is a sub division of S into non overlapping, non empty sub sets such that A1∪ A2∪ and Ai

∪An = S Aj∅ if i≠ j



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Equivalence Classes: Given any set A and an equivalence relation R on A, the equivalence class [x] of each element x of A is defined by [x] = { y∈ A | x R y } Note: i) We can have [x] = [y] even if ≠ y, provided

Ry

ii) Given any set A and any equivalence relation R on A, S = { [x] | x ∈ A } is a partition of A. Comparability: Two elements a and b in a posetA are said to be comparable under the relation <, if either a < b or b
∈

and b

∧

Least Upper Bound (lub) (Join or Supremum): The lub of two elements a and b is denoted by a

b.

If c = a V b then c satisfies i) a c and b ii) If a

c

d and b d then c

d

i.e..c is lub of (a , b) iii)

is the least element in upper bound of a an b

Lower bound:- lower bound of a and b in a poset A is defined as

( , )=

∈

∧

Greatest Lower Bound (glb)(meet or infimum) The glb of two elements a and b is denoted by a ∧b. If c = a ∧b then c satisfies i) c< a and c < b ii) If d < a and d < b then d < c iii)

is the greatest element in lower bound of a and b



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* Both join and meet operations are commutative and associative. Ex: In the poset |R; <|, a V b = max {a, b} and a ∧b = min{a, b} Ex: For the poset [D; | ], a V b= LCM of a and b aΛ b = G.C.D of a and b Ex: If S is any collection of sets, then for the poset [S; ⊆] A ∪ B is the lub of A and B, and A

B is the g b.

Hasse diagram ( A = 1, 2, 3, 4 };

Poset Diagram (Hasse diagram)

)

Let [A; <]be a poset

4 4 3 3 2 2 1

1

Hasse diagram

i) ii) iii)

There is a vertex for each element of A. All loops are omitted eliminating the explicit representation of reflexive property. An edge is not present in a poset diagram if it is implied by transitivity of the relation. iv) CoverAn edge connects a vertex x to a vertex y y covers x i.e...iff there is no element z such that x z and z y. Ex:

y cover

,

t

,

Join Semi lattice:A poset [A; ] in which each pair of elements a and b of A have a unique least upper bound is called join semi lattice. Meet Semi Lattice:A poset [A; ] inwhich each pair of elements a and b of A have a unique glb (meet) is called 'meet semi lattice".



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Lattice: A lattice is a poset in which each pair of elements has a unique lub and a glb. In other words, a lattice is both a join semi lattice and a meet semi lattice. A lattice is often denoted by [L, Λ, The following laws hold in L i) ii) iii) iv)

a b = b a and a Λ b = b Λ a (a Λ b) Λ c = a Λ( b Λ c) and (a b) a Λ (a b)= a and a (a Λ b) = a a Λ a = a and a a = a

c=a

(b

c)

Note: i) Let L be a lattice, then a Λ b = a ⇔ a V b = b ii) The Lub and Glb of any pair (a, b), if exists are unique.

Ex: f

f

h e

e c

g d

f

e

d c

b

d b

b a Lattice

c a

a

Lattice not a lattice [ Nolub for (b, c)]

Sub Lattice:Suppose M is a non empty subset of a lattice L. We say M is a sub lattice of L. if M itself is a lattice. Bounded lattice:A lattice L is said to have lower bound 0 if 0 < x ∀ x ∈ L. Similarly L is said to have an upper bound 1, If x <1 for all x ∈ L. We say L is bounded if L has both a lower bound 0 and upper bound 1. a

1 = 1, a Λ 1 = a, a

0 = a, a Λ 0 = 0

Note: Every finite lattice L is bounded. Distributive Lattice:A lattice (L, , ∧) is said to be distributive if the following dist. Laws hold. ∀a, b, c

a

(b ∧ c) = (a b) ∧ (a

a∧ (b

c) = (a ∧ b)

c)

(a ∧ c)

Complemented Lattice:Let L be a bounded lattice with lower bound 0 and upper bound 1. Let a be an element of L. An element x in L is called a complement of a, if a = 1 and a Λ x = 0.



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Note: In a lattice, compliments need not exist and need not be unique. Def: A lattice L, is said to be a complemented lattice, if L is bounded and every element in L has a complement. Note: Let L be a bounded distributive lattice. Then complements are unique if they exist. Lattices And Boolean Algebra Maximal Element: An element of a poset that is not less than any other element of the poset. Denoted as 1 Minimal Element:An element of a poset that is not greater than any other element of the poset. Denoted as 0 1) 2) 3) 4)

Every finite non empty poset has a minimal element. If a lattice has a universal lower bound (universal upper bound) it is unique. Every totally ordered set is a lattice. Every totally ordered set has a least element and a greatest element.

Note: The lub and glb of any pair (a, b) in a poset, if exists, are unique. Boolean Algebras (Boolean Lattice) A Lattice that contains the element 0 and 1, and which is both distributive and complemented is called a Boolean Algebra. Ex: Let be the set of propositions. is a Boolean algebra under the operations and ∧ with negation being the complement, a contradiction 0 is the zero element and a tautology 1 as the unit element. Boolean algebra can be represented by the system (B, + , . , - , 0, 1), where B is a set. The following properties hold: 2. There e ists at least two elements a, b ∈ B such that a ≠ b 3. ∀ , ∈ a) + ∈ closure property b) ∈ 4. ∀ , ∈ a) + = b) =

+

commutative laws

5. a) ∃ 0 ∈ such that +0 = a ∀ a ∈ B Identity element b) ∃ 1 ∈ B such that a 1 = a ∀ a ∈ B

6. ∀ , , ∈ a) + ( ) = ( + ) ( + ) )+( ) b) ( + ) = (

Distributive laws



7. ∀ , , ∈ a) + ( + ) = ( + ) + ) b) ( ) = (

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Associative laws

8. ∀ ∈ , ∃ ̅ ∈ such that a) + ̅ = 1 and Inverse element b) ̅ = 0 Note:1̅ = 0

0̅ = 1

1 + 0 = 1,

0 + 1 = 1,

1 + 1 = 1,

0 + 0 = 0,

0 0 = 0,

10 =0

Note: i) ∀ ∈ , + = = ii) The elements 0 and 1 are unique. iii) ∀ ∈ , + 1 = 1 0=0 iv) ∀ ∈ ̅is unique v) Let a, b ∈ B then +

=

Absorption Laws

( + )= vi) ∀ ,

∈ , (Demorgan’s Laws)

̅̅̅̅̅̅̅̅̅̅ ( + ) = ̅ ̅ ̅̅̅̅̅̅̅ ( )= ̅+̅ vii) ∀ , +̅

∈ , =

+

(̅ + ) =

Literal: A literal is a Boolean variable or its complement. A min term of the Boolean variables ( , )where for each variable.

=

,

,

is a Boolean product

̅ . Hence a min term is a product of n literals, with one literal

The sum of min terms that represents a given Boolean function is called the ‘sum of products e pansion’ or ‘Disjunctive normal form’ (DNF). Max terms: A Boolean expansion of the form of a disjunction (sum) of n literals is called max term. The product of max terms that represents a Boolean function is called the ‘product of sums e pansions’ or conjunctive normal form(CNF).



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Function: Let A and B be non-empty sets. A function from A to B element of B to each element of A.

:

is an assignment of exactly one

( )= B is the image of a & a is preimage of B A = Domain of f B = Co-domain of f The set of all image values is called the range of f and is denoted by One- To-One Function (Injection): A function : different elements in the domain A have distinct images. ( )

( ),

( )

( )

is said to be one-to-one function, if

co-domainof f for one-to-one function :

is one-to-one iff f (A) = f (B)

a=b

On-To Function: A function : is said to be an on-to function, if each element of B is the image of some element of A. i.e. Ran f=B or f (A) = B One- To-One Function & On-To Function (Bijection): [one-to-one correspondence] A function :

is said to be a bijection if f is one-to-one and on-to.

Inverse Of a Function: :

is invertible if its inverse relation

NOTE: A function :

is a function from B to A.

is invertible if and only if f is a bijection.

Constant Function: ( )= ∀ ,

∈ .

Identity Function: A mapping :

is called an identity function if = ( , )

∈

NOTE: Let m and n be positive integers with m>n then, there are ( , 1)(

1) + ( , 2)(

2)

+ ( 1)

( ,

1)1

On-to functions from a set with m elements to a set with n elements THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 53


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Groups Binary Operation: Let A and B be two sets. A function from A × A to B is called a binary operation on set A. Algebraic System: A set ‘A’ with one or more binary operations defined on it is called an algebraic system. Ex: ( , ), ( , , ), ( , +, ) Associativity: Let * be a binary operation on a set A. The operation * is said to be associative if (a*b)*c=a*(b*c) for all, a, b, c in A Identity: For an algebraic system (A,*), an element ‘e’ in A is said to be an identity element of A if a*e=e*a=a for all a∈A Inverse: Let (A,*) be an algebraic system with identity ‘e’ Let a be an element in A An element b is said to be inverse of A if a*b = b*a = e Closed operation: A binary operation is said to be closed operation on A if ,

∈ , ∀ ,

∈

Semi Group: An algebraic system (A,*) is said to be a semi group if 1. * is a closed operation on A 2. * is an associative operation, for all a, b, c in A Monoid: An algebraic system (A,*) is said to be a monoid if the following conditions are satisfied. 6. * is a closed operation 7. * is an associative operation 8. There is an identity in element in A Group: An algebraic system (A,*) is said to be a group if the following conditions are satisfied. 1. 2. 3. 4.

* is a closed operation * is an associative operation There is an identity in A Every element in A has inverse

AbelianGroup: A group (G,*) is said to be abelian or commutative if =

∀ ,

∈ [ commutative property]

Order of a group is equal to the number of elements in that group. Denoted as O(G) THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 54


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 In a group, the identity element is its own inverse. Order Of An Element Of A Group:  Let (G, ) be a group Let ‘a’ be an element of G. The smallest positive integer n such that an = e is called the order of ‘a’ If no such number e ists then the order is infinite Sub Group A non empty subset H of group (G,*) is a sub group of G if (H,*) is a group. NOTE: 1. A non empty subset H of group (G,*) is a sub group of G, iff i. ∈ ∀ , ∈ ii. ∈ ∀ ∈ iii. e ∈ 2. A necessary and sufficient condition for a non empty subset H of a group (G,*) to be a sub group is that ∈ , ∈ ∈ 3. A necessary and sufficient condition for a non empty finite subset H of a group (G,*) to be a sub group is that ∈ for all , ∈ 4. For any group {G,*},{e} and G are trivial sub groups. Cosets If H is a sub group of G and a ∈ G then the set Ha = {ha | h ∈ H} is called a right coset of H in G. Similarly, a H = {ah | h ∈ H} is called a left coset of H is G. Note:1. Any two left (right) cosets of H in G are either identical or disjoint. 2. Let H be a sub group of G. Then the right cosets of H form a partition of G. i.e., the union of all right cosets of a sub group H is equal to G. 3. Lagrange's theorem: The order of each sub group of a finite group is a divisor of the order of the group. ( ) [ = ∈ ] ( ) 4. The order of every element of a finite group is a divisor of the order of the group. 5. The converse of the lagrange's theorem need not be true. Cyclic Groups A group G is called cyclic, if for some a ∈G, every element of G is an integral power of a G=

, a-3, a-2, a-1, e, a, a2, a3

The element 'a' is then called a generator of G. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 55


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G = 1, ω, ω2 is a cyclic group w r t multiplication The generators are ωand ω2 . The set G = {1, -1, i, -i} is a cyclic group w.r.t multiplication. The generators are i. -i. Normal Sub Groups A sub group H of G is normal if a-1 H a ∈ H, ∀ a ∈ G, ∀ h ∈ H, oraH = Ha for all a ∈ G Note 1. A group having no proper normal sub groups is called a simple group. 2. Let H be a normal sub group of G, then the cosets of H form a group under coset multiplication, the group is called Quotient group and is denoted by G / H. Homomorphism and isomorphism: Consider the groups (G, *) and (G1, ⨁) A function f: G

G1 is called a homomorphism if f (a * b) = f(a) ⨁ f(b)

If f is a bijection then f is called isomorphism between G and G' and we write G ≅ G'



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2.4: Graph Theory A Graph G is a pair of sets (V, E) where V=a set of vertices (nodes) and E =a set of edges V(G) = Set of vertices in G E(G) = Set of edges in G | V(G) | = Number of vertices in graph G = Order of G | E(G) | = Number of edges in graph G = Size of the graph Directed Graph: In a digraph the elements of E are ordered pairs of vertices. In this case an edge (u, v) is said to be from u to v. UN (NON) directed graph: The elements of E are unordered pairs (sets) of vertices. In this case an edge {u, v} is said to join u and v or u is adjacent to v. LOOP: An edge drawn from a vertex to itself no multiple between 2 vertices. Simple Graph: A graph with no loops and edges [ no more than one edge between 2 vertices] Multi Graph: If one allows more than one edge to join a pair of vertices, the resultant graph is then called a multi graph. Out degree (deg V)) The number of edges “incident from” node ‘ ’ is called out degree (deg

(v)) of node V

Neighbours: If there is an edge incident from u to v, or incident on u and v, then u and v are said to be adjacent (or to be neighbors). [from here we usually term graph as “undirected graph” if we use directed, we will e plicitly mention it] Basic Terminology: Undirected graph b

a

a, b are adjacent is called “incident” the vertices a,b a , b are end points of Directed graph: a

b

{

incident to b



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a is said to be “adjacent to” b b is said to be “adjacent from” b a = initial vertex b = terminal or end vertex Degree: In an undirected graph degree of a node V, denoted as deg(V) is the number of edges “incident” with it, except that a vertex continue twice to degree In degree or (deg (V)) In a directed graph the no. of edges incident to a vertex is called the “in degree” (deg (V)) of a vertex . Complement of a graph: Let G be a graph with n vertices then the complement of G, denoted by is a simple graph with same vertices in G so that an edge is present in iff it is not present in G. If G has n vertices G∪ = Kn E( ) = ( ) ( ) D D C D C C ∀ A

G

B

= B

A

A

B

Note: In a simple graph with monotonically non- increasing degree sequence, this inequality Should held ∑

(

1) + ∑

n = no of vertices 1 k n : k =1,2

n

δ(G) = minimum of all the degrees of vertices in a graph G. (G) = Ma imum of all the degrees of vertices in a graph G

Regular Graph: In a graph G, ifδ(G) = (G) = k



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i.e..if each vertex of G has degree k, then G is said to be a regular graph of degree k. (kregular). Example: Polygon is a 2-regular graph.

Example: A 3-regular graph is a cubic graph.

Complete Graph: Asimple graph with ‘n’ mutually adjacent vertices is called a complete graph on n vertices and may be represented by Kn.

Note: A complete graph on n vertices has n(n-1)/ 2 edges, and each of its vertices has degree 'n-1'.

Cycle Graph: A cycle graph of order n is a connected graph whose edges form a cycle of length n. C

NOTE: A cycle graph 'Cn' of order n has n vertices and n edges.

Null Graph: A null graph of order n is a graph with n vertices and no edges. null graph of 3 vertices Wheel Graph: A wheel graph of order n is obtained by adding a single new vertex (the Wheel graph of order 4 hub) to each vertex of a cycle graph of order n. Vertices = 5, edges = 8 NOTE: A wheel graph Wn has ‘n+1’ vertices and 2n edges Bipartite Graph: A Bipartite graph is a non directed graph whose set of vertices can be partitioned into two sets M and N in such a way that each edge joins a vertex in M to a vertex in N. Complete Bipartite Graph: A complete Bipartite graph is a Bipartite graph in which every vertex of M is adjacent to every vertex of N. ,



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If |M| = m and |N| = n, then the complete Bipartite graph is denoted by Km,n. It has 'mn' edges. Degree Sequence: If v1,v2....... vn are the vertices of a graph G, then the sequence (d1, d2,......dn) where di = degree of vi , is called the degree sequence of G. Usually we order the degree sequence so that the degree sequence is monotonically non-increasing. Sum Of Degrees Theorem: If V = {v1, v2,........,vn} is the vertex set of a non directed graph G ∑ ( )=2 then For Digraph: ∑

( )= ∑

( )=

& ( )

[2|E| / |V|]

( )

Note: 1) An undirected graph, has an even number of vertices of odd degree. 2) If k = δ(G) is the minimum degree of all vertices in a undirected graph G, then ∑

( )=2

3) If G is a k - regular graph , then k |V| = ∑

( )=2

Connectivity Path: In a non directed graph G, a sequence p of zero or more edges of the form {v0, v1}, {v1, v2 , vn-1, vn} or {v0 – v1 – v2 - vn} is called a path from v0 to vn. v0 is called the initial vertex and vn is called the terminal vertex. v0and vnare called end points of the path. We denote p as v0 - vn path If v0 ≠ vn, then p is called an open path. If v0 = vn then p is called a closed path. Simple Path: A path p is simple if all edges and vertices on a path are distinct except possibly the end points.  An open simple path of length n has n distinct vertices and n -1 edges.  A closed simple path of length n has n distinct vertices and n distinct edges. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 60


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 A trivial path of {v0} is taken to be simple closed path of length zero. Circuit: A path of length ≥ 1 with no repeated edges and whose end points are equal is called a circuit.  A circuit may have repeated vertices other than its end points. c

a

b

e d

a b c is a cycle a b d e b c a is a circuit       

A cycle is a circuit with no other repeated vertices except its end points. A loop is a cycle of length 1. In a graph, a cycle that is not a loop must have length at least 3 edges. In a multi graph, there may be a cycle of length 2. Two paths in a graph are edge disjoint if they don't share a common edge. Two paths are vertex disjoint if they do not share a common vertex. An undirected graph is called connected if there is a path between every pair of distinct vertices.  A graph that is not connected is the union of two or more connected sub graphs each pair of which has no vertex in common. These disjoint connected sub graphs are called connected componentsof G.  If a graph G is connected and e is an edge such that G - e is not connected, then e is said to be a bridge or cut edge. e is bridge e  If v is a vertex such that G - v is not connected, then v is a cut vertex. G

is cut-vertex

,  A digraph is “weakly connected” if there is a non directed path between each pair of vertices.  A diagraph is “strongly connected” if there is a directed path from a to b and from b to a, for all vertices a, b in graph.  Two vertices u and v of a directed graph G are said to be “quasi strongly connected” if there is a vertex w from which there is a directed path to u and a directed path to v. a b a b a b c c

d Weakly connected

d c Strongly connected

d

f e Quasi-Strongly connected



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Note: 1) The graph G is said to be quasi strongly connected if each pair of Vertices in G is quasi strongly connected. 2) G is quasi strongly connected there is a vertex r in G such that there is a directed path r to all the remaining vertices of G. 3) If G is quasi strongly connected and 'G- e' is not quasi strongly connected for each edge e of G, then G is a directed tree. 4) G is a directed tree G is quasi strongly connected and contains a vertex r such that the in degree of r is zero and the out degree of all other vertices is 1. 5) G is a directed tree G is quasi strongly connected without circuit. 6) A diagraph G has directed spanning tree iff G is quasi strongly connected. 7) Any strongly connected graph is also weakly connected. 8) A connected graph with n vertices has atleast n - 1 edges. 9) In any simple graph there is a path from any vertex of odd degree to some other vertex of odd degree. Isomorphic Graphs: Two graphs G and G1 are isomorphic if there is a function f: V(G)

V(G1) such that

i) f is a bijection and ii) for each pair of vertices u and v of G, u, v ∈ E(G) iff f(u), f(v) ∈E(G) e G 2 1 G 5 a b 3

4

c

d

i.e..the function preserves adjacency. G & are isomorphic f(1) = a f (1) = a f (2) = b f (3) = c f (4) = d f (5) = e Homeomorphic Graphs: Given any graph G, we can obtain a new graph by dividing an edge of G with additional vertices. Two graphs G and G1 are said to homeomorphic if they can be obtained from the same graph. a

G e

b

a

c

b



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G& Euler Path: An Euler path in a multi graph is a path that includes each edge of the Multi graph exactly once and intersects each vertex of the multi graph atleast once.  A multi graph is traversable if it has an Euler path.  A non directed multi graph has an Euler path iff it is connected and has zero or exactly two vertices of odd degree.  A connected multi graph has an Euler circuit if and only if its vertices have even degree.  A directed multi graph G has an Euler circuit iff G is unilaterally connected and the in degree of every vertex of G is equal to its out degree.  A directed graph that contains an Euler circuit is strongly connected. Hamiltonian Graph: A Hamiltonian Graph is a graph with a closed path that includes every vertex exactly once. Such a path is a cycle and is called a Hamiltonian cycle.  An Eulerian circuit uses every edge exactly once but may repeat vertices, while a Hamiltonian cycle uses each vertex exactly once (except for the first and last) but may skip edges. Planarity A graph or a multi graph that can be drawn in a plane or on a sphere so that its edges do not cross is called a planer graph. Example: A complete graph on 4 vertices, K4,is a planar graph. Example: Tree is a planar graph B

A 3

1

2

4 = Regions C

D

Map, Connected map: A particular planar representation of a finite planar multigraph is called a map. We say that the map is connected if the under lying multigraph is connected. Region:A given map (planar graph) divide the plane into connected areas called regions. A

B Region 2

Region 3

Region 2

D

C



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Degree of a Region: The boundary of each region of a map consists of a sequence of edges forming a closed path. The degree of region ’r’ denoted by deg (r) is the length of the closed path bordering r. deg (Region 1) = 3 deg (Region 2) = 3 deg (Region 3) = 4  The sum of the degrees of the regions of a map M is equal to twice the number of edges in M. ∑

( )=

 Let G be a connected graph with ‘e’ edges and 'v ' vertices Let ‘r’ be the number of regions in a planar representation of G. Then r + v = e + 2 Euler's formula.  In a planar graph G, if the degree of each region > k then k <2|E|  In particular for a simple connected planar graph, R 

2E If G is a connected simple planar graph with |E| > 1 then,

(A) | E | < 3| V | 6 (B) There exists atleast one vertex v of G such that deg (v) < 5

Polyhedral Graph: A connected planar graph is said to be polyhedral if deg (v) ≥ v∈G

for all

 For any polyhedral graph (A) | V | > (2 + (B) |R | > (2 +

) ,

)

(C)(3.|R|-6) > |E|

,

are not planner graph

Kurtowski Theorem:    

A graph G is not planar iff G contains a sub graph homeomorphic to K3,3or K5 A complete graph Kn is planar iff n < 4 A complete Bipartite graph Km.n is planar iff m < 2 or n < 2 Any graph with 4 or fewer vertices is planar



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Coloring: A coloring of a simple graph is the assignment of color to each vertex of the graph so that no two adjacent vertices are assigned the same color.(vertex coloring) Chromatic Number: The minimum number of colors needed to paint a graph G is called 2 the chromatic number of G, denoted by x(G).Ex x ( ) = n , x ( ) = { x (Bipartite) = 2 Adjacent Regions: An assignment of colors to the regions of a map such that adjacent regions have different colors.  A map 'M' is n - colorable if there exists a coloring of M which uses n colors.  A planar graph is 5 – colorable.for 60th vertex coloring 2 Map coloring

 Four color Theorem: If the regions of a map M are colored so that adjacent regions have different colors, then no more than 4 colors are required.  Every planar graph is 4- colorable (vertex coloring). Matching Matching: Given an undirected graph G = (V, E), a matching is a subset of edges (M ⊆ E) such that for all vertices v ∈ , at most one edge of M is incident on V i e deg M(v) 1 ∀v∈  deg M(v) = 1 & vertex is matched , deg M(v) =0 , unmatched  A perfect matching is a matching in which every vertex is matched ( )= 1∀ ∀ B A A B C

D

C

D Perfect matching

   

A matching in a graph is a sub set of edges in which no two edges are adjacent. A single edge in a graph is obviously a matching. A maximal matching is a matching to which no edge in the graph can be added. A graph may have many different maximal matchings and of different sizes. Among these, the maximal matchings, with the largest number of edges are called the largest maximal matchings. The number of edges in largest maximal matching is called the matching number of the graph.  Off a graph has perfect matching, then the number of vertices in the graph must be even but the converse is not true, if n is even, the number of perfect matching in )( ) =( 1)( 1



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Complete Matching  In a bipartite graph having a vertex partition V1 and V2, a complete matching of vertices in set V1, into those in V2 is a matching in which there is one edge incident with every vertex in V1. Inother words, every vertex in V1 is matched against some vertex in V2.  A complete matching (if it exists) is a largest maximal matching, where as the converse is not necessarily true.  For the existence of a complete matching of set V1 in to set V2, first we must have atleast as many vertices in V2 as there are in V1. This condition however is not sufficient.  A complete matching of V1, into V2 in a bipartite graph exists if and only if every subset of r vertices in V1 is collectively adjacent to r or more vertices in V2 for all values of r(Hall's Theorem).  In a bipartite graph a complete matching of V1 into V2 exists if there is a positive integer 'm' for which degree of every vertex in V1> m > degree of every vertex in V2 NOTE: This condition is a sufficient condition and not necessary for the existence of a complete matching.  The maximal number of vertices in set V1 that can be mapped in to V2 is equal to number of vertices in V1 – δ (G) Edge Covering  In a graph G, a set of edges is said to cover G if every vertex in G is incident on atleast one edge in G.  A set of edges that covers a graph G is said to be an edge covering (a covering sub graph or simply a covering) of G. D e A

e

e

C e B

 e e e e is a covering  e e is a Minimal covering Minimal Edge Covering: A covering from which no edge can be removed without destroying its ability to cover the graph.  A covering exists for a graph iff the graph has no isolated vertex.  A covering of an 'n' vertex graph has atleast⌈ 2⌉edges.  Every covering contains a minimal covering.  No minimal covering can contain a circuit.  A graph, in general, has many coverings, and then may be of different sizes (i.e consisting of different number of edges). The number of edges in a minimal covering of smallest size is called the edge covering number of the graph denoted by THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 66


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 A covering g of a graph is minimal if g contains no paths of length three or more.  If a covering g contains no path of length three or more, all its components must be star graphs. (From a star graph no edge can be removed).  An Edge covering of n- vertex graph has at least [ ⁄2] edges  A sub-set S of V is called an edge independent set of G if no two vertices of S are adjacent in G. An independent set ‘S' is minimum if G has no independent set S1 such that S' ⊂ S  The number of vertices in a maximum edge independent set is called the edge independence number of G, denoted by D

+

= A

C

B

Edge Independent set = = , Independent No = 2 Vertex Covering: A vertex covering of a graph G = (V, E) is a subset K of V such that every edge of G is incident with a vertex in K. A covering K is called a minimum vertex covering if G has no covering K1 with |K1| <|K|. The number of vertices in a minimum covering of G is called the covering number of G, denoted by .  Vertex independent set:A subset S V is calls a vertex independent set if no 2 vertices of this set are adjacent  Maximum vertex independent set: A vertex independent set to which no other vertices of the graph can be added is called maximal vertex independent set , number of vertices in this set is vertex independent number denoted as + = Connectivity Any circuit in a graph must contain a cycle and that any circuit which is not a cycle contains atleast two cycles. Cut Set: Let G be connected graph. A cut set in G is a set of edges whose removal from G leaves the graph G disconnected provided no proper subset of these edges disconnects the graph G. Edge Connectivity: Let G be a connected graph. The edge connectivity of G is the minimum number of edges whose removal results in a disconnected graph. In other words, the number of edges in the smallest cut set of G is defined as the edge connectivity of G denoted by (G) THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 67


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If G is a connected graph and has a cut edge, then the edge connectivity of G is 1. Vertex Connectivity: Let G be a connected graph. The minimum number of vertices whose removal results in a disconnected graph is called the vertex connectivity of G and is denoted by K(G)  If G has a cut vertex then K(G) = 1  If G is complete graph Kn then K(G) = n-1  If Cn(n≥4) is a cycle graph, then (Cn)=2  If a graph G has a bridge then K(G)=1  The edge connectivity of graph G cannot exceed δ(G)  For any connected graph G, (G)

(G)

δ(G)

 If G is not connected then ̅ is connected  A simple graph with n vertices and k components can have at most [(n-k)(n-k+1)]/2 edges.  A simple graph G with n vertices is connected if it has more than [(n-1)(n-2)]/2 edges.  If a vertex v in a graph G then a Hamiltonian path if exists must contain atleast one edge incident on v and at most two edges incident on v.  A Hamiltonian cycle contains exactly two edges incident on v. In particular, both edges incident on a vertex of degree 2 will be contained in every Hamiltonian cycle. Finally there cannot be 3 or more edges incident with one vertex in a Hamiltonian cycle. Planarity  If G is a simple connected planar graph with

≥ and R regions then R

2

-4.

 If G is a planar graph with k connected components, each component having atleast 3 vertices then (3|V|-6k).  If G is a planar graph with k components then |V|- E + R = k+1 (for k=1, we get Euler’s formula). Isomorphism  Suppose G and G1 are two graphs and that f: V(G) V(G1) is a bijection. Let A be the , , adjacency matrix for the vertex ordering V1 V2 Vn of the vertices of G. Let A1 be the adjacency matrix for the vertex ordering f(V1), f(V2).......f(Vn). Then f is an isomorphism from V(G) to V(G1) iff the adjacency matrices A and A1 are equal.  If A≠A1, then it may still be the case that graph G and G1 are isomorphic under some other function.  Two simple graphs are isomorphic iff their complements are isomorphic.



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 If G is isomorphic to G1 then the following conditions must hold good.  |V(G)| = |V(G1)|  |E(G)| = |E(G1)|  The degree sequences of G and G1 are the same.  If {v, v} is a cycle in G, then {f(v), f(v)} is a loop in G1, and more generally, if v0 , v1, v2 a cycle of length k in G, then f(v0)-f(v1)-f(v2)-.....f(vk) is a cycle of length k in G1.

, -vn is

 If two graphs are isomorphic, then their corresponding subgraphs are isomorphic.  Induced Subgraph: If W is a subset of V(G), then the subgraph induced by W is the subgraph H of G obtained by taking V(H) = W and E(H) to be those edges of G that join pairs of vertices in W.  If G is isomorphic to ̅ then G is said to be self complementary.  If G is self complementary then G has 4n or 4n + 1 vertices.

Spanning Trees A sub graph H of a graph G is called a spanning tree of G if i) H is a tree and ii) H contains all vertices of G  In general, if G is a connected graph with n vertices and m edges, a spanning tree of G must have (n-1) edges. Therefore, the number of edges that must be removed before a spanning tee is obtained must be m - (n-1). This number is called circuit rank of G.  A non directed graph G is connected iff G contains a spanning tree.  The complete graph Kn has nn-2 different spanning trees (Calley’s formula)

Tree – traversal * Preorder (Right left Right) * In order (left Root Right ) *post order (left Right Root) Tree : A connected undirected graph with simple circuit is Tree Forest A set of tree irchoff’s Theorem: Let A be the adjacency matrix of a connected graph G and M be the matrix obtained from A by changing all 1’s into -1 and each diagonal element 0 to the degree of the corresponding vertex. Then the number of spanning trees of G is equal to value of any cofactor of M. *The number of different trees with vertex set {v1, v2, , vn} in which the vertices v1, v2, , vn have degrees d1, d2, dn respectively is (

(

) (

)

)

(

)



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Minimal Spanning Tree: Let G be a connected graph where each edge of G is labeled with a non negative cost. A spanning tree T where the total cost C(T) is minimum is called a minimal spanning tree. ruskal’s Algorithm: (For finding minimal spanning tree of a connected weighted graph) O (log e) Input: A connected graph G with non negative values assigned to each edge. Output: A minimal spanning tree for G. Method: 1) Select any edge of minimal value that is not a loop. This is the first edge of T(if there is more than one edge of minimal value, arbitrarily choose one of these edges). 2) Select any remaining edge of G having minimal value that does not form a circuit with the edges already included in T. 3) Continue step 2 until T contains (n-1) edges, where n=|V(G)|

4

D Ex:

7

2

6

C

A

D

5

9 B

C 2

4

D 2

B

A

A

C B

D A

2

Cycle 4 C 5

D

4 62

B

A

C B

rim’s Algorithm : (For finding a minimal spanning tree) ( O (e log v)) 1) Let G be a connected graph with non negative values assigned to each edge. First let T be the tree consisting of any vertex V1 of G. 2) Among all the edges not in T that are incident on a vertex in T and do not form a circuit when added to T, select one of minimal cost and add it to T. 3) The process terminates after we have added (n-1) edges, where n=|V (G)|

4

D 6

7 2

C 5

A

9

B

Suppose we start with A D D 6 A

.C

6 .B

.C

D 2

D 6

4

C 2

B A  Both prim’s and Kruskal’s will give same weighted minimum spanning tree [ Three structure way differ but weight of both MST will same ] A

B

 Depth – first search in a directed graph return a “forest”  Depth first search is used to find cut –vertices of a connected  Breadth first search is used to find all nodes within one connected component  Used for finding “shortest path” between 2 nodes u and v THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 70


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Part – 3: Data Structure and Algorithm 3.1 Data Structure and Algorithm Analysis Basic Terminology Algorithm: outline, the essence of a computational procedure, step-by-step instruction Program: An implementation of an algorithm in some language Data Structures: Organization of data, needed to solve the problem What is a good algorithm? A good algorithm is something which is efficient 

Efficient : Efficiency as a function of input size o In running time o Space used

Algorithm Analysis Mathematical Foundations of Algorithm Analysis 

Asymptotic Running Times



Solving Recurrences



Elements of Discrete Mathematics

Asymptotic Running Time of an Algorithm When we analyze the running time of an algorithm, it's usually not worth to compute the exact running time. For large input sizes only the order of growth of the running time - the asymptotic running time - is relevant. 

Sorting a billion numbers with the Bubblesort algorithm (running time O(n2)) on ASCII White, the world's fastest computer, takes about a week.



Sorting a billion numbers with Heapsort (running time O(n log n)) on a 100 MHz Pentium PC takes about 10 minutes.

Most times it does not matter if an algorithm is 2, 3 or 10 times faster than an other algorithm. Constant factors are dwarfed by the effect of the input size. The real question to ask when we analzye an algorithm's efficieny is: If we double the input size, how does this affect the running time of the program? Does it run twice as long? Or four times as long? Or even longer?



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Asymptotic Running Times - Example Which one is the more tedious job? (You will be told the value of n after accepting the assignment.) 

Add two n-digit numbers. (With pencil and paper, of course.)



Multiply two n-digit numbers. (Also with pencil and paper.)

Adding two n-digit numbers takes time proportional to n. Multiplying two n-digit numbers takes time proportional to n2. Asymptotic Notations When we talk about the running time of an algorithm, we use three different notations to give upper, lower or tight bounds of the asymptotic running time. 

O-notation: for asymptotic upper bounds



Omega-notation: for asymptotic lower bounds



Theta-notation: for asymptotically tight bounds

O-notation For asymptotic upper bounds. 

O(g(n)) = {f(n): there exist positive constants c and n0 such that 0 <= f(n) <= c*g(n) for all n>=n0 }



O(g(n)) = {functions that grow asymptotically slower than g(n)}



100n3 + 50n2 + 60n = O(n3)



100n3 + 50n2 + 60n = O(2n)



100n3 + 50n2 + 60n != O(n2)



2n = O(3n)

Omega-notation For asymptotic lower bounds. 

Omega(g(n)) = { f(n): there exist positive constants c and n0 such that 0 <= c*g(n) <= f(n) for all n>=n0 }



Omega(g(n)) = { functions that grow asymptotically faster than g(n) }



n2 log n = Omega(n2)



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Theta-notation For asymptotically tight bounds. 

Theta(g(n)) = { f(n): there exist positive constants c1, c2, n0 such that 0 <= c1*g(n) <= f(n) <= c2*g(n) for all n>=n0 }



Theta(g(n)) = { functions that grow asymptotically as fast as g(n)) }



(4n + sin n)*(5n - cos n) = 20n2 + O(n) = Theta(n2)

Examples 

3n2 + 6n + 17 = Theta(n2)= O(n2) = Omega(n2)



3n2 + 6n + 17 = O(n9)



3n2 + 6n + 17 = Omega(n log n)



2n+1 = Theta(2n)



22n = Omega(2n)

Recurrences When an algorithm contains a recursive call, its running time can often be described by a recurrence. Examples 

T(n) = n + T(n-1)

(Maxsort)



T(n) = 2 T(n/2) + O(n)

(Mergesort)



T(n) = T(n/2) + O(n)

(Finding the kth-largest element in a list)



T(n) = 20 T(n/5) + O(n2) (Is this still O(n2)?)

How do you solve such a recurrence equation? Solving Recurrences Four common ways to solve a recurrence are: 

Guess the solution, then use induction to prove it.



Expand the recurrence and convert it into a summation.



Visualize the recursion tree.



Use the master theorem.



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Guess the solution and then prove it. T(n) = 2 T(n/2) + n  We guess the solution is T(n) = Theta(n log n) 

Prove that T(n) <= c(n log n) for some constant c>0.



T(n) <= 2 (c n/2 log n/2) + n



= cn log n/2 + n



= cn (log n - log 2) + n



= cn log n - cn + n



<= cn log n



QED

The last step holds for c>=1. Expanding a Recurrence Doesn't require you to guess the solution, but it may require more algebra than guessing and then proving it by induction. 

T(n) = n + 2 T(n/3)



T(n) = n + 2 (n/3 + 2 T(n/9))



T(n) = n + 2 (n/3 + 2 (n/9 + 2 T(n/27)))



T(n) = n + 2/3 n + 4/9 n + 8/27 n + 16/81 n + ...



T(n) = n (1 + 2/3 + 4/9 + 8/27 + 16/81 + ...)



T(n) = n * (Sum (2/3)i from 0 to



T(n) = 3n



T(n) = Theta(n)

)

Visualize the Recursion Tree T(n) = 2 T(n/2) + n2 

Draw the recursion tree.



Label each node with its cost.



Compute the cost of each level of the tree.



Compute the total cost of all levels. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 74


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The Master Theorem To solve the recurrence T(n) = a T(n/b) + f(n), compare f(n) with 

If f(n) is smaller than



If f(n) and



If f(n) is larger than

.

, then T(n) = Theta

are the same, then T(n) = Theta(

log n).

, then T(n) = Theta(f(n)).

This is a rather sloppy definition, but it is good enough for most functions that we will encounter. There are some more technical formalities that should be understood, however. They are explained in "Cormen, Leiserson, Rivest: Introduction to Algorithms", pages 62-63.



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3.2 Elements of Discrete Mathematics for Data Structure We review the notations, definitions and elementary properties of the elements of discrete mathematics that are most relevant for the Algorithms course. 

Sets



Relations



Functions



Graphs



Trees

Sets, Relations, Functions 

A set is a collection of distinguishable objects.



A binary relation on two sets A and B is a subset of the Cartesian product A x B. Example: the "less than" relation on natural numbers is the set { (1,2), (1,3), (2,3), (1,4), (2,4),...) }



Formally, a function f is a binary relation on A x B such that for all elements of A there is exactly one element of B such that (a,b) is in f.



Intuitively, a function is a mapping that assigns an element of B to each element of A.

Graphs 

Formally, a graph is a pair (V,E) where V is a finite set and E is a binary relation on V.



Intuitively, a graph is a network consisting of vertices (V) and edges (E) that connect the vertices.



The edges can be directed or undirected. Dependent on that, the resulting graph is called a directed graph or an undirected graph.



By convention, edges from a vertex to itself - self-loops - are allowed in directed graphs, but forbidden in undirected graphs..



A directed graph without self-loops is called simple.



If (u,v) is an edge of a directed graph, we say that (u,v) leaves u and enters v.



If (u,v) is an edge of a graph, we say that u and v are adjacent to each other.



The degree of a vertex is the number of edges incident on it. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 76


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In case of a directed graph, the in-degree of a vertex is the number of edges entering it, the out-degree is the number of edges leaving it, and its degree is the sum of its in-degree and out-degree.



Graphs - Paths and Cycles 

A sequence of vertices (v0,v1,..., vk-1,vk) where all (vi,vi+1 are edges of the graph is called a path of length k that connects the vertices v0 and vk.



If there is a path p from u to v, we say that v is reachable from u via p.



A path is simple, if all vertices in the path are distinct.



A path (v0,v1,..., vk-1,vk) forms a cycle, if v0 = vk and the path contains at least one edge.



An undirected graph is connected, if every pair of vertices is connected by a path.



A directed graph is strongly connected, if every two vertices are reachable from each other.

Graph Problems 

Euler Circuit: find a cycle in a graph that visits each edge exactly once.



Hamiltonian Circuit: find a cycle in a graph that visits each vertex exactly once.



Determine the strongly connected components in a directed graph.



Test if two graphs are isomorphic.

Trees 

An undirected, acyclic, connected graph is called a tree.



An undirected, acyclic, but possibly disconnected graph is called a forest.



Any two vertices in a tree are connected by a unique simple path.



A tree with n nodes has n-1 edges.



If any edge is removed from a tree, the resulting graph is a forest of two trees.



If any edge is added to a tree, the resulting graph contains a cycle.

Linked Lists 

Like an array, a list also is a container datatype.



List elements consist of a value and a pointer to the next list element.



The list itself is represented by a pointer to the first list element.



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3.3 Abstract Data Type (ADT): ADT is mathematically specified entity that defines a set of its instances with: 

A specific interface - a collection of signatures of operations that can be invoked on an instance



A set of axioms – that defines the semantics of the operations

Examples of ADT 

Arrays, lists and trees are concrete datatypes. They are basic data structures typically provided by the computer language.



Stacks, queues and heaps are abstract datatypes. They are just ideas, i.e. "black boxes" with a defined behavior. To implement them, you have to choose a suitable concrete datatype.



In particular, stacks and queues can be implemented by arrays or linked lists.



A heap can be implemented by an array or a binary tree.



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3.4 Stacks 

A stack is a container of objects that are inserted and removed according to the last- infirst- out (LIFO) principle



Objects can be inserted at any time, but only the last object can be removed



Inserting an object is known as pushing on the stack and removing and item from the stack is known as popping

Stack is an ADT that supports the six main methods 

New() – Creates a new stack



Push(S,o) – I serts bject ‘ ’



Pop(S) - rem ves the t p e eme t f st ck ‘S’



Top(S) – Retur the t p e eme t f st ck ‘S’



Size(S) – Retur s the umber f e eme ts prese t i the st ck ‘S’



IsEmpty(S) – Returns true if the number of elements in the stack is 0, false otherwise

t the t p f st ck ‘S’

Stack ADT Implementations Linked List Implementation: This implementation uses linked list data structure. The most important advantage is that the size of the stack is not needed to be specified statically. The size of such stack is only limited by the heap memory allocated to process by the system. Push and pop operation are just insertion and deletion in the front of the linked list only hence it does not take more than constant time for implementing them. Array Implementation: The major drawback it has, as compared to the linked list implementation is that the size has to be declared statically. Stack Applications 

Being used to implement conversion efficiently from infix to postfix/prefix.



The stack is very most important data structure being used to implement function calls efficiently. When a function is called some housekeeping (register values, return address and etc.) is required before the program counter is assigned the address of called function. Thus in that case activation records is allocated and being used to store the required state and then this record is pushed onto the system stack



Being used to find balancing property of the program/expression. At the cost of O(n) this can be done.



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3.5 Queue 

A Queue differs from the stack that its insertion and removal routines follow the first-infirst-out (FIFO) principle.



Objects can be inserted at any time, but only the object which has been in the queue the longest may be removed



Objects are inserted from the rear and removed from the front



A double ended queue, or dequeue supports insertion and deletion from the front and back

Queue is an ADT that supports the three main methods 

New() – Creates anempty queue



EnQueue(Q,o) – Inserts bject ‘ ’ at the rear of the queue ‘Q’



DeQueue(Q) - removes object from the front of the queue ‘Q’



Front(Q) – Returnthe front element of Queue ‘Q’



Size(Q) – Returns the number of elements present in the Queue ‘Q’



IsEmpty(Q) – Returns true if the number of elements in the queue is 0, false otherwise

Queue ADT Implementations Array Implementation for circular queue: The queue consist of an N-element array and two integer variable      

f : index of front element r : index of the element after the rear one What does f = r ? (may be empty or full) Never add more than n-1 elements in the queue, in that case f = r means queue is empty Size() = N - f + r Disadvantage is fixed size

Linked List Implementation:    

Make the head to the front of the queue Make the tail to the rear of the queue Why re r c ’t be fr t f the queue bec use de eti is case The doubly linked list can be use to implement dequeue

t p ssib e i c

st

t time i this



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3.6 Trees Binary Trees Binary tree is a tree in which a node can have at most two children. Perfect Balanced Binary Tree In perfectly balanced binary tree each node has same number of nodes in its both subtrees. Height Definition Height in general measured as number of edges from the root to deepest node in the tree. Height can also be measured in terms of nodes in the longest path from the root to deepest node. Please use the first height definition unless until specified explicitly. Leaf nodes Vs Internal Nodes 

Le f



Internal Nodes have at least one child.

des d es ’t h ve chi d.

Theorem: A binary tree with n nodes will have n + 1 leaf nodes. Maximum Number of Nodes Let h is the height of a binary tree then the maximum number of nodes that binary tree can have is = 2h+1-1. Height Analysis of a binary tree The height of binary tree can vary from (log n) to n where n is number of nodes present in the binary tree. The following observation concludes this as follows: If the resulting tree of n nodes is almost a balanced binary tree then (2h+1 - 1) >= n; taking the log on both side 2h+1 >= n + 1 => h +1 >= log(n + 1) Which implies h is of O(logn). On the other hand if the resulting tree is having one node per level then h is of O(n). Binary Tree with Full Nodes (a full node has exactly two children) Theorem: A binary tree (in which each node has exactly two children except leaf nodes) with n internal nodes will have n + 1 leaf nodes and therefore total nodes would be 2n + 1.



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Tree Traversals Traversal is used to visit each node in the tree exactly once. A full traversal as a binary tree gives a linear ordering of the data in the tree. Inorder Traversal 1. Traverse the left sub tree of the root node R in inorder 2. Visit the root node R 3. Traverse the right subtree of the root node R in inorder for n nodes inorder takes O(n). Postorder Traversal 1. Traverse the left subtree of the root R in post order. 2. Traverse the right sub tree of the root R in post order 3. Visit the root node R. Running time is O(n) which can be derived as discussed for inorder running time. Binary Tree Construction Using Postorder A unique binary tree cannot be constructed using a postorder sequence as there can be many tree exists for the given sequence. Preorder Traversal 1. Visit the root node R 2. Traverse the left subtree of R in preorder 3. Traverse the right subtree of R in preorder Running time is O(n) which can be derived as discussed for inorder running time. Binary Tree Construction Using Preorder A unique binary tree cannot be constructed using a preorder sequence as there can be many tree exists for the given sequence. Levelorder Traversal This traverses the tree nodes in level by level order. Binary Tree Construction Using Levelorder A unique binary tree cannot be constructed using a level order sequence as there can be many tree existing for the given sequence. 

The generalized form of level order traversal is BFS.



The nodes can be visited in any order of a level however nodes from deeper level cannot be visited unless nodes from higher levels are visited.



Queue data structure can be used to implement level order traversal efficiently.



DSA

Binary Tree Construction Using inorder and postorder Inorder must be needed along with postorder for building a unique binary tree. The idea is very simple which is described as follows: Tree* createTree(intinorder[], intpostorder[]) { 1. newNode = Pick up the last key of the post order sequence and create a node of it; 2. newNode->left = create left subtree recursively using the values preceding root key in inorder. 3. newNode->right = create right subtree recursively using the values following root key in inorder. 4. return newNode; } Running Time analysis We need to consider the running time consumed on each node as usual. In worst case finding index of root key into inorder array takes L unit where L is the length of inorder array. Therefore, the following equation will best describe the running time; T(n) = (n-1) + (n-2) + (n-3) + …+ 1 = O( 2). The algorithm shows up worst case behaviour when postorder and inorder sequences are same. Binary Tree Construction Using inorder and preorder Inorder must be needed along with preorder for building a unique binary tree. The idea is very similar to above one which is described as follows: Tree* createTree(intinorder[], int preorder[]) { 1. newNode = Pick up the first key of the pre order sequence and create a node of it; 2. newNode->left = create left subtree recursively using the values preceding root key in inorder. 3. newNode->right = create right subtree recursively using the values following root key in inorder. 4. return newNode; } Running Time analysis The running time is O(n2). This can be derived in the same way as discussed in using post and in order section.



DSA

Binary Search Trees BST is a binary tree in which each node satisfies search property. The each node key value must be greater than all values in its left subtree and must be less than all values in its right subtree. Basic BST Operations Insert I sert per ti

i sert v ue if it d es ’t re dy exist.

Algorithm The new value is either inserted into left subtree or right subtree of the root node. Thus by comparing root and new value we identify the correct subtree to be chosen. And the same steps are applied to each subtree until a null subtree is found. Running Time Analysis The running time can be measured in terms of height of the tree which can vary from logn to n. Thus in worst case it would be n. However if the insertion is done into a perfect balanced tree then it would be logn. Delete This operation deletes a node from a given BST. Algorithm M i y 3 c ses sh u d be c

sidered. Let’s s y X is the de eti

de the ;

Case1: if (Node X is Null) Then traverse the tree until reaches the X parent node and make the parent pointer that is pointing to X, pointing to Null now. And free up the node X. (Running time --->O(n)) Case2: if (Node X has exactly one child subtree) Then traverse the tree until it reach the X parent node and make the parent pointer, that is p i ti t X, p i ti t X’s chi d subtree w. A d free up the de X. (Running time --->O(n)) Case3: if (Node X has exactly two children subtree(left and right)) This is somewhat complicated case as compare to other two. This will require immediate predecessor or successor in inorder sequence of the deleting node to be copied in place of it. The greatest key in left subtree is the required immediate predecessor and can be found as follows: Tree *maxNode = findMax(X->left);



DSA

OR The least key in right subtree is the required immediate successor and can be found as follows: Tree *minNode = findMin(X->right); Then copy minNode->val/maxNode->val into X->val; Thereafter call either case1 or case2 as minNode/maxNode will have either one child or none. (Running time --->O(n)) Running Time Analysis Only one of the cases (case1, case2 or case3) is executed for one deletion. Therefore running time is max (case1 RT, case2 RT, case3 RT) which is O(n). If the tree were a perfect balanced tree then it would be logn. The Other Operations The running time of the following operations is O(n) in worst case since a single path from root to leaf level is being traversed by all these operations.



3.7

DSA

Height Balanced Trees (AVL Trees, B and B+)

Tree structures support various basic dynamic set operations including Search, Predecessor, Successor, Minimum, Maximum, Insert, and Delete in time proportional to the height of the tree. To ensure that the height of the tree is as small as possible and therefore provide the best running time, a balanced tree structure like an AVL tree, or B-tree, or B+ tree must be used. AVL Trees AVL tree is a self-balancing binary search tree with height-balanced condition. That means for every node the height of left subtree and right subtree can differ by at most 1. The following tree a bad binary tree. Requiring balance at root is not enough. Minimum Number of Nodes in AVL Tree of Height h Let (h) be the umber f des i the sm est AVL tree f hei ht h. Si ce it’s AVL tree the definitely the height of its subtrees will be either (h-1 and h-2) or (h -1 and h -1). Now to have smallest AVL tree for the given height, its two subtree must also be smallest possible AVL tree. And that is possible only if subtrees are of height h-1 and h-2(if it was of h -1 height then it will have at least one extra node than the subtree of height h-2 has.). Then clearly the following recurrence can represent the smallest AVL tree. n(h) = n(h-1) + n(h-2) + 1 Maximum Number of Nodes in AVL Tree of Height h The largest AVL tree will have 2h+1 – 1 node. The derivation follows as discussed before in binary tree section. Height Analysis Since it is a balanced tree the height order is O(logn). Insertion After insertion it might be possible that only nodes that are on the path from the insertion point to the root might have their balance altered because only those nodes have their subtree altered. In that case we need to follow the path up to the root and update the balancing information; we may find a node whose new balance violates the AVL condition. At this moment using the rotations we can restore the AVL condition for this node and that rebalancing itself would be sufficient for the entire tree too, that means we will not need rebalancing on the further nodes on path to root. A violation might occur in the four cases 

An insertion into the left subtree of the left child of node X(where X violates the AVL condition).



An insertion into the right subtree of the right child of node X THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 86




An insertion into the right subtree of the left child of node X



An insertion into the left subtree of the right child of node X

DSA

1st two cases are symmetric and called as outside insertion that mean left-left or right-right and can be fixed by the simple single rotation. Last two are also symmetric and called as inside insertion that mean left-right or right-left and can be fixed by double rotation (two single rotations). Single Rotation The following rotation is called a right rotation. Here tree on the left side becomes imbalance because the insertion of a new node in x subtree causes k2 to violates AVL condition. This will fall into left-left case. NOTE: 

After the rotation, k2 and k1 not only satisfy AVL condition but also have subtree that are exactly the same height. Furthermore, the new height of the entire subtree is exactly the same as the height of the original one prior to the insertion that caused the X(Z) to grow up hence will not require any more rebalancing of the further nodes on the path to root. Double rotation is also retained the same height of the entire affected subtree what was there prior to insertion. Therefore the height of the entire tree is unchanged in these cases.

The main idea of rotation is to level the height of both left and right subtrees of the imbalance subtree by transferring the nodes from the deeper subtree to another subtree. After rotation deeper subtree height will get restored to the original one and another subtree height is increased by one. Thereby level the required heights. Double Rotation This rotation is required when inside insertion happens. As mentioned above that this can be implemented with left and right single rotation appropriately. Right-Left Double Rotation This case occurs when insertion happens into the left subtree of the right child of node X. A right-left rotation requires a single right rotation followed by a single left rotation on the appropriate keys. 

Step1: single right rotation on right child of X, where X node violates the property.



Step2: single left rotation on node X

Left-Right Double Rotation This case occurs when insertion happens into the right subtree of the left child of node X.



DSA

A left-right rotation requires a single left rotation followed by a single right rotation on the appropriate keys. 

Step1: single left rotation on left child of X, where X node violates the property.



Step2: single right rotation on node X

B - Tree 

In computer science, a B-tree is a tree data structure that keeps data sorted and allows searches, insertions, deletions, and sequential access in logarithmic amortized time.



The B-tree is a generalization of a binary search tree in that more than two paths diverge from a single node.



Unlike self-balancing binary search trees, the B-tree is optimized for systems that read and write large blocks of data in secondary storage such as a magnetic disk. Since disk accesses are expensive (time consuming) operations, a b-tree tries to minimize the number of disk accesses. It is most commonly used in databases and filesystems.

Overview 

In B-trees, internal (non-leaf) nodes can have a variable number of child nodes within some pre-defined range. When data is inserted or removed from a node, its number of child nodes changes. In order to maintain the pre-defined range, internal nodes may be joined or split. Because a range of child nodes is permitted, B-trees do not need re-balancing as frequently as other self-balancing search trees, but may waste some space, since nodes are not entirely full. The lower and upper bounds on the number of child nodes are typically fixed for a particular implementation. For example, in a 2-3 B-tree (often simply referred to as a 2-3 tree), each internal node may have only 2 or 3 child nodes.



A B-tree is kept balanced by requiring that all leaf nodes are at the same depth. This depth will increase slowly as elements are added to the tree, but an increase in the overall depth is infrequent.

Advantages B-trees have substantial advantages over alternative implementations when node access times far exceed access times within nodes. This usually occurs when the nodes are in secondary storage such as disk drives. By maximizing the number of child nodes within each internal node, the height of the tree decreases and the number of expensive node accesses is reduced. In addition, rebalancing the tree occurs less often. NOTE: By maximizing the number of child nodes within each internal node, the height of the tree decreases and the number of expensive node accesses is reduced. Definition The terminology used for B-trees is inconsistent in the literature. Unfortunately, the literature on B-trees is not uniform in its use of terms relating to B-Trees.



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A B-tree of order m is a tree which satisfies the following properties: 1. Every node has at most 2m children. 2. Every node (except root and leaves) has at least m children. 3. The root has at least two children if it is not a leaf node. 4. All leaves appear in the same level, and carry information. 5. A non-leaf node with k children contains k–1 keys. 6. Leaf nodes contains at least m-1 and at most 2m-1 keys. Each internal node's elements act as separation values which divide its subtrees. For example, if an internal node has three child nodes (or subtrees) then it must have two separation values or elements a1 and a2. All values in the leftmost subtree will be less than a1 , all values in the middle subtree will be between a1 and a2, and all values in the rightmost subtree will be greater than a2. Maximum Nodes In a B-Tree of Height h. As we discussed in binary tree chapter, we will follow the same instructions here too for getting the maximum number of nodes. We will get this only if each internal node has maximum number of children. Thus B-Tree of degree m will have maximum nodes at each level as follows: 1st level 2nd level 3rd level … Last level

(2m)0 (2m)1 (2m)2 (2m)h

Thus Nmax = (2m)0 + (2m)1 + (2m)2 …+(2m)h = ((2m)h+1 - 1 )/ (2m – 1) From this we will get maximum records also which is Nmax * (2m -1). Minimum Nodes In a B-Tree of Height h. This is somewhat interesting and important also. We will get this only if each internal node has minimum number of children. Thus B-Tree of degree m will have minimum nodes at each level as follows: 1st level (1) nd 2 level 2(m)0Remember root is allowed to 2 children minimally rd 3 level 2(m)1 th 4 level 2(m)2 … Last level 2(m)h -1 Thus Nmin = 1 + 2 *(mh – 1) / (m – 1)



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From this we will get minimum records also which is 1 + (Nmin – 1)* (m -1). Notice that not all node in Nminwill have m – 1 keys. The root node will have one key. NOTE: The best case height of a B-Tree is: O(log2mN) The worst case height of a B-Tree is: O(logmN) where m is the B-Tree degree. Maximizing B-Tree Degree A bad degree choice will lead to worst case height of B-Tree and hence its operations will become expensive. It is always good to choose a good degree to minimize the height this will not only utilizes the disk spaces but also reduces the access cost since height gets reduced in this case. The following derivation will let you enable for choosing the best degree.

Fig 4.12 As the diagram shows that each B-Tree node contains fields for child pointers, data pointers and keys. Let disk block size is = B bytes, and degree is = d (child pointers per node) And index pointer to data and child are of = b bytes, And key length is of = k bytes. Then a value of d would be best choice if it maximizes value of the left side of below equation such that result is closest to right side (B). (d – 1) * b + d * b + (d – 1)*k = B ----------- 1 Insertion All insertions start at a leaf node. To insert a new element Search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:



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1. If the node contains fewer than the maximum legal number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered. 2. Otherwise, the node it is full, so evenly split it into two nodes. 

A single median is chosen from among the leaf's elements and the new element.



Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value.



Insert the separation value in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the tree).

If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. Deletion The logic to delete a record is to locate and delete the item, then restructure the tree to regain its invariants There are two special cases to consider when deleting an element: 

The element in an internal node may be a separator for its child nodes.



Deleting an element may put its node under the minimum number of elements and children.

Each of these cases will be dealt with in order. Deletion from a leaf node Search for the value to delete. If the value is in a leaf node, it can simply be deleted from the node, perhaps leaving the node with too few elements; so some additional changes to the tree will be required. Deletion from an internal node Each element in an internal node acts as a separation value for two subtrees, and when such an element is deleted, two cases arise. In the first case, both of the two child nodes to the left and right of the deleted element have the minimum number of elements, namely L. They can then be joined into a single node with 2Lelements, a number which does not exceed allowed maximum number of elements per node. In the second case, one of the two child nodes contains more than the minimum number of elements. Then a new separator for those subtrees must be found. Note that the largest element in the left subtree is still less than the separator. Likewise, the smallest element in the right subtree is the smallest element which is still greater than the separator. Both of those elements are in leaf nodes, and either can be the new separator for the two subtrees. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 91


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

If the value is in an internal node, choose a new separator (either the largest element in the left subtree or the smallest element in the right subtree), remove it from the leaf node it is in, and replace the element to be deleted with the new separator.



This has deleted an element from a leaf node, and so is now equivalent to the previous case.

Rebalancing after deletion If deleting an element from a leaf node has brought it under the minimum size, some elements must be redistributed to bring all nodes up to the minimum. In some cases the rearrangement will move the deficiency to the parent, and the redistribution must be applied iteratively up the tree, perhaps even to the root. Since the minimum element count doesn't apply to the root, making the root be the only deficient node is not a problem. The algorithm to rebalance the tree is as follows: If the right sibling has more than the minimum number of elements. Add the separator to the end of the deficient node. Replace the separator in the parent with the first element of the right sibling. Append the first child of the right sibling as the last child of the deficient node otherwise, if the left sibling has more than the minimum number of elements. Add the separator to the start of the deficient node. Replace the separator in the parent with the last element of the left sibling. Insert the last child of the left sibling as the first child of the deficient node, if both immediate siblings have only the minimum number of elements. Create a new node with all the elements from the deficient node, all the elements from one of its siblings, and the separator in the parent between the two combined sibling nodes. Remove the separator from the parent, and replace the two children it separated withthe combined node. If that brings the number of elements in the parent under the minimum, repeat these steps with that deficient node, unless it is the root, since the root may be deficient. The only other case to account for is when the root has no elements and one child. In this case it is sufficient to replace it with its only child. B+ Tree A B+ tree is a type of tree, which represents sorted data in a way that allows for efficient insertion, retrieval, and removal of records, each of which is identified by a key. It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a "block" or "node"). NOTE: In a B+ tree, in contrast to a B-tree, all records are stored at the leaf level of the tree; only keys are stored in interior nodes.



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Fig 4.17. B+ Tree NOTE: 

The primary value of a B+ tree is in storing data for efficient retrieval in a blockoriented storage context. This is primarily because unlike binary search trees, B+ trees have very high fanout (typically on the order of 100 or more), which reduces the number of I/O operations required to find an element in the tree.



Notice that duplicates keys exist in B+ Trees.



B+ Trees are designed for having the range query running faster. The nodes in leaf level are connected in linked list fashion and that makes it possible.

Maximizing B+ Tree Degree The following derivation will let you enable for choosing the best degree.

Fig 4.18 As the diagram shows that each B+ tree internal node contains fields for child pointers, and keys. Let disk block size is = B bytes, and degree is = d (child pointers per node) And index pointer to child is of = b bytes, And key length is of = k bytes. Then a value of d would be best choice if it maximizes value of the left side of below equation such that result is closest to right side (B). d * b + (d – 1)*k = B ----------- 1 THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 93


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Insertion in B+ Tree 1. do a search to determine what bucket the new record should go in 2. if the bucket is not full, add the record. 3. Otherwise, split the bucket. 4. allocate new leaf and move half the bucket's elements to the new bucket 5. Insert the new leaf's smallest key and address into the parent. 6. if the parent is full, split it also 7. now add the middle key to the parent node 8. repeat until a parent is found that need not split 9. if the root splits, create a new root which has one key and two pointers.



DSA

3. 8 HASHING Hashtables 

A memory-efficient alternative for sparse arrays.



The set of posible keys (here: {0,1,2,...,K-1,K}) is called the universe.



If the universe U is much larger than the number of keys actually stored (here: N), then a hashtable might be better than a standard array.



In our problem we have |U| = 109 and N <= 106.

The basic idea inthashtable[M]; // M > N void store (int key, int value) { hashtable[h(key)] = value; }

voidget_value (int key) { returnhashtable[h(key)]; } int h (int key) { return h % M; } 

Instead of using key as the index, we use h(key).



h is a function that maps the universe U of possible keys to the indices of the hashtable.



h : U -> {0,1,2,...,M-2,M-1}



Usual hash functions:



o

h(k) = k mod "a prime not too close to a power of 2"

o

h(k) = floor(m*(k*c mod 1)) with 0
Problem: Collisions. Two keys might be mapped to the same index.



DSA

How likely are collisions? 

In a room with 35 people, how likely is it that two people have the same birthday?



|U| = 365



N = 35

Ways to resolve collisions 

Open Hashing



Linear Probing



Quadratic Probing



Double Hashing

Open Hashing 

Each element of the hashtable is a linked list of elements that got mapped to this location.



If you store N keys in a table of size M, the lists will have an average length of N/M.

Linear Probing 

If a cell is already in use, take the next one.



hi(key) = h(key) + i



Problem: Clustering

Quadratic Probing 

If cell x is already taken, try x+1, x+4, x+9, x+16,..., until you find an empty cell.



hi(key) = h(key) + i*i



At least: No clustering anymore.



But if many keys are mapped to the same cell x, each key runs through the same sequence of cells until an empty cell is found.

Double Hashing 

On the ith try to find an empty cell for key, use the hash function: hi(key) = (h(key) + i*h'(key)) mod m.



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Universal Hashing 

One problem remains: No matter which hash function you choose, there will always be a worst-case input where each key is hashed to the same slot.



Solution: Randomize the hash function!



Have a set of hash functions ready and choose from them after the program started.



A collection H of hash functions is called universal, if for each pair of distinct keys x,y, the number of hash functions for which h(x)=h(y) is exactly |H|/m.



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3.9: Graph Algorithms Important Definitions Connected Graph An undirected graph is connected if there is path from every vertex to every other vertex. A directed graph with this property is called strongly connected. If a directed graph is not strongly connected however the underlying graph (without directions to the arcs) is connected, then graph is said to be weakly connected. Complete Graph A complete graph is a graph in which every vertex is connected every other vertex. That means e ch vertex’s de ree i u directed r ph is n – 1. Total number of edges are = n(n - 1) / 2. A d e ch vertex’s i

d ut de rees re n – 1 in case of directed graph.

Total number of edges in directed complete graph are = n(n - 1) . Acyclic Graph An acyclic graph is a graph which does not have cycles. Maximum Number of Graphs for n Vertices Let Nmaxis maximum number of undirected graphs for n vertices. This is c e r y sum f r phs with 0ed es, 1ed es, 2ed es, …,

d ( -1)/2 edges.

Graphs with n(n-1)/2 edges = C(n(n-1)/2, n(n-1)/2). Thus, Nmax n(n-1)/2)

=

C(n(n-1)/2, 0) + C(n(n-1)/2, 1) + C(n(n-1)/2, 2) + C(n(n-1)/2, 3) + C(n(n-1)/2,

= 2n(n-1)/2 In case of directed graph without self loopNmax= 2n(n-1). Representation of Graphs Adjacency matrix: Here two dimensional arrays are used. For each edge, (u,v) a[u,v] = true ,otherwise it will be false. The space complexity to represent a graph using this is O(|V|2). Adjacency list: Array of lists is used, where each list stores adjacent vertices of the corresponding vertex. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 98


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The space complexity to represent a graph using this is O(|V|+ |E|). Single Source Shortest Path Algorithm The problem is about to compute shortest path to every other vertex from a given starting vertex in a directed graph. Unweighted Shortest Path Algorithm This is efficiently done using BFS algorithm. RT of the whole algorithm is = O(|V| + |E|). If the graph were represented using adjacency matrix then the RT would be O(|V|2). This is bec use ‘RT f pr cessi e ch vertex’ corresponding row has to be searched). Wei hted Sh rtest P th A

rithm (Dijkstr ’s A

w wi t ke O(|v|) (e ch rr y e eme t f the rithm)

Dijkstr ’s rithm pr ceeds i st es. At e ch st e, this rithm se ects vertex v, which h s the smallest dv among all the unknown vertices, and declares that the shortest path from s to v is known. The remainder of a stage consists of updating the values of dw. RT is = |V|log|V| + |E|log|V| = (|E| + |V|)log|V|. However, if graph is represented using adjacency matrix, then RT is = |V|log|V| + |V|2log|V| = O (|V|2log|V|).

Minimum Spanning Tree Let G V , E  be an undirected graph where V  set of vertices & E  set of edges, then a spanning tree T of graph can be defined as a subgraph of G. A spanning tree of a graph must include every vertex and if the graph contains n vertices, spanning tree must contain exactly (n -1) edges. A minimum cost spanning tree is a spanning tree such that the sum of all the weights of edges in spanning tree is minimum. Two greedy algorithms exists, to solve this problem, namely as Prims and Krushkal algorithm. Prims Algorithm This algorithm computes the minimum spanning tree by including appropriate one vertex and thus one edge into existing partially constructed tree in successive stages. At any point in the algorithm, we can observe that we have a set of vertices that have already been included in the tree; the rest of the vertices have not. The algorithm then finds, at each stage, a new vertex to add to the tree by choosing the edge (u, v) such that the cost of this edge is the smallest among all edges where u is in the tree and v is not. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 99


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The Prims algorithm is essentially identical to Dijkstra only, except the update rule. The new update rule is as follows: dw= min(dw, cw,v). Thus, the RT analysis of Dijkstra algorithm will also remain applicable here too. Krushkal Algorithm Initially from the graph G, consider a forest of all the vertices without any edge. 1. Sort all the edges in ascending order according to their costs. 2. Include the edge with minimum cost into the forest if it does not form a cycle in the partially constructed tree. Repeat step (2) until no edge can be added to the tree. Running Time Analysis Total number of edges considered by this algorithm in worst case is |E|. That means, the maximum number of stages is |E|. Now we need to compute the RT of each stage. In each stage an edge with lowest cost is selected and being checked for not causing cycle. Thus, if minHeap of edges is used then finding next edge takes up O(log|E|). Therefore, total RT is O(|E|log|E|). Notice that if E=O(|V|2), then RT is O(|E|log|V|). Remarks: 

If the weight of all the edges of a graph G is unique, then only one minimum spanning tree exists of that graph.

If the weights of all the edges of a graph G are not unique, then the graph might have only one minimum spanning tree or more than one also which are structurally different.



DSA

3.10: Sorting Algorithms Sorting algorithms used in computer science are often classified by: Bubble sort Bubble sort is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. First pass bubble out the largest element and places in the last position and second pass place the second largest in the second last position and so on. Thus, in the last pass smallest items is placed in the first position. Because, it only uses comparisons to operate on elements, it is a comparison sort. Performance The running time can be bound as the total number of comparison being made in the entire execution of this algorithm. Thus, the worst case (input is descending order) comparisons in the respective passes are as follows: Total comparisons= n(n -1) / 2; which implies O(n2) time complexity. Bubble sort has worst-case and average complexity both О(n²), where n is the number of items being sorted. Performance of bubble sort over an already-sorted list (best-case) is O(n). Remember using a flag it can easily be determined that if no swaps were performed in the current pass that would mean the list is sorted and bubble sort can come out then itself rather than going through all the remaining passes. Imp Points: 

It is comparison based sorting method.



Worst case space complexity is of O(1).



It is adaptive sorting method as presortedness affects the running time.



A stable sorting method.

Insertion Sort Insertion sort is a simple a comparison sorting algorithm. Every iteration of insertion sort removes an element from the input data, inserting it into the correct position in the alreadysorted list, until no input elements remain. The choice of which element to remove from the input is arbitrary, and can be made using almost any choice algorithm. Performance The worst case input is an array sorted in reverse order. In this case every iteration of the inner loop will scan and shift the entire sorted subsection of the array before inserting the next element. For this case insertion sort has a quadratic running time (i.e., O(n2)). The running time can be bound as the total number of comparison being made in the entire execution of this algorithm. Thus, the worst case comparisons in the respective passes are as follows: THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 101


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Total comparisons = n(n -1) / 2; which implies O(n2) time complexity. The best case input is an array that is already sorted. In this case insertion sort has a linear ru i time (i.e., Θ(n)). During each iteration, the first remaining element of the input is only compared with the right-most element of the sorted subsection of the array. While insertion sort typically requires more writes because the inner loop can require shifting large sections of the sorted portion of the array. In general, insertion sort will write to the array O(n2) times, whereas selection sort will write only O(n) times. For this reason selection sort may be preferable in cases where writing to memory is significantly more expensive than reading. Remarks: It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages: 

Simple implementation.



Efficient for (quite) small data sets.



Adaptive, i.e. efficient for data sets that are already substantially sorted: the time complexity is O(n ).



Stable, i.e. does not change the relative order of elements with equal keys.



Only requires a constant amount O(1) of additional memory space.

Selection Sort Selection sort is also a simple a comparison sorting algorithm. The algorithm works as follows: 1. Find the minimum value in the list 2. Swap it with the value in the first position 3. Repeat the steps above for the remainder of the list (starting at the second position and advancing each time) Effectively, the list is divided into two parts: the sublist of items already sorted, which is built up from left to right and is found at the beginning, and the sublist of items remaining to be sorted, occupying the remainder of the array. Performance The all inputs are worst case input for selection sort as each current element has to be compared with the rest of unsorted array. The running time can be bound as the total number of comparison being made in the entire execution of this algorithm. Thus, the worst case comparisons in the respective passes are as follows: Total comparisons = n(n -1) / 2; which implies O(n2) time complexity.



DSA

Remarks: 

It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort.



Simple implementation.



Efficient for (quite) small data sets.



Not being adaptive.



Its stability depends on the implementation of choosing minimum.



In-place, i.e. only requires a constant amount O(1) of additional memory space.



Insertion sort is very similar in that after the kth iteration, the first k elements in the array are in sorted order. Insertion sort's advantage is that it only scans as many elements as it needs in order to place the k + 1st element, while selection sort must scan all remaining elements to find the k + 1st element. Selection s rt w ys perf rms Θ(n) swaps.



Merge Sort Merge sort is an O(n log n) comparison-based divide and conquer sorting algorithm. Conceptually, a merge sort works as follows: 1. If the list is of length 0 or 1, then it is already sorted,otherwise: 2. Divide the unsorted list into two sublists of about half the size. 3. Sort each sublist recursively by re-applying merge sort algorithm. 4. Merge the two sublists back into one sorted list. Performance In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n units taken to merge the resulting two sorted lists). Thus, after simplifying the recurrence relation; T(n) = O(nlogn). Remarks: 

Its not adaptive.



Merge sort is a stable sort as long as the merge operation is implemented properly.





DSA

Not a in-place sorting method, requires O(n) of additional memory space. The additional n locations were needed because one couldn't reasonably merge two sorted sets in place.

Heap Sort Heap sort is a comparison-based sorting algorithm which is much more efficient version of selection sort. It also works by determining the largest (or smallest) element of the list, placing that at the end (or beginning) of the list, then continuing with the rest of the list, but accomplishes this task efficiently by using a data structure called a heap. Once the data list has been made into a heap, the root node is guaranteed to be the largest(or smallest) element. When it is removed(using deleteMin/deleteMax) and placed at the end of the list, the heap is rearranged so the largest element of remaining moves to the root. Using the heap, finding the next largest element takes O(log n) time, instead of O(n) for a linear scan as in simple selection sort. This allows Heapsort to run in O(n log n) time. Remarks: 

Its not adaptive.



It is in-place sorting method as utilized the same input array for placing the sorted subarray.



Not a stable sorting method as during deleteMin/deleteMax the order is not preserved for the same key values. Consider an input that having all the same key values. The deleteMin will pickup the last heap element as to place in the root location. Thereby, the order is changed because in the sorted output later values appears before.

Quick Sort Quick sort sorts by employing a divide and conquer strategy to divide a list into two sub-lists. The steps are: 1. Pick an element, called a pivot, from the list. 2. Reorder the list so that all elements which are less than the pivot come before the pivot and so that all elements greater than the pivot come after it (equal values can go either way). After this partitioning, the pivot is in its final position. This is called the partition operation. 3. Recursively sort the sub-list of lesser elements and the sub-list of greater elements.

The base case of the recursion are lists of size zero or one, which are always sorted. Performance If the running time of quick sort for a list of length n is T(n), then the recurrence T(n) = T(size of s1) + T(size of s2) + n follows from the definition of the algorithm (apply the algorithm to two sets which are result of partioning, and add the n units taken by the partioning the given list).



DSA

Worst Case Time Complexity The quick sort shows its worst case behaviour when one of the subset after partioning is always empty for all subsequent recursive quick sort calls. This could have happened because of a bad choice of pivot. Then the above recurrence relation can be written as follows to represent the worst case behaviour. T(n) = 0 + T(n -1) + n = T(n -1) + n Therefore, T(n) = n + (n - 1) + (n - 2) + … 1 = O( 2). Best Case Time Complexity The quick sort shows its best case behaviour when the size of each subset after partioning is always close to same for all subsequent recursive quick sort calls. This could have happened because of a good choice of pivot. Then the above recurrence relation can be written as follows to represent the best case behaviour. T(n) = 2T(n/2) + n Therefore, T(n) = O(nlogn). O(nlogn) is also the average case time complexity of quick sort. Remarks: 

The known fastest in-place algorithm in practice.



Each call finalizes the correct position of pivot element which can never be changed by any subsequent calls.



It is an adaptive sorting method.



Can be implemented as a stable sort depending on how the pivot is handled.



Operating System

Part – 4: Operating System 4.1: Introduction to Operating System An operating system is a program (system software) that manages the computer hardware resources. It also provides a basis for application programs and acts as an intermediary between a user of a computer and the computer hardware. The secondary goal of operating system is to allocate the resources among various application programs as efficient as possible. Also, it enforces security through abstraction- it transforms physical world of devices, instruction, memory etc. into virtual world. 1. Computer system Architecture (i) Single-Processor system There is one main CPU capable of executing a general-purpose instruction set, including instructions from user processes. (ii) Multi-Processor System These systems have two or more than two processors in close communication, sharing the computerresources. Multiprocessor system has three main advantages. a. Increased throughput b. Economy of scale c. Increased reliability (iii)Clustered System Like multiprocessor systems, clustered systems gather together multiple CPUs to accomplish computational work. Clustered systems differ frommultiprocessor systems, however, in that they are composed of two or more individual systems coupled together. 2. Operating System Structure One of the most important aspects of operating systemis the ability of multiuser and multiprocessing. A single user cannot, in general, keep either the CPU or the I/O devices busy at all times. Multiprogramming increases CPU utilization by organizing jobs (code or data) so that the CPU always has one job to execute.



Operating System

Time sharing (or multitasking) is a logical extension of multiprogramming. In time sharing systems, the CPU executes multiple jobs by switching among them, but the switches occurs so frequently that the users can interact with each program while it is running. 3. Operating System Responsibility (A) Process Management: The operating system provides supervisor call for managing processes and must manage the allocation of resources to processes.. (B) Memory Management: The operating system must manage the allocation of memory to process and control the memory management hardware that determine which memory location can a process may access. (C) File System Management:A file system object is an abstract entity for storing or transmitting a collection of information. The file system is an organized collection of file object. The operating system must provide primitive to manipulate those object. (D) Device Management: A computer communicates information through its input-output devices. Processes access those devices through the operating system supervisor call provided for that purpose. 4. Operating System Operations Traps and interrupts are events that disrupt the normal sequence of instructions executed by the CPU. A trap is an abnormal condition detected by the CPU that usually is an indicative of an error. Trap conditions can occur in following ways: (a) dividing by zero, (b) trying to access a memory location that does not exist or for which the program does not have access, (c) executing an instruction with an undefined opcode, (d) trying to access a nonexistent I/O device. An interrupt is a signal sent to the CPU by an external device, typically an I/O device. It is the CPU’s equivalent of a pager, a signal requesting the CPU to interrupt its current activities to attend to the interrupting devices needs. A CPU will check interrupts only after it has completed the processing of one instruction and before it fetches a subsequent one. The CPU responds to traps and interrupts by saving the current value of the program counter and resetting the program counter to a new address. This allows the CPU to return, to executing at the point, the trap or interrupt occurred, after it has executed a procedure for handling the trap or interrupt. (i) Dual-Mode Operation Two separate modes of operation: user mode and kernel mode (also called supervisor mode, system mode, or privileged mode). A bit, called the mode bit, is added to the computer harware to indicate the current mode: Kernel (0) or user (1). 5. Operating System Services The Operating system services are provided for the convenience of the programmer, to make programming task easier. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 107


Operating System

One set of operating system services provides functions that are helpful to the user. (i) User interface (ii) Program executions (iii) I/O operation (iv) File system manipulation (v) Communications (vi) Error detection (vii) Resource allocation (viii) Protection and security 6. User Operating System Interface There are two fundamental approaches for users to interface with the operating system. One technique is to provide a command line interface or command interpreter that allows operations to be performed by the operating system. The second approach allows the user to interface with the operating system via a Graphical User Interface (GUI). 7. System Calls System calls provide an interface between the user process and the Operating System. Basically, it provides the interface to the services made available to user processes by an operating system. System calls can be grouped roughly into five major categories: (i) (ii) (iii) (iv) (v)

Process control File manipulation Device manipulation Information maintenance communications.



Operating System

4.2: Process Management Process: A process is a program in execution. A process is more than the program code, which is sometimes known as the text section. Rather a process is also considered to be as an ‘active’ entity and a program as a ‘passive’ entity. A process is sequence of instructions in context of process states. One process cannot affect state of another process directly In addition to static program text (code) process comprises of following (i) Current value of program counter (ii) The content of the processor registers (iii) Process static (temporary data, sub-routine parameters return address etc)i.e stack. (iv)Global variables. (v) Open file tables. Process state: As a process executes, it changes state. The state of a process is defined by the current activity of that process. Long-term scheduling adds the following state. Held (New, Start, Born): A process that has been considered for loading into memory or for execution. Once a process leaves the held state, it does not return to it. Medium-term scheduling adds the following state Swapped out: A process that is running may go in swapped out state, it depends on scheduling algorithm. The state diagram corresponding to these states is presented in Fig.4.2.1 Importance Of Scheduling 1. Scheduling can effect the performance of the system in a big way.



Operating System

New/Start Job scheduler long term scheduling I/O completion

Waiting for I/O Waiting

Ready

Short term scheduler Running

exit Indefinite waiting is Deadlock

Medium-term scheduling

Swapped out

STOP

Fig. 4.2.1.State diagram Following are the states in state diagram 4.2.1 New/Start: This is the state before the process is first created. It waits to be admitted in “ready” state. Ready: The process is waiting to be assigned to a processor. Running: Instructions are being executed. Waiting: The process is waiting for some input to be completed Stop: The process has finished execution Swapped: Process blocked is placed in swapped state. 

A process that has halted its execution but a record of the process is still maintained by the operating system (on UNIX). Such process referred as zombie process.

Schedulers: On multiprocessing systems, scheduling is used to determine which process is given to the control of the CPU. Scheduling may be divided into three phases: long-term, medium-term, and short-term.  Long-term scheduler: controls the degree of multi programming- the number of processes in memory.  Short term scheduler: Decides which process in ready state showed to be allocated CPU. It can also prempt the process in running state.  Mid term Scheduler: It swaps out the blocked process from running state to swapped state. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 110


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The long-term scheduler must make a careful selection. In general, most processes can be described as either I/O bound or CPU bound. An I/O bound process spends more of its time doing I/O operations than it spend doing computations. A CPU bound process using more of its time doing computations The long-term scheduler should select a good process mix of I/O bound and CPU bound processes. If all processes are I/O bound, the ready queue will be almost always empty, and the short-term scheduler will have little to do. If all processes are CPU bound, the I/O waiting queue will almost always be empty and device will go unused, and again the system will be unbalanced. The system with the best performance will have a combination of CPU bound and I/O bound processes. Context Switching Switching the CPU to another process requires saving the content of the old process and loading the saved content for the new process. This task is known as a context switch. The context of process is represented in PCB of a process; it includes the value of the CPU register and the process state. a. The context switching part of the scheduler ordinarily uses conventional load and store operation to save the register contents. This means that each context switch requires (n + m) b × k time unit, to save the state of a processor with n general registers and m status registers, assuming b store operation are required to save a single register and each store instruction requires K instruction time unit. b. Total time in context switching = time requires in saving the old process in its PCB + time requires in loading the saved state of the new process schedule to run. The context of a process is represented in the PCB of a process; it includes the value of the CPU registers, the process state (figure 4.2.2) and memory management information. pointer process state Process identifier PID program counter registers memory limits list of open files

Fig. 4.2.2 PCB Each process is represented in operating system by a process control block (PCB) also called task control block. A PCB is shown in Fig. 4.2.2 THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 111


Operating System

It contains many pieces of information associated with a specific process, including these:  

Process identifier (unique in the system) Process State: The state may be new, ready, running, waiting, stop and so on. Program Counter: The program counter indicates the address of the next instruction to be executed for this process. CPU Registers: Information about CPU registers. CPU Scheduling: It includes a process priority, pointers to scheduling queues etc. When a context switch occurs, the kernel saves the context of the old process in its PCB and loads the saved context of the new process scheduled to run. Context–switch time is pure overhead, because the system does not do useful work while switching. Most number of content switching is done by short term-scheduler.

Each subsequent selection involves one less choice. The total number of possibilities is computed by multiplying all the possibilities at each point, making the answer n! Inter-process communication, concurrency and synchronization. The concurrent processes executing in the operating system may be either independent processes or cooperating processes. A process is independent if it cannot affect or be affected by the other processes executing in the system. A process is cooperating if it can affect or be affected by the other processes executing in the system. Clearly, any process that shares data with other processes is a cooperating process. Process cooperation is used for information sharing, computation speedup, modularity and convenience. Concurrent execution of cooperating processes requires mechanisms that allow processes to communicate with one another and to synchronize their actions.

Inter-process communication: Cooperating processes can communicate in a shared memory environment and another way to achieve the same effect is for the operating system to provide the means for cooperating processes to communicate with each other via an interprocess communication (IPC) facility. IPC provides a mechanism to allow processes to communicate and to synchronize their actions without sharing the same address space. IPC is particularly useful in a distributed environment where the communicating processes may reside on different computers connected with a network. An example is a chat program used on the World Wide Web. IPC is best provided by a message-passing system, and message passing systems can be defined in many ways. Different issues when designing message passing systems are as follows:



Operating System

1. Message Passing system: Communication among the user processes is accomplished for message passing facility provides at least the two operations: i. Send (message) ii. Receive (message) Messages sent by process can be either fixed or variable size. If process P and Q want to communicate, they must send message to and receive message from each other; a communication link must exist between them. Here are several methods for logically implementing a link and the send( )/ receive( ) operations: (i) direct or indirect communication (ii) symmetric or asymmetric communication (iii) automatic or explicit buffering (iv)send by copy or send by reference (v) fixed – sized or variable – sized message 2. Naming Processes that want to communicate must have a way to refer to each other. They can use either direct or indirect communication. (i) Direct communication: With direct communication, each process that works to communicate must explicitly name the recipient or sender of the communication. In this scheme, the send and receive primitives are defined as: a) Send (P, message) – send a message to process P. b) Receive (Q, message) – receive a message from process Q. (ii) Indirect communication: With indirect communication the message are sent to and received from mailboxes, or ports. A mailbox can be viewed abstractly as an object into which messages can be placed by processes and from which messages can be removed. Two process can communicate only if they share a mailbox. The send and receive primitives are defined as follows: a) Send (A, message) – send a message to mailbox A. b) Receive (A, message) – receive a message from mailbox A.



Operating System

3. Synchronization Communication between processes takes place by calls to send( ) and receive( )primitives. Message passing may be either blocking or non-blocking also known as synchronous and asynchronous. I.

Blocking Send: The sending process is blocked until the message is received by the receivingprocess or by the mailbox. II. Non blocking receive: The receiver retrieves either a valid message or a null. III. Non blocking send: The sending process sends the message and resumes operation IV. Blocking Receive : The receiver blocks until a message is available 4. Buffering: Whether the communication is direct or indirect, message exchanged by communicating processes reside in a temporary queue. Basically, such a queue can be implemented in three ways: I.

Zero capacity: The queue has maximum length of 0; thus the link cannot have any message in it II. Bounded capacity: The queue has finite length n; thus, atmost n messages can reside in it. III. Unbounded capacity: Any number of messages can wait in it. Example of IPC systems: POSIX shared memory, mach and windows XP

Process Synchronization A situation where several processes access and manipulate the same data concurrently and the outcome of the execution depends on the particular order in which the access takes place, is called a race condition. To guard against the race condition, we need to ensure that only one process at a time can manipulate the variable counter. To make such a guarantee, we require same form of synchronization of the processes.

1. The critical section problem: Critical section: A section of code within a process that requires access to shared resources and that must not be executed while another process is in a corresponding section of code. The execution of critical sections by the processes is mutually exclusive in time. The critical-section problem is to design a protocol, that the processes can use to cooperate. Each process must request permission to enter its critical section. The section of code implementing this request is the entry section.



Operating System

The critical section may be followed by an exit section. The remaining code is the remainder section. The general structure of a typical process Pi is shown in following code. do{ entry section Critical section exit section remainder section } while (TRUE); A solution to the critical section problem must satisfy the following 3 requirements. I. Mutual exclusion: If process Pi is executing in its critical section, then no other processes can be executed in their critical sections. II. Progress: If no process is executing in its critical section and some processes wish to enter their critical sections, then only those processes that are not executing in their remainder sections can participate in the decision on which will enter its critical section next, and this section cannot be postponed indefinitely. III. Bounded waiting: There exists a bound, or limit, on the number of times that other processes are allowed to enter their critical section after a process has made a request to enter its critical section and before that request is granted. Peterson’s solution A classic software-based solution to the critical section problem known as Peterson’s solution We restrict our attention to algorithms that are applicable to only two processes at a time the processes are numbered P0 and P1. For convenience, when presenting Pl we use Pj to denote the other process; i.e. j = 1 – i. Note: Below is an algorithm for two processes. It can be generalized for n processes. Algorithm 1 

Shared variables: int turn; initiallyturn = 0 turn = I; Pi can enter its critical section





Operating System

Process Pi do{ while (turn != i) ; critical section turn = j; reminder section } while (1);



Satisfies mutual exclusion, but not progress

Algorithm 2 

Shared variables boolean flag[2]; initiallyflag [0] = flag [1] = false. flag [i] = true; Pi ready to enter its critical section



Process Pi

do { flag[i] := true; while (flag[j]) ; critical section flag [i] = false; remainder section } while (1); 

Satisfies mutual exclusion, but not progress requirement.

Algorithm 3 

Combined shared variables of algorithms 1 and 2.



Process Pi

do{ flag [i]:= true; turn = j;



Operating System

while (flag [j] and turn = j) ; critical section flag [i] = false; remainder section } while (1); 

Meets all three requirements; solves the critical-section problem for two processes

2. Synchronization hardware Many machines provide special hardware instructions that allow us either to test and modify the content of a word or to swap the contents of two words, atomically – that is, as one uninterruptable unit. The Test And Set instruction can be defined as follows: boolean Test And Set (boolean& lock) { boolean r; r = lock; lock = true; return r; } If lock is true then we can’t enter in to critical section Initially lock = false. while (Test And Set (lock)) critical section lock = false ; In above method problem is busy waiting, because when lock is true then process continues execute that instruction. (i) Swap instruction void swap (boolean a, boolean b) { boolean c; c = a; a = b; THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 117


Operating System

b = c; return b; } A global Boolean variable lock is declared and is initialized to false. lock = false ; while (swap (lock, true)) critical section lock = false ; In above also problem of busy waiting 3. Semaphores A semaphores S is an integer variable that, apart from initialization, is accessed only through two standard atomic operations: wait ( )and signal( ). These operations were originally termed P (for wait) and V (for signal) The classical definition of wait in pseudo code is: wait (s) { while ( s

0)

// keep waiting s = s – 1; } The classical definitions of signal in pseudo code is: signal (s) { s = s + 1; } A binary semaphore is a semaphore where count may only take on the value of 1 or 0. A semaphores which can take on any non negative value may be referred to as general semaphores or counting semaphores. Let S be a counting semaphore. To implement it in term of binary semaphores we need the following data structure:



Operating System

binary semaphore S1, S2; int C ; Initially S1 = 1, S2 = 0, and the value of integer C is set to the initial value of the counting semaphore S. The wait operation on the counting semaphore S can be implemented as follows: Wait (S1); C--; if(C

) Signal (S1); Wait (S2); }

else Signal(S2); The signal operation on the counting semaphore s can be implemented as follows: Wait (S1); C + +; if (C < = 0) Signal(S2) ; else Signal(S1); The main disadvantage of the mutual-exclusion solutions and the semaphores definition given here, is that they all require busy waiting. Busy waiting: While a process is in its critical section, any other process that tries to enter its critical section must loop continuously in the entry code. This continual looping is clearly a problem in a real multiprogramming system, where a single CPU is shared among many process. Busy waiting wastes CPU cycle that some other process might be able to use productively. This type of semaphore is also called a spinlock. Spinlocks are useful in multiprocessor systems. When two or more process are waiting indefinitely for an event that can be caused only by one of the waiting processes, these processes are said to be deadlocked.



Operating System

Another problem related to deadlocks is indefinite blocking or starvation, a situation where processes wait indefinitely within the semaphore. Classic Problem of Synchronization I.

Classical Producer consumer problem with bounded buffer. semaphore mutex = 1 semaphore can_produce = n = 5 = buffer size semaphore can_consume = 0 Producer while (1) { wait (can_produce) ; wait (mutex) ; -------------------------Produce data -------------------------Signal (mutex); Signal ( can_consume); }

Consumer while (1) { wait ( can_consume) ; wait (mutex) ; ----------------------------------Consume data ----------------------------------Signal (mutex); Signal ( can_produce);

}

For unbounded buffer remove wait ( can produce) from produce code.

(i) Readers- writers problems (a) If reader come and reader available then no problem. (b) If writer available then reader and writer have to wait. Reader_count = 0 semaphore s = 1,mutex=1; Reader wait(mutex); reader_count++;

Writer wait (s) ; Perform writing Signal (s) ;

If (reader_count = 1) wait (s) ; Signal(mutex); Perform Reading



Operating System

Wait(mutex); reader count - - ; If( reader count = = 0) Signal (s) ; Signal(mutex);

II.

The Dining – Philosophers problem: Consider five philosophers who spend their lives thinking and eating. The philosophers share a common circular table surrounded by five chairs, each belonging to one philosopher. In the center of the table is a bowl of rice, and the table is laid with five single chopsticks.

When a philosopher thinks, she does not interact with her colleagues. From time to time, a philosopher gets hungry and tries to pick up the two chopsticks that are closest to her. When a hungry philosopher has both her chopsticks at same time, she eats without releasing her chopsticks. When she finishes eating, she puts down both of her chopsticks and starts thinking again. One simple solution is to represent each chopstick by a semaphore. A philosopher tries to grab the chopsticks by executing a wait operation on that semaphore; she releases her appropriate semaphore. Thus the shared data are Semaphore chopsticks [5]; Where all the element of chopsticks are initialized to 1. do { wait (chopstick [i]); wait (chopstick[(i+1)%5]); …… eat Signal (chopstick [i]); Signal (chopstick [ (i+1)% 5]); …… Think …… THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 121


Operating System

} while (1); Although this solution guarantees that no two neighbors are eating simultaneously, it nevertheless must be rejected because it has the possibility of creating a deadlock. Suppose that all five philosophers become hungry simultaneously, and each grabs her left chopsticks. All the elements of chopsticks will now be equal to 0. When each philosopher tries to grab her right chopstick, she will be delayed forever. Solution that ensure freedom from deadlocks: (i) Allow atmost four philosophers to be sitting simultaneously at the table. (ii) Allow a philosopher to pick up her chopsticks only if both chopsticks are available. (iii) A symmetric solution: An odd philosopher picks up first her left chopsticks and then her right chopstick, whereas an even philosopher pick up her right chopstick and than her left chopstick. (iv) One should be right handed (first pick right chopsticks) and all other left handed or vice versa.

Monitors   

A high-level abstraction that provides a convenient and effective mechanism for process synchronization. Only one process may be active with in the monitor at a time Monitor {

Monitor name || shared procedure

variable p1 (…)

declarations …}

…… procedure

pn(…)

…}

initialization

code (…)

…}

……. }



Schematic view of a monitor







Operating System

Condition Variables  Condition x, y:  Two operations on condition variables:  x.wait( ) –a process that invoked the operation is suspended  x. signal ( ) – resumes one of processes that invoked x.wait ( ) Monitor wit condition variables

,



Operating System

4.3: Threads A thread is a single sequence stream within a process. It inhibits some properties of the process, therefore it is sometimes termed as a light weight process (LWP) which is a basic unit of CPU utilization. It comprises of a thread ID, a program counter, a register set, and a stack. It shares with other threads belonging to the same process its code section, data section, and other operating-system resources, such as open files and signals. If the process has multiple threads of control, it can do more than one task at a time. Code register s

data

file s stack

Code register s stack

data register s stack

file s register s stack

Sharable resources Non-Sharable resources

Thread Thread Single Threaded

Multi Threaded

Fig. 4.3.1 Single and multi threaded processes Benefits of multithreaded programming 1. Responsiveness: Multithreading is an interactive application which may allow a program to continue running even if part of it is blocked or is performing a lengthy operation. 2. Resource sharing: Threads share the memory and the resources of the process to which they belong. 3. Economy: Threads share resources, so it is more economical to create and context switch threads. 4. Utilization of multiprocessor architecture: The benefits of multithreading can be greatly increased in a multiprocessor architecture, where each thread may be running in parallel on a different processors. Table 4.3.1 Comparison of User-level Thread & Kernel-level Thread S. No User – level Thread Kernel – level Thread i. It is managed and supported by It is managed and supported by the the user. There is no operating system. Here, kernel intervention of Kernel. manages the thread. ii. It is fast to create and manage It is slower to create and manage than user-level threads. iii. If any thread in user level A thread in kernel – mode cannot makes a blocking system call block entire system by making a and kernel run in a single blocking system call. thread mode, then entire system will be blocked.



Libraries:

Operating System

User – level Thread

Kernel – level Thread

POSIX Pthread, mach C-thread, and solaries 2 U1 – threads.

window (NT, 2000) solaries 2, BeOS, and Tru 64.

Multithreading Models: 1. Many to one model : The many-to-one model maps many user-level threads to one kernel thread. Thread management is done in user space. So, it is efficient, but the entire process will block if a thread makes a blocking system call. Green threads – a thread library available for Solaris 2 uses this model. User thread

K

Kernel thread

Fig. 4.3.2 2. One-to-one model: The One-to-One model maps each user thread to a kernel thread. It provides more concurrency than the many-to-one model by allowing another thread to run when a thread makes a blocking system call. User thread

K

K

K

Kernel thread

Fig. 4.3.3 3. Many-to-many model: This model multiplexes many user-level threads to a smaller or equal number of kernel threads. User thread

K

K

K

Kernel thread

Fig. 4.3.4



Operating System

4.4: CPU Scheduling CPU Scheduling is the basis of multi-programmed operating systems. By switching the CPU among processes, the operating system can make the computer more productive. 1. Basic concepts: Scheduling is a fundamental operating system functions. Almost all computer resources are scheduled before use. The CPU is, of course, one of the primary computer resources. Thus, its scheduling is central to operating-system design. CPU – I/O Burst Cycle: Process execution consists of a cycle of CPU execution and I/O wait. Processes alternate between these two state. CPU scheduling decisions may take place under the following two ways: (i) Non preemptive: In this case, once a process is in the running state, it continues to execute until (a) it terminates or (b) it blocks itself to wait for I/O or to request some operating system service. (ii) Preemptive: The currently running process may be interrupted and moved to the ready state by the operating system. The decision to preempt may be performed when a new process arrives; when an interrupt occurs that places a blocked process in the ready state; or periodically, based on a clock interrupt. 2. Scheduling Criteria Many criteria have been suggested for comparing CPU scheduling algorithms. The characteristics used for comparison can make a substantial difference in the determination of the best algorithm. The criteria include the following:  Throughput: The number of processes completed per unit time, called throughput. It should be maximized  Turn around Time (TAT): The interval from the time of submission of a process to the time of completion is turn around time. TAT = Completion time – Arrival time It should be minimized  Waiting time: Waiting time is the sum of the periods spent waiting in the ready queue. Waiting time = TAT – CPU Burst Or time Waiting time = Completion time – Arrival time – CPU Burst time It should be minimized.  Response time: Time between submission and first response It should be minimized.



Operating System

Scheduling algorithms: CPU scheduling deals with the problem of deciding which of the processes in the ready queue are to be allocated to the CPU. There are following types of scheduling: 1. First-Come, First-Served (FCFS) scheduling: Processes are scheduled in the order they are received. Advantage: Easy to implement. Disadvantage: If big process comes first, then small process suffer. This is called convey effect. Process Execution time Arrival time P1 20 0 P2 25 15 P3 10 30 P4 15 45 P 0

P 20

P 45

P 55

70

2. Shortest-Job-First (SJF) scheduling: Selects the process with the shortest expected processing time, and do not preempt the process. Advantage:Minimum average waiting time Disadvantage:(1) There is no way to know the length of the next CPU burst. (2) The constant arrival of small jobs can lead to indefinite blocking (Starvation)of a long job. 3. Shortest-Remaining-Time-First (SRTF) scheduling: Selects the process with the shortest expected remaining process time. A process may be preempted when another process becomes ready. 4. Priority scheduling: Priority scheduling requires each process to be assigned, a priority value. The CPU is allocated to the process with the highest priority. FCFS can be used in case of a tie. Priority scheduling can be either preemptive or non-preemptive A major problem with priority scheduling algorithm is indefinite blocking (or starvation) A solution to the problem of indefinite blocking of low- priority processes is aging. Aging is a technique of gradually increasing the priority of processes that wait in the system for a long time. 5. Round-Robin (RR) Scheduling: This algorithm is designed especially for time sharing systems. It is similar to FCFS scheduling, but preemption is added to switch between processes. A small unit of time, called a time quantum (or time slice) is defined. A time quantum is generally from 10 to 100 millisecond.      

To implement RR scheduling, we keep the ready queue as First In First Out (FIFO) queue of processes. New processes are added to the tail of the ready queue The RR scheduling algorithm is preemptive. If there are n processes in the ready queue and the time quantum is q, then each process gets 1/n of the CPU time in chunk of at most q time units. Each process must wait no longer than (n-1)×q time units until its next time quantum The performance of the RR algorithm depends heavily on the size of the time quantum At one extreme, if the time quantum is very large, the RR policy is the same as the FCFS policy THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 127




Operating System

If the time quantum is very small (say 1 micro second) the RR approach is called processor sharing

6. Multilevel queue (MLQ) scheduling: A multilevel queue scheduling algorithm partitions the ready queue into several separate queues. The processes are permanently assigned to one queue, generally based on some property of the process, such as memory size, process priority, or process type. Each queue has its own scheduling algorithm. Foreground – RR Back ground- FCFS Highest priority System process Interactive process

Interactive editing process

Batch process

Student process Lowest priority Fig: 4.4.1. Multilevel Queue Scheduling 7. Multilevel Feedback Queue Scheduling: In MLQ (Multilevel Queue) algorithm, processes are permanently assigned to a queue on entry to the system. Processes do not move between queues. But in multilevel feedback queue scheduling, however, it allows a process to move between queues. The idea is to separate processes with different CPU burst characteristics. If a process uses too much CPU time, it will be moved to a lower priority queue. This scheme levels I/O bound and interactive processes in the higher-priority queues. Similarly, a process that waits too long in a lower-priority queue. This form of aging prevents starvation. Queue

0

1

2

Quantum = 8

Quantum = 16

FCFS Fig. 4.4.2.Multilevel feedback queues



Operating System

4.5: Deadlocks A process requests resources: If the resources are not available at that time the process enters a wait state. Waiting process may never again change state, because the resources they have requested are held by other waiting processes this situation is called a deadlock. A deadlock situation can arise if the following four condition hold simultaneously in a system. (i) (ii)

Mutual exclusion: At a time only one process can use the resource. Hold and wait: A process must hold atleast one resource while waiting, to acquire additional requested resources that are currently being held by other processes. (iii) Circular wait: In circular wait, the processes in the system form a circular list or chain where each process in the list is waiting for a resource held by the next process in the list.

R &P R.

P signifies, Resource R held by Process P. R signifies Process P is waiting for Resource Fig 4.5.1

(iv)No preemption: We can’t take resources forcefully from any process. Deadlock can be described more precisely in terms of a directed graph called a system Resource Allocation Graph. This graph consists of a set of vertices V and a set of edges E. The set of vertices V is partitioned into two different types of nodes P={P1, P2 , . . . .Pn} the set consisting of all the active processes in the system and R={ R1, R2 , . . . .Rn} resources. Directed edge Pi Rj : process Pi requested an instance of resources type Rj Rj Pi: an instance of resource type Rj has been allocated to process Pi For a single instance: cycle is necessary and sufficient condition for deadlock. For multiple instances: cycle is necessary but not sufficient condition.



R3

R1

P1

P2

R1

P3 R2

R4

Resources allocation graph with a deadlock Fig. 4.5.2

P2

P3 P1

R2

Operating System

P4 Cycle but no deadlock

The request for R by R may now be granted and all the arcs pointing towards P may be erased. Similarly, the arcs for P can be earased and the reduced graph has no arcs.

Methods for handling deadlocks (i) (ii) (iii) (iv)

Deadlocks prevention Deadlock avoidance Deadlock detection. Recovery from deadlock.

(i) Deadlock prevention: By ensuring that at least one of these conditions can not hold we can prevent the occurrence of deadlock. (a) Mutual exclusion: The mutual- exclusion condition must hold for non sharable resources. (b) Hold and wait : Two possibilities are there: (i) The process request to be granted all resources it needs at once, prior to execution. (ii) It can request any additional resources, however, it must release all the resources that it is currently allocated. Disadvantage: (i) waiting time is high. (ii) resources utilization is low. (iii) Request resources only when the process has none before it can request any additional resources however it must release all the resources that it is currently allocated. (iv) starvation is possible. (c) Circular wait: Each resource is numbered here, and all the processes are forced to request the resources in a numerical order. This protocol ensures that RAG (Resource Allocation Graph) can never have a cycle. (d) No preemption: To prevent deadlock, no preemption ensures that this condition should not hold to make following protocols. (i) if a process holding one resource Ri requests for another resource suppose Rjthen it should first release its previously held resource Ri. (ii) if a process requests for resources and resources are available, allocate them to process. If resources are not available and allocated to other process that is waiting for other resources in such situation, preempt the resources from waiting process and allocate them to requesting. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 130


Operating System

(ii) Deadlock avoidance: It requires additional information about how resource will be requested by a process in its lifetime, by this information, it takes decision whether resources should be allocated to the process or it should wait. Here, all available resources allocated currently and resources will be requested in future kept in account. (1) Safe state: A state is safe if the system can allocate resources to each process (upto its maximum) in some order and still avoid a deadlock. More formally, a system is in a safe state only if there exists a safe sequence. A sequence of a processes < p1, p2 , . . . .pn> is a safe sequences for the current allocation state if, for each Pi, the resources that Pi can still request can be satisfied by the currently available resources plus the resources held by all the Pj with j
safe

Fig 4.5.3 Note:  A safe state is not a deadlock state. Conversely, a deadlock state is an unsafe state.  Not all unsafe states are deadlocks, however an unsafe state may lead to a deadlock. (2) Banker’s Algorithm Assumption for Banker’s Algorithm are as follows: (a) Every process tells in advance, the number of resources of each type it may require. (b) No process asks for more resources than what the system has. (c) At the termination time every process will release resources.

Example :- Consider a system with 5 processes (P0 …..P4) and 4 resources (R0 …..R3).



Operating System

Total available resources are R0 –6 , R1 –4, R2 – 4, R3 – 2. Resource allocation state at some time t0: Process Max R0 , R1, R2, R3 P0 3 2 1 1

Allocation R0 , R1, R2, R3 2 0 1 1

Need R0 , R1, R2, R3 1 2 0 0

P1

1

2 0

2

1 1 0

0

0

1 0

2

P2

1

1 2

0

1

1 0

0

0

0 2

0

P3

3

2 1

0

1

0 1

0

2

2 0

0

P4

2 1 0

1

0 1 0

1

2 0 0

0

Available R0 , R1, R2, R3 1 1 2 0

Q: Consider process P3 requests one instance of R1 and R0 . How do you ensure/know system is in a safe state? (Use Banker’s algorithm). So request [P3]= ( 1,1,0,0) Check if request [P3] Available (yes) Process Max Allocation R0 , R1, R2, R3 R0 , R1, R2, R3 P0 3 2 1 1 2 0 1 1

Need R0 , R1, R2, R3 1 2 0 0

P1

1

2 0

2

1 1 0

0

0

1 0

2

P2

1

1 2

0

1

1 0

0

0

0 2

0

P3

3

2 1

0

2

1 1

0

1

1 0

0

P4

2 1 0

1

0 1 0

1

2 0 0

0

Now,

Available R0 , R1, R2, R3 0 0 2 0

is a safe sequence.

(iii) Deadlock detection: If a system does not employ either a deadlock-prevention or a deadlock-avoidance algorithm, then a deadlock situation may occur. In this environment, the system must provide: (a) An algorithm that examines the state of the system to determine whether a deadlock has occurred. (b) An algorithm to recover from the deadlock. (iv) Recovery from deadlock: (1) Process termination (i) Abort all deadlocked processes. (ii) Abort one process at a time until the deadlock cycle is eliminated.



Operating System

(2) Resource preemption In this scheme recovery from deadlock is done by resource preemption. (i) Select a resource or process for preemption so that cost should be minimum. (ii) If a resource is preempted from a process, then system is roll backed to some safe state. (iii) It is necessary to avoid starvation (i.e infinite preemption of resource from same process).



Operating System

4.6: Memory Management & Virtual Memory Task of memory management system (i) Allocation of memory (ii) Relocation: Convert from logical address to physical address. (iii) Sharing: How to share data (iv) Protection (v) 1. Memory allocation methods The main memory must accommodate both the operating system and the various user processes. (i) Two Partitions: One for the resident operating system, and one for the user processes. Before discussing memory allocation we must discuss the issue of memory protection, protecting the operation system from user process, and protecting user process from one another. We can provide this protection by using a relocation register with limit register. Limit Register or Size Register

CPU

Logical Address

Relocation Register

Yes

<

+

Physical Address

Memory

no

Trap; Addressing Error Fig.4.6.1

Advantages: (a) Simple memory management scheme. (b) Memory is allocated entirely to one job. (c) All the memory is available after job finished. (d) No need of special Hardware support, only this method needs protection of user programs from the operating system.



Operating System

Disadvantages: (a) Poor utilization of memory. (b) Poor utilization of process. (c) Inefficient for multi-programming . (d) User job being limited to the size of available main memory. (ii) Multiple-partition (a) Multiple-partition with fixed size. (b) Multiple-partition with variable-size. (a) Multiple-partition with fixed size: One of the simplest methods for memory allocation is to divide memory into several fixed-sized partitions. Each partition may contain exactly one process. Thus, the degree of multiprogramming is bounded by the number of partitions. In this multiple-partition method, when partition is free, a process is selected from the input queue and is loaded into the free partition. Initially all memory is available for user processes, and is considered as one large block of available memory, a hole. When a process arrives and needs memory, we search for a hole large enough for this process. If we find one, we allocate only as much memory is needed. Keeping the rest available to satisfy the future request. The set of holes is searched to determine which hole is best to allocate. The first-fit, bestfit, and worst- fit strategies are the most common ones used to select a free hole from the set of available holes. First fit : (i) Allocate the first hole that is big enough. Advantage: Searching time is less Disadvantage: More internal fragmentation. For the final request, the first hole not smaller than 5K starts at location 60K. Internal fragmentation:Memory that is internal to a partition but not being used. Best fit: Allocating the smallest hole that is big enough. Advantage: (i) Less internal fragmentation Disadvantage: (i) It takes more searching time (ii) Program can’t grow dynamically Worst fit: Allocate the largest hole. Advantage: (i) program can grow dynamically Disadvantage: (i) Suffer from more internal fragmentation



Operating System

(b) Multiple-partition with variable – size: Partitions created according to program arrival. When a process is allocated space, it is loaded into memory and it can then compete for the CPU. When a process terminates, it releases it’s memory which the operating system may then fill with another process from the input queue. The set of holes is searched to determine which hole is best to be allocated. The first-fit, bestfit and worst-fit strategies are the most common ones used to select a free hole from the set of variable holes. No matter which algorithm is used external fragmentation will be a problem. External fragmentation exists when enough total memory space exists to specify a request but it is not contiguous. Storage is fragmented into a large number of small holes. This fragmentation problem can be serve in the worst case, we could even, have a block of free memory between two processes. If all this memory were in one big free block we might be able to run several more processes. One solution to the problem of external fragmentation is compaction. The goal is to shuffle the memory contents to all free memory together in one large block. (iii) Paging: Physical memory is broken into fixed sized blocks called frames, logical memory is also broken into blocks of the same size called pages. When a process is to be executed, its page is loaded into any available memory frames from the backing store. Frame size= page size Every address generated by the CPU is divided into two parts a page number (p) and a page offset (d). The page number is used as an index into a page table. The page table contains the base address of each page in physical memory. This base address is combined with the page offset to define the physical memory address that is sent to the memory unit.     

Logical address space is divided into equal size pages. Page size is generally power of 2. No. of pages (N) = N= No. of bits for pages (b) = log Page offset depends on page size number of bits for page offset (d) = log (page size) Similarly, physical address space is divided into equal size frames. Frame size = page size



Each process has its own page table. Page table size = No. of entries X size of each entry

Example: logical address is of 32 bits. Physical address space is 128MB. Page size is 8 KB. What is the size of page table in bytes?



logical adress space page si e ( ogical address space conssts of

umber of pages

= umber of frames

Operating System

bits i. e. ,

entries)

physical adress space frame si e (same as page si e)

= =  Each page table entry si e

14 bits ≈ bytes (Assume)

 Page table size = = = 1MB. Index of page table Logical address CPU

f

Physical address f0000 . . . . . . 0000

P

d

offset

f

d

f1111 . . . . . . 1111

f

Physical memory Page table

Fig 4.6.2 The page size (like the frame size)is defined by the hardware . The size of a page is typically a power of 2. When we use a paging scheme, we have no external fragmentation: Any free frame can be allocated to a process that needs it. However, we may have some internal fragmentation. The hardware implementation of the page table can be done in several ways: (i) The page table is implemented as a set of dedicated registers.



Operating System

(ii) If page table is very large then it is kept in main memory and a page table base register (PTBR) points to the page table. (iii) The standard way is to use a special, small, fast-lookup hardware cache, called Translation Looks aside Buffer(TLB). The TLB is associative, high-speed memory. Each entry in TLB consists of two parts a key(or tag) and a value. If the page number is in the TLB, it is known as a TLB hit. If not then TLB miss. The percentage of times that a particular page number is found in the TLB is called the hit ratio. TLB hit ratio h= Now effective memory access time with TLB = h(t1 +t2) +( 1-h) (t1 +2t2) Where t1 –TLB access time t2 – Memory access time. Logical address p CPU

d

frame page number number TLB hit

TLB p TLB miss

______ ______ f ______ ______

f

d

Physical address

Physical memory

Page table

Fig. 4.6.3 Memory protection in a paged environment is accomplished by protection bits that are associated with each frame. Normally, these bits are kept in the page table. One bit can define a page to be read-write or read only. One more bit is generally attached to each entry in the page table: a valid- invalid bit. Valid bit indicates that the associated page is in the process logical address space, and is thus a legal (or valid)page If the bit is set to “invalid” this value indicates that the page is not in the process logical-address space.



Operating System

Hierarchical Paging: For large logical address space we can divide the page table into smaller pieces, division can be two, three or four level paging scheme. Hashed Page Table: A common approach for handling address spaces larger than 32 bits is to use a hashed page table, hash value being the virtual page number. Each entry in the hash table contains a linked list of elements of three fields. (a) The virtual page number (b) The value of the mapped page frame and (c) A pointer to the next element in the linked list Inverted Page Table: An inverted page table has one entry for each real page of memory. Each entry consists of virtual address of the page stored in that real memory location with information about the process that own that page. 

Each virtual address in the system consists of the triple < process-id, page-number, offset> Each inverted page table entry is a pair < process-id, page number>



(iv) Segmentation It is a memory-management scheme that supports user view of memory. A logical address space is a collection of segments. Each segment has a name and a length. The addresses specify both the segments name and the offset within the segment. The user specifics each address by two quantities: a segment name and an offset. For simplicity, segments are numbered and are referred to by a segment number, rather than segment name. s Limit

base

Segment table CPU

S d ddd

yes

< no

Physical memory

Trap; addressing error

Fig. 4.6.4 .



Operating System

Advantages of segmentation (a) Allows growing and shrinking of each segment independently. (b) Provides sharing procedures on data between several processes. (c) Procedure is put in segment , then it is very easy to modify and recompile that procedure without distributing others Disadvantage:External fragmentation (v) Segmentation with paging If segmentation is combined with paging then this combination is very useful in many situations. Virtual Memory: Virtual memory is a technique that allows the execution of process that may not be completely in memory. The instructions being executed must be in physical memory. The first approach to meet this requirement is to place the entire logical address space in physical memory overlays and dynamic loading can help to ease this restriction, but they generally require special precautions and extra work by the programmer. Overlays: The idea of overlays is to keep in memory only those instructions and data that are needed at the given time. When other instructions are needed, they are loaded into space occupied previously by instructions that are no longer needed. Dynamic loading and linking: The concept of dynamic linking is similar to that of dynamic loading. Rather than loading being postponed until execution time, linking is postponed. This feature is usually used with system libraries such as language subroutine libraries. Virtual memory is the separation of user logical memory from physical memory. This separation allows extremely large virtual memory to be provided for programmer when only a smaller physical memory is available. Virtual memory is commonly implemented by demand paging. It can also be implemented in a segmentation system. Several systems provide a paged segmentation scheme, where segments are broken into pages. (i) Demand paging A demand paging system is similar to a paging system with swapping. Processes reside on secondary memory. When we want to execute a process, we swap it into memory. Rather than swapping the entire process into memory, however we use a lazy swapper. A lazy swapper never swaps a page into memory unless that page is needed. A swapper manipulates entire processes whereas a page is concerned with the individual pages of a process. We thus use pages, rather than swapper in connection with demand paging. With this scheme we need some form of hardware support to distinguish between these pages that are in memory and these that are on the disk. The valid invalid bit scheme can be used for this purpose valid means page is legal and in memory. Invalid means page either is not valid or is valid but currently is on the disk.



Operating System

0 1 2 Valid –invalid bit

Frame 0 1 2 3 4 5 6 7

A B C D E F G H

4 0 1 2 3 4 5 6 7

4

v i v i i v i i

6

9

A

5 6

C

7 8 9

6

v

10

9

v

11

Page table Logical memory

3

A B a C Da E F

F

12 13 14 Physical memory Fig. 4.6.5 Pure demand paging: Never brings a page into memory until it is required. Effective access time for a demand paged memory is: Effective access time = (1-P)

ma + P

page fault time

Where, P = probability of a page fault. ma = memory – access time Page Replacement algorithm When a page fault occurs , the operating system has to choose a page to remove from memory to make space for the page that has to be brought in. If the page to be removed has been modified while in memory, it must be re- written to the disk to bring the disk copy upto date. We can reduce this over-head by using a modify bit (or dirty bit). If the bit is set, we know that the page has been modified since it was read in from the disk. If the modify bit is not set, however, the page has not been modified since it was read into memory. Therefore, if the copy of the page on the disk has not been overwritten then we can avoid writing the memory page to the disk. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 141


Operating System

Below we will describe some of the most important algorithms: (a) FIFO page replacement algorithm: It is the simplest page replacement-algorithm. The idea behind FIFO (first-in first-out) is that when page is brought into memory, then the oldest page will be replaced by new page. FIFO has problem known as Belady’s anomaly Belady’s anomaly: For some page replacement algorithms the page fault rate may increase as the number of allocated frames increases. Reference 3 2 1 0 3 2 4 3 2 1 0 string Newest page 3 2 1 0 3 2 4 4 4 1 0 3 2 1 0 3 2 2 2 4 1 Oldest page 3 2 1 0 3 3 3 2 4 Total page 1 2 3 4 5 6 7 7 7 8 9 faults Reference string Newest page

Oldest page Total page faults

4 0 1 4 9

3

2

1

0

3

2

4

3

2

1

0

4

3

2 3

1 2 3

1

2

3

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

4 0 1 2 5

3 4 0 1 6

2 3 4 0 7

1 2 3 4 8

0 1 2 3 9

4 0 1 2 10

(b) Optimal page replacement Replace the page that will not be used for the longest period of time. Use of this page replacement algorithm guarantees the lowest possible page-fault rate for fixed number of frames. Unfortunately, the optimal page-replacement algorithm is difficult to implement, because it requires future knowledge of the reference string. Reference string

3 3

2 2 3

Total page faults

1

2

1 1 2 3 3

0 0 2 3 4

3 0 2 3 4

2 0 2 3 4

4 4 2 3 5

3 4 2 3 5

2 4 2 3 5

(c) LRU page replacement The page that has not been used for the longest time period will be replaced. Reference string 3 2 1 0 3 2 4 3 2 Most recently used 3 2 1 0 3 2 4 3 2 3 2 1 0 3 2 4 3 Least recently used 3 2 1 0 3 2 4 Total page faults 1 2 3 4 5 6 7 7 7

1 4 1 3 6

1 1 2 3 8

0 4 1 0 7

0 0 1 2 9

4 4 1 0 7

4 4 0 1 10



Operating System

LRU approximation algorithms (i) Using reference bit  Associate a bit with each page, initially 0.  When page is referenced, update it to 1.  Replace the page whose reference bit is 0. (if one exists ) (ii)     

Second chance or clock algorithm Variant of FIFO algo It needs one reference bit. Associate a bit with each page, initially 0. If it is 0, replace it. If it is 1, give it a 2nd chance and move onto next FIFO page, reset reference bit to 0 & reset arrival time to current time.  A page given a second chance will not be replaced until all other pages have been replaced or given second chance.

(iii) Enhanced second chance algorithm  Use 2-bits (reference bit & dirty bit).  While replacing consider four cases. (0,0) – neither recently wed nor modified (best page to replace) (0,1) – not recently used but modified. (1,0) – recently used, but clean (1,1) – recently used and modified. (iv)Counting – based page replacement algorithm (i) Least frequently used (LFU) – page with smallest count is replaced. (ii) Most frequently used (MFU) – page with the largest count is replaced . Thrashing The condition in which, process spends it’s more time in paging than in execution called thrashing. In order to increase CPU utilization, degree of multiprogramming is increased but if by increasing degree of multi programming, CPU utilization is decreased then such a condition is called thrashing.

CPU Utilization

thrashing

Degree of multiprogramming Fig. 4.6.6 . THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 143


Operating System

We can limit the effects of thrashing by using a local replacement algorithm (or priority replacement algorithm). With local replacement, if one process starts thrashing, it cannot steal frames from another process and cause the latter to thrash also. Pages are replaced with regard to the process of which they are part Working –Set model: The working-set model is based on the assumption of locality. This model uses a parameter , to define the working-set window. The idea is to examine the most recent page reference. The set of pages in the most recent

page references is the working set.

If a page is in active use, it will be in the working set. If it is no longer being used, it will drop from the working set time units after its last reference. Thus, the working set is an approximation of the program’s locality. Example

= 10 memory reference

Reference string ……… 6 1 5 7 7 7 7 5 1 6 6 4 1 2 3 4 4 4 3 4 3 4 4 4 1 3 2 3 4 4 3 4 4 4

WS(t1 ) = { 1 , 2, 5 , 6, 7}

WS(t2 ) = { 3 , 4}

Fig. 4.6.7 . The accuracy of the working set depends on the selection of encompass the entire locality. If is too large, it may overlap several localities. In the extreme, if the set of pages touched during the process execution.

. If

is too small, it will not

is infinite, the working set is



Operating System

4.7: File System The file system consists of two distinct parts: a collection of files, each storing related data, and a directory structure, which organizes and provides information about all the files in the system. Some files systems have a third part, partitions, which are used to separate physically or logically large collection of directories. (1) Access Methods: Files store information. When it is used, this information must be accessed and read into computer memory. There must be some methods that provide reading from or writing to a file. Such methods are known as access methods. (i) Sequential Access Methods: This is a simple access method where information is processed in a sequential order. Here, the basic operations on a file are reads and writes. A read operation reads the next portion of the file and automatically advances a file pointer, which tracks the I/O location. Similarly, a write appends to the end of the file and advances to the end of the newly written material (the new end of file). Sequential access is based on a tape model of a file. Current position End

Beginning

Rewind

Read or write Fig 4.7.1

(ii) Direct /Random / Relative Access method: A direct access files are made up of collection of records, each of which is identified by a number called record number. In direct access file, one can access any record at any time for reading or writing. Database are often of this type. (iii)Index Sequential Access Method (ISAM): This method is developed by slight modification in the direct access method. The index contains pointer to the various blocks or records. To find an entry in the file, we first search the index and find the pointer of the record to access file entry directly. (2) Directory Structure: One-level Directory or Single-level-Directory: If system supports only a single directory and all files are stored in the same directory, then it is known as one-level directory.



Operating System

It is simplest directory structure but it has own limitations. One is, if number of files increases, then it is difficult to remember names of files any-where conflict can occur in file names. Each file must have unique name. Root Directory

A B

C

D

E

files

Fig. 4.7.2 Two level Directory: In this type of directory system, a different directory is given to a different user. Each of such directories is related to a user which has a group of files. Tree-Structured Directory: This is most common directory structure in use. In this type of structure, there is a root directory and all intermediate directories are known as subdirectories. Each subdirectory contains information of files and subdirectories within it. Acyclic- Graph Directory Structure: This type of directory structure allows sharing files or directories among users. This is graph like structure where no cycle exists. The problems with acyclic-graph directory system are as follows: First, a file may have more than one absolute address. Second is the problem of deletion. If a user wanted to delete shared file, then after deletion other user will point to garbage value that contain dangling pointers. General-Graph Directory Structure: The directory system supports graph like structure. In graph, any node can connect to any other node. Similarly, in graph-directory system, directories and files can be connected in any form. The prerequisite of acyclic-graph directory system is that there should not be any cycle. But general-graph directory system supports cycles also. (3) Allocation methods: An allocation method refers to show disk blocks are allocated for files. (i) Contiguous Allocation method: It stores each file as a contiguous blocks of data in the disk. Thus if full size is n KB and the block size on disk is 1 KB, then file utilized n consecutive blocks of the disk in contiguous allocation as shown below:



Operating System

Directory 0

1

2

3

4

5

6

7

8

9

10

11

File

Start

Length

Count

0

2

New

8

3

List

5

2

Fig 4.7.3 Contiguous Allocation of disk space Advantages: a) It is simplest allocation scheme and can be implemented easily. Only file size and first starting block number are necessary to remember entire file. b) Since entire file can be read in a single operation, so we can say that it gives excellent performance. Disadvantages: a) The size of file up to maximum limit must be known at the creation time, otherwise this policy cannot be implemented. b) It is the cause of fragmentation. So some disk space is wasted due to fragmentation. Compaction is required to remove fragmentation. c) Memory is not utilized properly. Therefore, more memory is wasted. d) This algorithm suffer from the problem of external fragmentation. (ii) Linked Allocation method: To resolve problems related with contiguous allocation, linked allocation is used. In linked allocation scheme, each file is stored in memory blocks in the form of a linked list. Each block has two parts-one is to contain pointer to the next block and second is to store data. In directory entry only two pointers are required for a file: first starting block address and second end block address. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 147


Operating System

Advantage: a) There is no wastage of memory. Every block can be used. b) There is no external fragmentation. Only internal fragmentation is possible in last block. c) Directory needs only two values to remember file i.e. starting block address and end block address Disadvantages: a) A block can be accessed directly. Only sequential reading is possible, since every block contains address of next block of file. b) Pointer uses same disk space for storing. This space is wasted totally, since it is not a part of information. If data part of file is larger, then space used for pointer consumes more space than data itself. c) Access time, seek time, and latency time are higher than contiguous allocation. (iii)Index Allocation/Linked Allocation Using Index: In this allocation scheme, a table is maintained that contains pointer to next block of disk. This table is known as index table. This index table is prepared to speed up access to file. This is also known as FAT (File Allocation Table). Index allocation support direct access, without suffering from external fragmentation. Advantages: a) Entire block is available for data. b) Random access is much easier. c) Like linked list allocation, it requires only the starting block number to find entire file. d) Index table is used to find next block entry. Since index table always remains in memory, so no disk reference is necessary for each access. Disadvantages: Extra space is required to keep index table in memory all the time. Free space management a) Bit vector or Bit map: Each block is represented by 1 bit. If the block is free, the bit is 1, if block is allocated, the bit is 0. Advantage: It is simple and efficient in finding the first free block, or n consecutive free blocks on the disk. Disadvantages: Difficult to maintain 0,1 bits. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 148


Operating System

b) Linked free space management: It just store free blocks entry, remaining free block connected to each other through linked list. Advantages: (i)Less size of information. Disadvantages: I. II.

Direct access not possible. If one will corrupt then all will corrupt.

c) Index free space management: OS maintain block which maintain free blocks entry. This block called indexed block. If it is not possible into one block then we will do two level indexing.



Operating System

4.8: I/O Systems The role of the operating system in computer I/O is to manage and control I/O operations and I/O devices. 1.

Kernel I/O subsystem: Kernels provide many services related to I/O. Several services scheduling, buffering, caching, spooling, reservation and error handling are provided by the kernel I/O subsystem and build on the hardware and device driver infrastructure.

(i) I/O scheduling: To schedule a set of I/O requests means to determine a good order in which to execute them. The order in which application issue system calls rarely is the best choice. Scheduling can improve overall system performance, can share device access fairly among processes and can reduce the average waiting time for I/O to complete. (ii) Buffering: A Buffer is a memory area that stores data while they are transferred between two devices or between a device and an application. Buffering is done for three reasons: One reason is to cope with a speed mismatch between the producer and consumer of a data stream. A second use of buffering is to adapt between devices that have different data transfer sizes. Such disparities are especially common in computer networking where buffers are used widely for fragmentation and reassembling of message. A third use of buffering is to support copy semantics for application I/O. (iii)Caching: A cache is a region of fast memory that holds copies of data. Access to the cached copy is more efficient than access to the original. Caching and buffering are distinct functions, but sometimes a region of memory can be used for both purposes. (iv) Spooling and device Reservation: A spool is a buffer that holds output for a device, such as a printer, that cannot accept interleaved data streams. Although a printer can serve only one job at a time, several application may wish to print their output concurrently, without having their output mixed together. The operating system solves this problem by intercepting all output to the printer. Each application’s output is spooled to a separate disk file. When an application finishes printing the spooling system queues the corresponding spool file for output to the printer.



Operating System

2. Disk structure Modern disk drives are addressed as large one dimensional arrays of logical blocks, where the logical block is the smallest unit of transfer. The size of a logical block is usually 512 bytes, although some disks can be low-level formatted to choose a different logical block size, such as 1024 bytes. The one dimensional array of logical blocks is mapped onto the sectors of the disk sequentially. Sector 0 is the first sector of the first track an the outer most cylinder. The number of sectors per track is not a constant on same drives. Constant linear velocity (CLV): The density of bits per track is uniform. The greater its length, so the more sectors it can hold. Alternatively, the disk rotation speed can stay constant, and density of bits decreases from inner track to outer track to keep the data rate constant. This method is used in hard disks and is known as constant angular velocity (CAV). Disk Scheduling The Seek time is the time for the disk arm to move the heads to the cylinder containing the desired sector. The Rotational latency is the additional time waiting for the disk to rotate the desired sector to the disk head. The Disk bandwidth is the total number of bytes transferred divided by the total time between the first request for service and the completion of the last transfer. We can improve both the access time and the bandwidth by scheduling the servicing of disk I/O requests in good order. (i) FCFS : ( First-Came First-Serve) Scheduling This is the simplest disk scheduling. As its name FCFS, the request for block that comes first is serviced first. For example ,the request for I/O to blocks are on following cylinder 4, 34, 10, 7, 19, 73, 2, 15, 6, 20 that order. If the disk head is initially at cylinder 50, 0

2

4

6

7

10 15

19

20

34

50

73 100



Operating System

Total head movements = (50 – 4) + (34 – 4) + (34 – 10) + (10 – 7) + (19 – 7) + (73 – 19) + (73 – 2) + (15 – 2) + (15 – 6) + (20 – 6) = 276 (ii) SSTF (Shortest-Seek-Time-First) scheduling The SSTF algorithms select the request with the minimum seek time from the current head position For above example 0

2

4

10 15

7

6

19

20

34

50

73 100

Total head movements = 119 This strategy reduces the total head movement to 208 cylinders. So, it is more improved form of scheduling than FCFS. (iii) SCAN ALGORITHM In the SCAN algorithm, the disk arm starts at one end of the disk, and moves towards the other end, servicing requests until it gets to other end of the disk. At the other end, the direction of head movement is reversed, and servicing continues. The head continuously scans back and forth across the disk. We again use our example Before applying SCAN to schedule the requests on cylinders 4, 34, 10, 7, 19, 73, 2, 15, 6, 20 we need to know the direction of head movement, in addition to the head’s current position (50). If the disk arm is moving toward 100 then 0

2

4

6

7

10 15

19

20

34

50

73 100

(iv) C-SCAN Scheduling Circular scan(C-SCAN) scheduling is a variant of SCAN designed to provide a more uniform wait time. Like SCAN, C-SCAN moves the head from one end of the disk to other, servicing request along the way. When the head reaches the other end, however, it immediately returns to the beginning of the disk, without servicing any requests on the return trip. (v) LOOK and C-LOOK Scheduling As we have described, both SCAN and C-SCAN move the disk arm across the full width of the disk. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 152


Operating System

In practice, neither algorithm is implemented this way. More commonly, the arm goes only as far as the final request in each direction. Then, it reverses direction immediately, without going all the way to the end of the disk. These versions of SCAN and C-SCAN are called LOOK and C-LOOK scheduling, because they look for a request before continuing to move in a given direction. Disk Management (i) Disk Formatting: Before a disk can store data, it must be divided into sectors that the disk controller can read and write. This process is called low level formatting (or physical formatting). Low-level formatting fills the disk with a special data structure for each sector. The data structure for a sector typically consists of a header, a data area (usually 512 bytes in size), and a trailer. Logical formatting (or creation of a file system): In this step, the operating system stores the initial file-system data onto the disk. These data structures may include maps of free and allocated space (a FAT or inodes) and an initial empty directory. Boot Block It contains code required to boot the operating system. For example, MS DOS uses one 512-byte block for its boot program. Sector 0

Boot block

Sector 1

FAT Root directory Data blocks sub Su (directories)

Fig. 4.8.4 FAT: File allocation table, which stores the position of each file in the directory tree. Root directory: Every file within the directory hierarchy can be specified by giving its path name from the top of the directory hierarchy, the root directory. Such absolute path names consist of the list of directories that must be traversed from the root directory to get the file, with slashes separating the components. The leading slash indicates that the path is absolute that is, starting at the root directory.



Operating System

4.9: Protection and Security Protection refers to a mechanism for controlling the access of programs, processes, or users to the resources defined by a computer system. Security: The information stored in the system (both data and code), as well as the physical resources of the computer system, need to be protected from unauthorized access, malicious destruction or alteration, and accidental introduction of inconsistency. We need to provide protection for several reasons. The most obvious is the need to prevent mischievous, intentional violation of an access restriction by a user. Protection can improve reliability by detecting latent errors at the interfaces between component subsystems. (1) Domain of Protection (i) Domain Structure A process operates within a protection domain, which specifies the resources that the process may access. Each domain defines a set of objects and the types of operations that may be invoked on each object. The ability to execute an operation on an object is an access right. A domain is a collection of access rights, each of which is an ordered pair For example, if domain D has the access right , then a process executing in domain D can both read and write file F; it cannot, however, perform any other operation on that object. Domain may share access rights. For example

D1

D2

Fig. 4.9.1

The access right is shared by both D1 and D2, implying that a process executing in either of these two domain can print object O4. Domain may be a user, process and procedure.



Operating System

In the UNIX operating System, a domain is associated with the use. In the MULTICS system, the protection domains are organized hierarchically into a ring structure. Each ring corresponds to a single domain. Ring 0 Ring 1

Ring n-1

Fig. 4.9.2 Let Di and Dj be any two domain rings. If j


Operating System

User Authentication Generally, authentication is based on user possession (a key or card) , user knowledge(a user identifier and password), and/or a user attribute(finger print, retina pattern or signature). (A) Passwords: If the user supplied password, matches the password stored in the system, the system assumes that the user is legitimate. (B) Encrypted Passwords: It is difficult to keep the password secret within the computer. UNIX system uses encryption to avoid the necessity of keeping its password list secret. (C) One Time Password: When a session begins, the system randomly selects and presents one part of a password pair; the user must supply the other part. (D) Biometrics: Palm or hand readers are common to secure physical access. Finger readers have become accurate and cost-effective, and should become more common in the future. These devices read your finger’s ridge pattern and convert them into a sequence of numbers. Program Threats: When a program written by one user may be used by another user, misuse and unexpected behavior may ensure. Some common methods by which such behavior may occur are: Trojan horse, trap doors, and stack and buffer overflow. (i) Trojan Horse: Many systems have mechanisms for allowing programs written by users to be executed by other users. If these programs are executed in a domain that provides the access rights of the executing users, the other users may misuse these rights. A code segment that misuses its environment is called a Trojan Horse. (ii) Trap Door: The designer of a program or system might leave a hole in the software that can be only used by them. This type of security breach (or trap door) was shown in the movie War Games. (iii) Stack and Buffer overflow: Exploits a bug in a program System Threats System threats create a situation in which operating-system resources and user files are misused. (i) Worms A worm is a process that uses the spawn mechanism to clobber system performance. The worm spawns copies of itself, using up system resources and perhaps locking out system used by all other processes.



Operating System

(ii) Viruses Viruses are designed to spread into other programs and can wreak havoc in a system, including modifying or destroying files and causing system crashes and program malfunctions. Denial of service It does not involve gaining information or stealing resources, but rather disabling legitimate use of a system or facility. This check would be expensive and needs to be performed every time the object is accessed.



DBMS

Part 5: Database Management System Part 5.1: ER Diagrams Points to emphasize     

How to use Entity-Relationship (ER) modeling in database design. The basic concepts associated with the Entity-Relationship (ER) model. A diagrammatic technique for displaying an ER model using the Unified Modeling Language (UML). How to identify and resolve problems with ER models called connection traps. How to build an ER model from a requirements specification.

What To Model? Static Information Data -- Entities Associations -- Relationships among entities Dynamic Information Processes -- Operations/transactions Integrity constraints -- Business rules/regulations and data meanings What is Data Model? A collection of tools for describing:- data, data relationships, data semantics, data constraints Data Model:- A data model is a collection of concepts that can be used to describe the structure of database. Schema:- The description of a database is called the database schema. System model tools: Data Flow Diagram (DFD) Hierarchical Input Process and Output (HIPO) State Transition Diagrams (STD) Entity Relationship (ER) Diagrams Entity-Relationship Model (ER Model) A data model in which information stored in the database is viewed as sets of entities and sets of relationships among entities and it is diagram-based representation of domain knowledge, data properties etc...., but it is more intuitive and less mechanical. Entity – Relationship is a popular high-level conceptual data model.



DBMS

Components of E R model a. Entity

b. Relationship

c. Attributes

Entity:- The basic object that the ER model represents is an entity, which is a “thing” in the real world with an independent existence. Entity set:-A set of entities of the same type. • Entity sets need not be disjoint. Example: A person entity could be in both the customer and employee sets Types of Entities • Entities with Physical existence Example: Student, Customer, Book etc • Entities with Conceptual existence Example: Sale, University course etc Relationship:An association among two or more Entities. Example:The Relationship between a Faculty and Student i.e. Faculty take course for Student Relationship Set A set of Relationships of the same type Attribute:The particular properties of entity that describe it Example: A student entity might have attributes such as: roll number, name, age, address etc • As all entities in an entity set have the same attributes, entity sets also have attributes - the attributes of the contained entities. The value of the attribute can be different for each entity in the set. Types of Attributes:i) Composite attributes ii) Simple attributes iii) Single-valued attributes iv) Multivalued attributes v) Stored attributes vi) Derived attributes Relationship Degree:The number of entity types associated with that relationship (fig. 5.1.1).


Quick Refresher Guide Unary

DBMS Binary

Employee

Work

Employee

Department

Supervise n-ary

Ternary

project

Part

Supply

Supplier

Fig. 5.1.1

Types of Relationships (fig. 5.1.2):Connectivity one – to one

1 Department

is managed by

1 Employee

The one-to-one relationship has the cardinality or degree of one and only one in both direction. 1 one – to - many

N

Department

has

Employee

The one tomany or-to-one relationship has a cardinality in one direction of one or more and in the other direction of one and only one. M many – to - many

Employee

N works-on

The many –to-many relationship has a cardinality in both directions

Fig. 5.1.2

Project

task-assignment

start-date



DBMS

Multiplicity: Multiplicity constrains the way that entities are related - it is a representation of the policies (or business rules) established by the user or enterprise. Multiplicity actually consists of two separate constraints. Cardinality: Cardinality describes the maximum number of possible relationship occurrences for an entity participating in a given relationship type i.e. How many relationship instances is an entity permitted to be linked to. Participation: Participation determines whether all or only some entity occurrences participate in a relationship i.e. how is an entity linked to the relationship. Total participation (indicated by double line): Every entity in the entity set participates in atleast one relationship in the relationship set. Partial participation: Some entities may not participate in any relationship in the relationship set Note: Cardinality limits can also express participation constraints. Weak and Strong Entity Set 

A Strong Entity set has a primary key. All tuples in the set are distinguishable by that key.



A Weak entity set has no primary key unless attributes of the strong entity seton which it depends are included. Tuples in a weak entity set are partitioned according to their relationship with tuples in a strong entity set. Tuples within each partition are distinguishable by a discriminator, which is a set of attributes. The discriminator (or partial key) of a weak entity set is the set of attributes that distinguishes among all the entities of a weak entity set. Weak Entity set is represented by double rectangles. Underline the discriminator of a weak entity set with a dashed line. The primary key of the Strong entity set is not explicitly stored with the Weak entity set, since it is implicit in the identifying relationship.

     

Reasons to have weak entities    

We want to avoid the data duplication and consequent possible inconsistencies caused by duplicating the key of the strong entity. Weak entities reflect the logical structure of an entity being dependent on another entity. Weak entities can be deleted automatically when their strong entity is deleted. Weak entities can be stored physically with their strong entities.

Existence Dependencies If the existence of entity x depends on the existence of entity y, then x is said to be existence dependent on y. y is a dominant entity, x is a subordinate entity. If y entity is deleted, then all its associated x entities must be deleted.



DBMS

ER diagram symbols (fig. 5.1.3) Entity Set

E

E

Weak Entity Set

R

Relationship Set

R

A

Identifying Relationship Set for Work Entity Set

one-to one Relationship

many to one Relationship

Attribute

A

Multivated Attribute

A

Derived Attribute

R

Primary Key

E

A

Total participation of Entity Set in Relationship Discriminating Attribute of Weak Entity Set

E

Entity set E with attributes A1, A2, A3 and primary key A1

many to many Relationship

A

A1 A2 A3

*

*

R

R

1

1

R

1

R

R

* R

Fig. 5.1.3



DBMS

Keys   

A super key of an entity set is a set of one or more attributes whose values uniquely determine each entity. A candidate key of an entity set is a minimal super key. Although several candidate keys may exist, one of the candidate keys is selected to be the primary key.

Aggregation Aggregation is an abstraction through which relationships are treated as higher-level entities. Thus the relationship between entities A and B is treated as if it were an entity C. Utility of E-R Model   

It maps well to the relational model. The constructs used in the ER model can easily be transformed into relational tables. It is simple and easy to understand with a minimum of training. Therefore, the model can be used by the database designer to communicate the design to the end user. In addition, the model can be used as a design plan by the database developer to implement a data model in specific database management software.

E-R Design Decisions      

The use of an attribute or entity set to represent an object. Whether a real-world concept is best expressed by an entity set or a relationship set. The use of a ternary relationship versus a pair of binary relationships. The use of a strong or weak entity set. The use of specialization/generalization - contributes to modularity in the design. The use of aggregation - can treat the aggregate entity set as a single unit without concern for the details of its internal structure

Tools to create E-R Diagrams Several Computer Aided Software Engineering (CASE) tools exist to help in creating E-R Diagrams and the resulting physical database elements. Products include: 

IEW, IEF, DEFT, ER-WIN, Visio



DBMS

Part 5.2: Functional Dependencies & Normalization Points to emphasize:  Various design steps  The problems associated with redundant data.  The identification of various types of update anomalies such as insertion, deletion, and modification anomalies.  How to recognize the appropriateness or quality of the design of relations.  The concept of functional dependency, the main tool for measuring the appropriateness of attribute groupings in relations.  How functional dependencies can be used to group attributes into relations that are in a known normal form.  The purpose of normalization.  How to undertake the process of normalization.  How to identify the most commonly used normal forms, namely first (1NF), second (2NF), and third (3NF) normal forms, and Boyce-Codd normal form (BCNF).  How to identify fourth (4NF) and fifth (5NF) normal forms. Database Application life cycle:i) System definition ii) Database design iii) Database implementation iv) Loading or data conversion v) Application conversion vi) Testing and validation vii) Operation viii) Monitoring and maintenance 1. Requirement specification Entities, attributes, associations, constraints, security, performance, various types of retrieval and manipulation requests & their frequencies. 2. Semantic modeling   

Structural modeling: Model data entities, their attributes and association among entities. Constraint specification: Model security and integrity-constraints. Operations: Specify Meaningful operations associated with entities/objects and their associations.

3. DBMS specific logical design  

Use the data model (relational, object oriented etc) of a DBMS to specify the structural properties and constraints. Determine the structural properties, constrains and operations which are not captured by the data model and are to be implemented in application programs



DBMS

4. DBMS specific physical design  

Determining the access methods, data clustering and partitioning strategies Data distribution and replication in distributed systems.

A Relation has the following properties:        

Has a name that is distinct from all other relation names in the relational schema. Each cell contains exactly one atomic (single) value. Each attribute has a distinct name. The values of an attribute are all from the same domain. Each tuple is distinct; there are no duplicate tuples. The order of attributes has no significance. The order of tuples has no significance, theoretically. (however, in practice, the order may affect the efficiency of accessing tuples.) A relation is defined as a set of tuples

Placing of proper attribute into a relation The problem in relational model is to place the proper attributes into a relation and of grouping attributes in different relations. This problem can be solved by the following two methods: 1. By intuition and semantic. But problems are... a. No semantic distinction between the two attributes, Ex. T (a, b) b. Duplication of attribute values c. Semantic ambiguity d. Storage anomalies 2. Relational normalization using the notion of functional dependency. Three design goals for relational databases 1. Avoid redundancies and resulting update, insertion, and deletion anomalies, by decomposing schemes as necessary. 2. Ensure that all decompositions are lossless-join. 3. Ensure that all decompositions are dependency-preserving. The types of update anomalies that may occur on a relation that has redundant data. A major aim of relational database design is to group attributes into relations so as to minimize information redundancy and thereby reduce the file storage space required by the base relations. Another serious difficulty using relations that have redundant information is the problem of update anomalies. These can be classified as insertion, deletion, or modification anomalies. The process of Normalization can reduce all these anomalies. Types of anomalies Insertion anomalies: An independent piece of information cannot be recorded into a relation unless an irrelevant information must be inserted together at the same time. Update anomalies: The update of a piece of information must occur at multiple locations, not required by the referential integrity rule.



DBMS

Deletion anomalies: The deletion of a piece of information unintentionally removes other information. Purpose of Normalizing data: Normalization can be achieved with the help of keys and functional dependencies(FD).  

Normalization process, as first proposed by codd (1972). Initially, codd proposed three normal forms which he called first, second and third normal form

Normalization of data can be looked upon as a process of analyzing the given relation schemas based on their FDs and primary keys to achieve the desirable properties of (1) minimizing redundany and (2) minimizing the insertion, detection, and update anomalies. The Concept of Functional Dependency:  Functional dependency describes the relationship between attributes in a relation.  If A and B are attributes of relation R, B is functionally dependent on A (denoted by A B), if each value of A in R is associated with exactly one value of B in R.  Given a table T with at least two attributes A and B, we say that A B (A functionally determines B, or B is functionally dependent on A)if it is the intent of the designer that for any set of rows that might exist in the table, two rows in T cannot agree on A and disagree on B. More formally, given two rows r and r in T, if r (A)= r (A) then r (B) = r (B). Inference Rules for Functional Dependencies The following six rules IR1 through IR6 are well-known inference rules for functional dependencies.        

IR1 (reflexive rule): if X ⊇ Y, then X Y IR2 (augmentation rule): {X Y}|= X Z YZ IR3 (transitive rule): {X Y, Y Z} |= X Z. IR4 (decomposition or projective rule) : {X YZ} |= X Y IR5 (union, or additive rule): {X Y, X Z} |= X YZ IR6 (pseudotransitive rule): {X Y, WY Z} |= WX Z A functional dependence X Y is trivial if X ⊇ Y; otherwise it is nontrivial. Inference rules IR1 through IR3 are known as Armstrong’s inference rules.

Closure of set of attributes Suppose {A , A , A A } is a set of attributes and S is the set of FD’s. The closure of {A , A , A A } under the FD’s in S is the set of attributes B such that every relation satisfies all the FD’s in set S, also satisfies A , A A →B Determining X

, the closure of X under F:-

X;

repeat old X

X ;

for each functional dependency Y→Z in F do ifX ⊇ Y then X : Until (X

X

Z

old X )



DBMS

Equivalence of sets of functional dependencies Two sets of functional dependencies E and F are equivalent if E+ = F+. Hence equivalence means that every FD in E can be inferred from F, and every FD in F can be inferred from E, that is, E is equivalent to F if both the conditions E covers F and F covers E hold. Minimal sets of functional dependencies A set of functional dependencies F is minimal if it satisfies the following conditions 1. Every dependency in F has a single attribute for its right hand side. 2. We cannot replace any dependency X→ A in F with a dependency Y→A, where Y is a proper subset of X, and still have a set of dependencies that is equivalent to F. 3. We cannot remove any dependency from F and still have a set of dependencies that is equivalent to F We can thick of a minimal set of dependencies as being a set of dependencies in a standard or canonical form & with no redundancies. Algorithm: Finding a Minimal Cover F for a Set of Functional Dependencies E 1) Set F: = E 2) Replace each functional dependency X {A , A A } in F by the n functional dependencies X A ,X A .X A 3) For each functional dependency X A in F for each attribute B that is an element of X if if {{F (X A)} {(X {B}) A}}is equivalent to F, then replace X A with (X {B}) A in F 4) For each remaining functional dependency X A in F if {F {X A}} is equivalent to F, then remove X A from F Normalization and database Design  

E – R Diagram provides macro view , determines entities Normalization provide micro view of entities, focuses on characteristics of specific entities, may yield additional entities

Essence of Normalization When a relation has more than one theme, break it in to two (or more) relations, each having a single theme. Functional dependency and the process of normalization Three normal forms were initially proposed, which are called first (1NF), second (2NF) and third (3NF) normal form. Subsequently, a stronger definition of third normal form was introduced and is a referred to as Boyce- Codd normal form (BCNF) .All these normal form are based on the functional dependencies among the attribute of a relation.



DBMS

Steps in Normalization UNNORAMALIZED RELATION (non NF) Remove repeating group NORMALIZED RELATION (1NF) Remove partial dependencies SECOND NORMAL FORM (2NF) Remove transitive dependencies THIRD NORMAL FORM (3 NF) Remove overlapping candidate keys BOYCE –CODD NORMAL FORM (BCNF) Remove multi-valued dependencies FOURTH NORMAL FORM (4NF) Remove “non-implied “joindependencies FIFTH NORMAL FORM (5NF)

DOMAIN KEY NORMAL FORM (DKNF)

Every constraint on the relation is as consequence of the definition of keys and domain (subsumes 4NF and 5NF).

First Normal Form:  A relation in 1NF if the values in the domain of each attribute of the relation are atomic.  It also disallows multivalued attributes that are themselves composite. They are called nested relations. Second Normal Form:  It is based on full functional dependency  A relation schema R is in 2NF if it is in 1NF and every non-pirme attribute A in R is fully functional dependent on any key of R.



DBMS

Third Normal Form: A relation R is in 3NF, if it is in 2NF and whenever a non trivial FDX → A holds in R either (i) X is a super key of R or (ii) A is a prime attribute of R must be satisfied. i.e., It disallows the transitive dependency i.e., relation should not have a FD in which a nonprime attribute depends on a non prime attribute. Boyce Codd Normal Form:   

A relation scheme R is in BCNF if it is in 3NF and whenever a nontrivial functional dependency X → A holds then X is a super key of R and it does not allow A to be prime. This is more restrictive than 3NF While decomposing relation to make them in BCNF we may loose some dependencies i.e. BCNF does not gurqntee the dependency preservation property.

Multi-Valued Dependency (MVD) represents a dependency between attributes (for example, A, B, and C) in a relation, such that for each value of A there is a set of values for B and a set of values for C. However, the set of values for B and C are independent of each other.  A multi-valued dependency can be defined as being trivial or nontrivial. A MVD A -- >> B in relation R is defined as being trivialif B is a subset of A or (b) A B R.  A MVD is defined as being nontrivial if neither (a) nor (b) are satisfied. A trivial MVD does not specify a constraint on a relation, while a nontrivial MVD does specify a constraint. Fourth Normal Form:  

A relation schema R is in 4NF whit respect to a set of dependencies F (including both functional dependency and multivalued dependency) if it is in BCNF and for every nontrivial multivalued dependency X→Y in F , X is a super ey for R Every relation in 4NF is in BCNF.

Join dependencies A join dependency (JD), denoted by JD(R , R . R ), specified on relation schema R, specifies a constraint on the states r of R. The constraint states that every legal state r of R should have a non-additive (lossless) join decomposition into R , R . R that is for every such r we have (

(r),

(r),

.

(r))

r.

A join dependency JD (R , R . R ), specified on relation schema R, is a trivial JD if one of the relation schemas R in JD (R , R . R ) is equal to R. Fifth Normal Form A relation schema R is in 5NF with respect to a set F of functional multivalued & join dependencies if, for every non-trivial dependency JD(R , R . R ),isF every R is a super key of R



DBMS

Properties of Normal Forms of Their Decompositions Property

3NF

BCNF

4NF

Eliminates redundancy due to FD’s

Yes

Yes

Yes

Eliminates redundancy due to MVD’s

No

No

Yes

Preserve FD’s

Yes

Maybe Maybe



DBMS

Part 5.3: Relational Algebra &Relational Calculus

The Relational Algebra The relational algebra is a procedural query language. It consist of a set of operations that take on one or two relations as input and produce a new relation as their result. The fundamental operations in the relational algebra are select, project, union, set difference, cartesian product and rename. In addition to the fundamental operations, there are several other operations – namely, set intersection, natural join, division and assignment.

Relational Operators We call relational operators as a set of operators that allow us to manipulate the tables of the database. Relational operators are said to satisfy the closure property, since they operate on relations to produce new relations. When a relational operator is used to manipulate a relation, we say that the operator is “applied” to the relation. The Selection Operator When applied to a relation r, this operator produces another relation whose rows are a subset of the rows of r that satisfy a specified condition. The resulting relation and r have the same attributes. Definition: Let r be a relation, A an attribute of r, and a is an element of the Domain (A). The Selection of r on attribute A is the set of tuples t of r such that t (A) = a. The Selection of r on A is denoted (r). The selection operator is a unary operator. That is, it operates on one relation at a time. The Projection Operator The projection operator is also a unary operator. The selection operator chooses a subset of the rows of the relation, whereas the projection operator chooses a subset of the columns. Definition:The projection of relation r onto a set X of its attributes, denoted by (r), is a new relation that we can obtain by first eliminating the columns of r not indicated in X and then removing any duplicate tuple. The columns of the projection relation are the attributes of X. The Equijoin Operator The equijoin operator is a binary operator for combining two relations on all their common attributes. That is, the join consists of all the tuples resulting from concatenating the taples of the first relation with the tuples of the second relation that have identical values for a common attribute X. Definition:Let r be a relation with a set of attributes R and let s be another relation with a set of attributes S. In addition, let us assume that R and S have some common attributes, and let X be that set of common attributes. That is, R S = X.



DBMS

The join of r and s, denoted by r Join s, is a new relation whose attributes are the elements of R S. In addition, for every tuple t of the r Joins relation, the following three conditions need to be satisfied simultaneously: (1) t(R) = t for some tuple t of the relation r, (2) t(S) = t for some tuple t of the relation s, and (3) t (X) = t (X). Set Operators on Relations Union:- The result of this operation, denoted by R S, is a relation that includes all tuples that are either in R or in S or in both R and S. Duplicate tuples are eliminated. Intersection:-The result of this operation, denoted by R S, is a relation that includes all tuples that are in both R and S Set difference (or MINUS):The result of this operation, denoted by R – S, is a relation that includes all tuples that are in R but not in S.   

Both UNION and INTERSECTION are commutative operations R S = S R, and R S = S R R (S T) = (R S) T and (R S) T = R (S T) R–S≠S–R

Cartesian Product (or Cross Product) :This operation is used to combine tuples from two relation in a combinational fashion. In general, the result of R (A , A -----A ) S (B , B ------B ) is a relation Q with degree n + m attributes Q (A , A ----- A , B , B ------ B ) is that order 

If R has n tuples and S has n tuples, then R × S will have n

n tuples.

Natural Joins The simplest sort of match is the natural join of two relations R and S, denoted R ∞ S, in which we match only those pairs between R and S that agree in whatever attributes are common to the schemas of R and S. A Complete Set of Relational Algebra operations:The set of relational algebra operations { , , , , x} is a complete set; that is any of the other original relational algebra operation can be expressed as a sequence of operation from this set. Tuple Relational Calculus The tuple relational calculus is based on specifying a number of tuple variables.Each tuple variable usually ranges overa particular database relation, meaning that the variable may take as its valuefrom any individual tuple in that relation. A simple tuple relational calculus query is of the form {t|COND (t)}



DBMS

where t is a tuple variable and COND(t) or FORMULA is a conditional expression involving t. The result of such a query is the set of all tuples t that satisfy COND (t). For example, to find all employees whose age is above 30, we can write the following tuple calculus expression: (t | EMPLOYEE (t) and t.age> 30} The condition EMPLOYEE (t) specifies that the range relation of tuple variable t is EMPLOYEE. Each EMPLOYEE tuple t that satisfies the condition t.age> 50000 will be retrieved. A (well-formed) condition or formula is made out of one or more atoms, where an atom has one of the following forms:  



R(Si), where Si is a tuple variable and R is a relation Si a1 θ Sj a2, where Si and Sj are tuple variables, a1 is an attribute of the relation over which Si ranges a2 is an attribute of the relation over which Sj ranges and θ is one of the comparison operators (<, ≤, >, ≥, , ≠); the attributes a1 and a2 must have domains whose members can be compared by θ. Si.a1 θ c, where Si is a tuple variable, a1 is an attribute of the relation over which Si ranges, c is a constant from the domain of attribute a1, and θ is one of the comparison operators.

We recursively build up condition or formulae from atoms using the following rules:   

An atom is a formula; If F1 and F2 are formulae, so are their conjunction F1 ∧ F2, their disjunction F1 ∨ F2 and the negation ~F1 are also formula; If F is a formula with free variable X, then (∃X)(F) and (∀X)(F) are also formula.

The Existential and Universal Quantifiers In addition, two special symbols called quantifiers can appear in formulas; these are the universal quantifier (∀) and the existential quantifier (∃). Truth values for formulas with quantifiers must be described. A tuple variable t is bound if it is quantified, meaning that it appears in an (∃t) or (∀t) clause; otherwise, it is free. Ex: All free occurrences of a tuple variable t in F are bound in a formula F’ of the form F’ (∃t)(F) or F’ (∀t)(F). The tuple variable is bound to the quantifier specified in F’. For example.consider the formulas. F1 :d.DNAME

‘Research’

F2 : (∃t) (d.DNUMBER F3 : (∀d) (d.MGRSSN

t.DNO) ‘333445555’)

The tuple variable d is free in F1, whereas it is bound to the universal quantifier (∀) in F3. Variable t is bound to the (∃) quantifier in F2.



DBMS

The Domain Relational Calculus There is another type of relational calculus called the domain relational calculus, or simply, domain calculus. The domain calculus differs from the tuple calculus in the type of variables used in formulas: rather than having variables range over tuples, the variables range over single values from domains of attributes. To form a relation of degree n for a query result, we must have n of these domain variables – one for each attributes. An expression of the domain calculus is of the form {x1,x2,

,xn| COND (x1, x2,

..,xn, xn+1, xn+2,

.,xn+m)}

where x1, x2, xn,xn+1,xn+2, , xn+m are domain variables that range over domains (of attributes) and COND is a condition or formulaof the domain relational calculus. A formula is made up of atoms.The atoms of a formula are slightly different from those for the tuple calculus and can be one of the following: 1. An atom of the form R (x1,x2, ...... ,xj), where R is the name of a relation of degree j and each xi, 1 ≤ i ≤ j, is a domain variable. This atom states that a list of values of must be a tuple in the relation whose name is R, where xi, is the value of the ith attribute value of the tuple. To make a domain calculus expression more concise, we drop the commas in a list of variable; thus we write {x1, x2......,x„| R(x1x2x3) and----} instead of: {x1, x2 . . .., xn | R(x1,x2,x3) and ... } 2. An atom of the form xi op xj, where op is one of the comparison operators in the set {=,>,≥, <, ≤≠} and xi and xj are domain variables. 3. An atom of the form xi op c, where op is one of the compressionoperators in the set { , <, ≤, >, ≥, ≠}, x and x are domain variables and c is a constant value.



DBMS

Part 5.4: SQL What is SQL        

Structured Query Language. A declarative query language. Its prototype, SEQUEL, was developed at IBM Research. Consists DDL, DML and DCL. Basic elements -- statements/commands. Many other functions: defining views, indexes, embedded in a host language, etc. Available in almost all commercial DBMSs. SQL is case insensitive.

DDL is Data Definition Language statements. Some examples:       

CREATE - to create objects in the database ALTER - alters the structure of the database DROP - delete objects from the database TRUNCATE - remove all records from a table, including all spaces allocated for the records are removed COMMENT - add comments to the data dictionary GRANT - gives user's access privileges to database REVOKE - withdraw access privileges given with the GRANT command

Create table:CREATE TABLE “table name” (Column 1” “data-type-for-column – 1”, “Column 2” “data-type-for-column – 2”, ) Alter table: The ALTER TABLE statement is used to add, delete, or modify column in an existing table. To add column in a table – ALTER TABLE table-name ADD column-name datatype To delete a column in a table – ALTER TABLE table-name DROP COLUMN column-name



DBMS

To change the data type of a column in a table – ALTER TABLE table-name ALTER COLUMN column-name datatype DML is Data Manipulation Language statements. Some examples:      

SELECT - retrieve data from the a database INSERT - insert data into a table UPDATE - updates existing data within a table DELETE - deletes all records from a table, the space for the records remain CALL - call a PL/SQL or Java subprogram LOCK TABLE - control concurrency

INSERT statement allow user to insert a single record or multiple records into a table INSERT INTO table (Column-1, Column-2, . . . Column-n) (Value-1, Value-2, . . . . Value-n); DELETE statement is used to delete rows in a table. DELETE FROM table-name WHERE some-column = some-valve UPDATE statement is used to update existing records in a table. UPDATE table-name SET Column 1 = Value, column 2 = Value 2, - - WHERE some-column = some- value DCL is Data Control Language statements. Some examples:    

COMMIT - save work done SAVEPOINT - identify a point in a transaction to which you can later roll back ROLLBACK - restore database to original since the last COMMIT SET TRANSACTION - Change transaction options like what rollback segment to use

SQL DML - SELECT SELECT [DISTINCT|ALL] {* | [column expression (AS newname)] [, ...]} FROM table-name [alias] [,...]

- Specifies the table or tables to be used

[WHERE condition]

- Filters the rows subject to some condition



[GROUP BY column list]

-Forms groups of rows with the same column value

[HAVING condition]

- Filters the groups subject to some condition.

[ORDER BY column list]

DBMS

- Specifies the order of the output.

Simple SELECT• SELECT attributes (or calculations: +, -, /, *) FROM relation •SELECT DISTINCT attributes FROM relation •SELECT attributes (or * wild card) FROM relation WHERE condition SELECT - WHERE condition •AND

OR

•NOT

IN

•NOT IN

BETWEEN

•IS NULL

IS NOT NULL

•SOME

ALL

•NOT BETWEEN •LIKE ‘%’ multiple characters •LIKE ‘_’ single character •Evaluation rule: left to right, brac ets, NOT before AND & OR, AND before OR SELECT - aggregate functions •COUNT •SUM •AVG •MIN •MAX THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 177


DBMS

SELECT - ORDER BY •ORDER BY •ORDER BY ... DESC SELECT - JOIN Tables• Multiple tables in FROM clause MUST have join conditions!!! Compatible Operations •UNION (Combine the result of two quaries together) •EXCEPT (Return any distinct values from the left query that are not found on the right query) •INTERSECT(Return only rows that appear in both result sets) •Union compatible operator [ALL] [CORRESPONDING][BY column...] (ALL duplicated rows in the result)

includes

COLUMN Alias •SELECT prodid, prodname, (salesprice - goodofcost) profit FROM product ORDER BY prodid; •SELECT prodid, prodname, (salesprice - goodofcost) AS profit FROM product ORDER BY prodid; EXIST:-EXIST simply tests whether the inner query returns any row. If it does than the outer query proceeds. NOT EXIST:-NOT EXIST subquery is used to display cases where a selected column does not appear in another table. SOME:-Compare a scalar value with a single – column set of value. ANY: Compare a scalar value with a single – column set of value. Find stuid, stuname, major, and credits of the student whose credits are greater than any mis student s credits •SELECT stuid, stuname, major, credits FROM student WHERE credits > ANY (SELECT credits FROM student WHERE major='mis'); THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 178


DBMS

ALL: The ALL key word specifies that the search condition is true if the companion is true for every value, that the sub query returns. Grouping  

Partition the set of tuples in a relation into group based on certain criteria and compute aggregate functions for each group All tuples that agree on a set of attributes (i.e., have the same value or each of these attributes) called grouping attributes are put into a group.

Example: Determine that maximum of the GATE CS marks obtained by students in each city for all cities. SELECT City, MAX(Marks) As Max marks FROM Gate Marks WHERE Branch

‘CS’

GROUP BY City; Result: City

Max Marks

Hyderabad

87

Chennai

84

Mysore

90

Bangalore

82

Join operation Join operation takes two relations and return another relation as a result. Join Types 1. Inner Join (default) (r inner join r on < >) use of just ‘join’ is equivalent to inner join Example: loan INNER JOIN borrower ON loan. loan Number = borrower loan Number. 2. Left Outer Join (r left outer join r on < >) Example : loan LEFT OUTER JOIN borrower on loan. loan Number = borrower loan Number 3. Right Outer Join (r right outer join r on < >) Example : loan RIGHT OUTER JOIN borrower ON loan.loanNumber = borrower loan Number. The right outer join is symmetric to the left outer join. Tuples from the right hand-side relation that do not match any tuples in the left hand-side relation are padded with nulls and are added to the result of the right outer join.



DBMS

4. Full Outer Join (r full outer join r on < >) It is a combination of the left & right outer join types. Natural Join: The condition ‘natural’ can be used with any of the join types of specify natural join. (r natural < > r on < > [using < , >])  Natural join by default considers all common attributes  A subset of common attributes can be specified in an optional using phrase Views Views provide virtual relations which contain data spread across different tables. Views are mainly used for  Simplified query formulations  Data hiding  Logical data independence Creating view CREATEVIEW V AS Create view V with structure defined by outcome of the query expression Tables involved in the view definition are called BASE tables. Operations in views:  Querying is allowed  Update operations are usually restricted because 1. Updates on a view may modify many base tables 2. There may not be unique way of updating the base tables to reflect the update on view 3. View may contain some aggregate values 4. Ambiguity where primary key of a base table is not included in view definition  Update on views defined on joining of more than one table are not allowed  Updates on views defined with ‘group by’ cause and aggregated functions is not permitted as a tuple in view will not have a corresponding tuple in base relation.  Update on views which do not include primary key of base table are also not permitted.  Updates to view are allowed only if:  Defined on single base table  Not defined using group by’ clause and aggregate functions  Include primary key of base table



DBMS

Part 5.5: Transactions and Concurrency Control

Transactions A transaction T is a logical unit of database processing that includes one or more database access operations  

Embedded within application program Specified interactively (e.g., via SQL)

ACID properties Atomicity: Either all operations of the transaction are reflected properly in the database, or none. This property is maintained by transaction management component of DBMS. Consistency: Execution of a transaction in isolation (that is, with no other transaction executing concurrently) preserves the consistency of the database. This is typically the responsibility of the application programmer who codes the transactions. Isolation: When multiple transactions execute concurrently, it should be the case that, for every pair of transactions Ti and Tj, it appears to Ti that either Tj finished execution before Ti started or Tj started execution after Ti finished. Thus, each transaction is unaware of other transactions executing concurrently with it. The user view of a transaction system requires the isolation property and the property that concurrent schedules take the system from one consistent state to another. It is enforced by concurrenty control component of database Durability: After a transaction completes successfully, the changes it has made to the database persist, even if there are system failures. It is enforced by recovery management component of the DBMS. Transaction States (fig. 5.5.1)     

BEGIN_TRANSACTION: marks start of transaction. READ or WRITE: two possible operations on the data. END_TRANSACTION: marks the end of the read or write operations; start checking whether everything went according to plan. COMIT_TRANSACTION: signals successful end of transaction; changes can be “committed” to DB. ROLLBACK (or ABORT): signals unsuccessful end of transaction; changes applied to DB must be undone.



DBMS

BEGIN TRANSACTION

Active END TRANSACTION COMMIT Partially committed READ WRITE

Committed

ABORT ABORT Failed

Terminated

Fig. 5.5.1

Schedule When transactions are executing concurrently in an interleaved fashion, then the order of execution of operations from the various transactions is known as Schedule. Types of schedules: Serial, Non serial, Recoverable and Non recoverable schedules Serial schedule:A schedule where the operations of each transaction are executed consecutively without any interleaved operations from other transactions. Nonserial schedule: A schedule where the operations from a set of concurrent transactions are interleaved. Recoverable Schedule: A schedule which recovers from aborts by itself Non-recoverable Schedule: A schedule which is not recoverable Serializability A schedule'S' of 'n' transactions is Serializable if it is equivalent to some serial schedule of the same 'n' transactions, i.e If interleaved schedule produces the same result as that of the serial schedule, then the interleaved schedule is said to be serializable. For two schedules to be equivalent, the operations applied to each data item affected by the schedules should be applied to that item in both schedules in the same order. There are two types of equivalences they are conflict equivalence and view equivalence and they lead to (a) Conflict Serializability (b) View Serializability. Conflict Serializabilty 

Schedule S is conflict serializable if it is a schedule which is conflict equivalent to some serial schedule S'. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 182






DBMS

 Can reorder the non-conflicting operations to improve efficiency Non-conflicting operations:  Reads and writes from same transaction  Reads from different transactions  Reads and writes from different transactions on different data items Conflicting operations:  Reads and writes from different transactions on same data item

View Serializability Consider two schedules S and S1, they are said to be view equivalent if the following conditions are met. 1. For each data item Q if transaction Ti reads the initial value of Q in schedule S, then transaction Timust in schedules S1 must also read the initial value of Q. 2. For each data item Q, if transaction Ti executes read Q in schedule S and that value was produced by Tj (if any), then transaction Ti must in schedule S1, also read the value of Q that was produced by transaction Tj. 3. For each data item Q, the transaction (if any) that performs the final write (Q) operation in schedule S must perform the final write (Q) operation is schedule S1. Conditions 1 and 2, ensure that each transaction reads the same values in both schedules and therefore performs the same computation. Condition 3 ensures that both schedules result in the same final system state. The concept of view equivalence leads to the concept of view serilizability. Concurrency control Protocols Two-phase-locking protocol 

Basic 2PL: Transaction is said to follow the two-phase-locking protocol if all locking operations precede the first unlock operation  Two phases  Expanding or growing: New locks on items can be acquired but none can be released.  Shrinking: Existing locks can be released but no new locks can be acquired.



Conservative 2PL (static) 2PL: Lock all items needed BEFORE execution begins by predeclaring its read and write set  If any of the items in read or write set is already locked (by other transactions), transaction waits (does not acquire any locks)  Deadlock free but not very realistic



Strict 2PL: Transaction does not release its write locks until AFTER it aborts/commits  Not deadlock free but guarantees recoverable schedules  Most popular variation of 2PL



Rigorous 2PL: No lock is released until after abort/commit  Transaction is in its expanding phase until it ends





DBMS

Gurantees Strict schedules (strict schedule: Transaction can neither read/write X until last transaction that wrote X has committed/aborted)

Graph based protocols The simplest graph based protocol is tree locking protocol which is used to employ exclusive locks and when the database is in the form of a tree of data items. In the tree locking protocol, each transaction Ti can lock a data item at most once and must observe the following rules. a. b. c. d. e.

All locks are exclusive locks. The first lock by Ti can be any data item including the root node. Ti can lock a data item Q only if Ti currently locks the parent of Q Data items may be unlocked at any time. Ti cannot subsequently lock a data item that has been locked and unlocked by Ti.

A schedule with a set of transactions, that uses the tree locking protocol can be shown to be serializable. The transactions need not be two phase. Advantage of tree locking control: a. Compared to the two phase locking protocol, unlocking of the data item is earlier. So it leads to the shorter waiting times and increase in concurrency. Disadvantages of tree locking control: a. A transaction may have to lock data items that it does not access, because to access descendants we have to lock its parent also. So the number of locks and associated locking overhead is high. b. A risk of deadlock: One problem that is not solved by two-phase locking is the potential for deadlocks, where several transactions are forced by the scheduler to wait forever for a lock held by another transaction Timestamp based protocols The use of locks, combined with the two phase locking protocol, guarantees serializability of schedules. The order of transactions in the equivalent serial schedule is based on the order in which executing transactions lock the items they require. If a transaction needs an item that is already locked, it may be forced to wait until the item is released. A different approach that guarantees serializability involves using transaction timestamps to order transaction execution for an equivalent serial schedule. Time Stamps A Timestamp is a unique identifier created by the DBMS to identify a transaction. Timestamp values are assigned in the order in which the transactions are submitted to the system. So a timestamp is considered as the transaction start time with each transaction Ti in the system, a unique timestamp is assigned and it is denoted by TS(Ti). When a new transaction Tj enters the system, then TS(Ti) < TS(Tj), this is known as timestamp ordering scheme. To implement this scheme, each data item (Q) is associated with two timestamp values. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 184


DBMS

1. "W-timestamp(Q)"denotes the largest timestamp of any transaction that executed write(Q) successfully. 2. "R-timestamp(Q)"denotes the largest timestamp of any transaction that executed read(Q) successfully. These timestamps are updated whenever a new read (Q) or write (Q) instruction is executed. Timestamp ordering protocol The timestamp ordering protocol ensures that any conflicting read and write operations are executed in timestamp order. This protocol operation is as follows. a. Suppose transaction Ti issues read(Q) a) If TS(Ti) < W- timestamp(Q) then Ti needs to read a value of Q that was already overwritten. Hence, the read operation is rejected and Ti is rolled back. b) If TS(Ti) > W-timestamp(Q), then the read operation is executed and R-timestamp(Q) is set to the maximum of R-timestamp(Q) and TS(Ti). b. Transaction issue a write(X) a) If TS(Ti) < read-timestamp(X), this means that a younger transaction is already using the current value of the item and it would be an error to update it now. This occurs when a transaction is late in doing a write and younger transaction has already read the old value. b) If TS(Ti) < write-timestamp(X). This means transaction T asks to write any item(X) whose value has already been written by a younger transaction, i.e., T is attempting to write an absolute value of data item X. So T should be rolled hack and restarted using a later timestamp c) Otherwise, the write operation can proceed we set write-timestamp (X)-TS(Ti) This scheme is called basic timestamp ordering and guarantees that transaction are conflict serializable and the results are equivalent to a serial schedule. Advantages of timestamp ordering protocol 1. Conflicting operations are processed in timestamp order and therefore it ensures conflict serializability. 2. Since timestamps do not use locks, deadlocks cannot occur. Disadvantages of timestamp ordering protocol: 1. Starvation may occur if a transaction is continually aborted and restarted. 2. It does not ensure recoverable schedules. Thomas's write rule: A modification to the basic timestamp ordering protocol is that it relaxes conflict serializability and provides greater concurrency by rejecting absolute write operation. The extension is known as Thomas's write rule. Suppose transaction Ti issues read(Q): no change , same as Time stamp ordering protocol.



DBMS

If transaction issue a write(X): a) If TS(T) < read-timestamp(X), this means that a younger transaction is already using the current value of the item and it would be an error to update it now. This occurs when a transaction is late in doing a write and younger transaction has already read the old value. b) If TS(Ti) < write-timestamp(X). This means that a young transaction has already updated the value of the item and the value that the older transaction is writing must be based on an absolute value of the item. In this case the write operation can safely be ignored. This is sometimes known as the ignore absolute write rule and allows greater concurrency. Problems of Dirty Data There is a problem that the commit bit is designed to help with the dirty read problem. A transaction T reads , and was written by another transaction . The time stamp of is less than that of T, and the read by T occurs write by in real time, so the events are physically realizable. However it is possible that after T reads , the transaction will abort. The value of read by T becomes invalid. In this case it is better to delay T; read of until commits or aborts. (See Fig. below) writes

start

T reads

T start

aborts

Multiple Granularity:     

Allow data items to be of various sizes and define a hierarchy of data granularities, where the small granularities are nested within larger ones. Can be represented graphically as a tree. When a transaction locks a node in the tree explicitly, it implicitly locks all nodes descendants in the same lock mode. Granularity of locking (level in tree where locking is done):  Fine granularity (lower in tree): High concurrency high locking overhead  Coarse Granularity (higher in tree): Low locking overhead, low concurrenty. Example of granularity hierarchy:



DBMS

DB

Fa

r





r

r

A1

A2

Fb

Fc

r

r

r

r

The highest level in the example hierarchy is the entire database. The levels below are of type area, file and record in that order. Intention Lock Modes:  In addition to shred and exclusive lock modes, there are three additional lock modes with multiple granularity.  Intention shared (IS) indicates explicit locking at a lower level of the tree but only with shared locks  Intention exclusive (IX) indicates explicit locking at a lower level with exclusive or shared locks.  Shared and intention exclusive (SIX)The subtree rooted by that node is locked explicitly in shared mode and explicit locking is being done at a lower level with exclusive mode locks. Compatibility matrix with intention lock modes IS IX S SIX X  IS       IX     S       SIX         X

Multi version Schemes In multi version database systems, each write operation on data item say Q creates a new version of Q. When a read (Q) operation is issued, the system selects one of the versions of Q to read. The concurrency control scheme must ensure that the selection of the version to be read is done in a manner that ensures serializability. There are two multi version schemes 1. Multi version Timestamp ordering. 2. Multi version Two-phase locking. Multi version timestamp ordering In this technique, several versions Q1Q2 ......Qkof each data item Q are kept by the system. For each version the value of the version Qk and the following two timestamps are kept. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 187


DBMS

1. W-timestamp(Qk) is the timestamp of the transaction that created version Qk. 2. R-timestamp(Qk) is the largest timestamp of any transaction that successfully read version Qk. The scheme operates as follows: When transaction Ti issues a read(Q) or write(Q) operation. Let Qk denote the version of Q whose write timestamp is the largest write timestamp less than or equal to TS(Ti). 1. It transaction Ti issues a read(Q), then the value returned is the content of version Qk 2. It transaction Ti issues a write(Q), and if TS(Ti) < R-timestamp(Qk), then transaction Ti is rolled back. Otherwise if TS(Ti) = W-timestamp(Qk) the contents of Qk are overwritten, otherwise a new version of Q is created. Advantages 1. The read request never fails and is never made to wait. Disadvantages 1. It requires more storage to maintain multiple versions of the data item. 2. Reading of a data item also requires the update of the R-timestamp field, resulting in two potential disk accesses. 3. Conflicts between transactions are resolved through roll backs, rather than through waits. This may be expensive. Multi version Two Phase Locking The multi version two phase locking protocol attempts to combine the advantage of multi version concurrency control with the advantages of two phase locking. In the standard locking scheme, once a transactions obtain a write lock on an item, no other transactions can access that item. So here it allows other transactions T1 to read an item X while a single transaction T holds a write lock on X. This is accomplished by allowing two versions of each item of X. When an update transaction reads an item it gets shared lock on the item and reads the latest version of that item. When an update transactions wants to write an item, it first gets an exclusive lock on the item and then creates a new version of the data item. The write is performed on the new versions the timestamp of the new version is initially set to a value . Advantages 1. Reads can proceed concurrently with a write operation but it is not permitted in standard two phase locking. 2. It avoids cascading aborts, since transactions are only allowed to read the version that was written by a committed transaction. Disadvantages 1. It requires more storage to maintain multiple versions of the data item. 2. Dead locks may occur.



DBMS

Part 5.6: File Structures (Sequential files, indexing, B and B+ trees)

Dynamic Multilevel Indexes Using B – Trees and

-Trees

B-trees and B+ trees are special cases of the well – known tree data structure. We introduce very briefly the terminology used in discussing tree data structures. A tree is formed of nodes. Each node in the tree, except for a special node called the root, has one parent node and several – zero or more – child nodes. The root node has no parent. A node that does not have any child nodes is called a leaf node; a nonleaf node is called an internal node. The level of a node is always one more than the level of its parent, with the level of the root node being zero. A sub tree of a node consists of that node and all its descendant node – its child nodes, the child nodes of its child nodes, and so on. A precise recursive definition of a sub tree is that it consists of a node n and the sub trees of all the child nodes of n. In this figure the root node is A, and its child nodes are B, C, and D. Nodes E, J, C, G, H, and K are leaf nodes. Usually, we display a tree with the root node at the top, as shown in Figure 5.6.1. One way to implement a tree is to have an many pointers in each node as there are child nodes

SUBTREE FOR NODE B

nodes at level 1

D

B

E

root node (level 0)

A

C

F

G G

J

H

I

nodes at level 2

K

nodes at level 3

(nodes E,J,C,G,H and K are leaf nodes of the tree) Figure 5.6.1 A tree data structure that shows an unbalanced tree. of that node. In some cases, a parent pointer is also stored in each node. In addition to pointers, a node usually contains some kind of stored information. When a multilevel index is implemented as a tree structure, this information includes the values of the file’s indexing field that are used to guide the search for a particular record. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 189


DBMS

Search Trees and B-Trees Search Trees. A search tree is slightly different from a multilevel index. A search tree of order p is a tree such that each node contains at most p – 1 search values of p pointers in the order
, where q ≤ p; each P is a pointer to a child node (or a null pointer); and each K is a search value from some ordered set of values. All search values are assumed to be unique. Figure 5.6.2 illustrates a node in a search tree. Two constraints must hold at all times on the search tree: 1. Within each node, K < K < P

K

. K

P

K

K

P

P X X
X K

<

X K

<

Figure 5.6.2 A node in a search tree with pointers to subtrees below it. 2. For all values X in the subtree pointed at by P , we have K for i = 1; and K < for i = q (see Figure 5.6.2).

<

< K for 1 < < ; X
Whenever we search for a value X, we follow the appropriate pointer P according to the formulas in condition 2 above We can use a search tree as a mechanism to search for records stored in a disk file. The values in the tree can be the values of one of the fields of the file, called the search field (which is the same as the index field if a multilevel index guides the search). Each key value in the tree is associated with a pointer to the record in the data file having that value. Alternatively, the pointer could be to the disk block containing that record. The search tree itself can be stored on disk by assigning each tree node to a disk block. When a new record is inserted, we must update the search tree by inserting an entry in the tree containing the search field value of the new record and a pointer to the new record. Algorithms are necessary for inserting and deleting search values into and from the search tree while maintaining the preceding two constraints. In general, these algorithms do not guarantee that a search tree is balanced, meaning that all of its leaf nodes are at the same level. The tree in figure 5.6.1 is not balanced because it has leaf nodes at levels 1, 2, and 3. Keeping a search tree balanced is important because it guarantees that no nodes will be at very high levels and hence require many block accesses during a tree search. Keeping the tree balanced yields a uniform search speed regardless of the value of the search key. Another problem with search trees is that record deletion may leave some nodes in the tree nearly empty, thus wasting storage space and increasing the number of levels. The B-tree addresses both of these problems by specifying additional constraints on the search tree. B – Trees: The B-tree has additional constraints that ensure that the tree is always balanced and that the space wasted by deletion, if any, never becomes excessive. The Algorithms for insertion and deletion, though, become more complex in order to maintain these constraints. Nonetheless, most insertions and deletions are simply processes; they become complicated only under special circumstance – namely, whenever we attempt an insertion into a node that is already full or a THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 190


DBMS

deletion from a node that makes it less than half full. More formally, a B-tree of order p, when used as an access structure on a key field to search for records in a data file, can be defined as follows: 1. Each internal node in the B-tree (figure 5.6.3a) is of the form < P , < K , P >, P , < K , P >, , < K , Pr >, P > where q ≤ p. Each P is a tree pointer – a pointer to another node in the B-tree. Each Pr is a data pointer – a pointer to the record whose search key field value is equal to K (or to the data file block containing the record). 2. Within each node, K < K < < K . 3. For all search key field values X in the subtree pointed at by P (the ithsubtree, see Figure 6.4a), we have: K < < K for 1 < < ; X < K for i = 1; and K < for i = q. 4. Each node has at most p tree pointers. 5. Each node, except the root and leaf nodes, has at least ⌈(p 2)⌉ tree pointers. The root node has at least two tree pointers unless it is the only node in the tree. 6. A node with q tree pointers, q ≤ p, has q – 1 search key field values (and hence has q – 1 data pointers). 7. All leaf nodes are at the same level. Leaf node have the same structure as internal nodes except that all of their tree pointers P are null. A B-tree starts with a single root node (which is also a leaf node) at level 0 (zero). Once the root node is full with p – 1 search key values and we attempt to insert another entry in the tree, the root node splits into two nodes at level 1. Only the middle value is kept in the root node, and the rest of the values are split evenly between the other two nodes. When a nonroot node is full and a new entry is inserted into it, that node is split into two nodes at the same level, and the middle entry is moved to the parent node along with two pointers to the new split nodes. If the parent node is full, it is also split. Splitting can propagate all the way to the root node, creating a new level if the root is split. .

(a) tree pointer

data pointer

X X<

tree pointer

. tree data pointer pointer

X

<

tree data pointer pointer

data pointer

X <

<

(b)

Tree node pointer 5 0

8 0

0 Data pointer Null node pointer

1 0

3 0

6 0

7 0

9 0

1 2

0

Figure 5.6.3 B-tree structures. (a) A node in a B-tree with q – 1 search values. (b) A B-tree of order p = 3. The values were inserted in the order 8, 5, 1, 7, 3, 12, 9, 6. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 191


DBMS

If deletion of a value causes a node to be less that half full, it is combined with its neighboring nodes, and this can also propagate all the way to the root. Hence, deletion can reduce the number of tree levels. It has been shown by analysis and simulation that, after numerous random insertions and deletions on a B-tree, the nodes are approximately 69 percent full when the number of values in the tree stabilizes. This is also true of B+-trees. If this happens, node splitting and combining will occur only rarely, so insertion and deletion become quite efficient. If the number of values grows, the tree will expand without a problem – although splitting of nodes may occur, so some insertions will take more time. B-trees are sometimes used as primary file organizations. In this case, whole records are stored within the B-tree nodes rather than just the
entries. This works well for files with a relatively small number of records, and a small record size. Otherwise, the fan-out and the number of levels become too great to permit efficient access. In summary, B-trees provide a multilevel access structure that is a balanced tree structure in which each node is at least half full. Each node in a B-tree of order p can have at most p – 1 search values. -Trees Most implementations of a dynamic multilevel index use a variation of the B-tree data structure called a -tree. In a B-tree, every value of the search field appears once at some level in the tree, along with a data pointer. In a B -tree, data pointers are stored only at the leaf nodes of the tree; hence, the structure of leaf nodes differs from the structure of internal nodes. The leaf nodes have an entry for every value of the search field, along with a data pointer to the record (or to the block that contains this record) if the search field is a key field. For a nonkey search field, the pointer points to a block containing pointers to the data file records, creating an extra level of indirection. The leaf nodes of the B -tree are usually linked together to provide ordered access on the search field to the records. These leaf nodes are similar to the first (base) level of an index. Internal nodes of the B -tree correspond to the other levels of a multilevel index. Some search field values from the leaf nodes are repeated in the internal nodes of the B - tree to guide the search. The structure of the internal nodes of a B -tree of order p (Figure 5.6.4a) is as follows: 1. Each internal nodes is of the form
Where q ≤ p and each P is a tree pointer. 2. Within each internal node, K < K < < K . 3. For all search field values X in the subtree pointed at by P , we have K < ≤ K for 1 < i < q; X ≤ K for i = 1; and K < for i = q (see Figure 5.6.4a). 4. Each internal node has at most p tree pointers. 5. Each internal node, except the root, has at least ⌈(p 2)⌉ tree pointers. The root node has at least two tree pointers if it is an internal node. 6. An internal node with q pointers, q ≤ p, has q – 1 search field values. The structure of the leaf nodes of a B -tree of order p (Figure 5.6.4b) is as follows: 1. Each leaf node is of the form << K , Pr >, < K , Pr >,

,< K

, Pr

>, P

>



(a)

…

…

tree pointer X

X

<

X≤

….

data pointer

data pointer

Figure 5.6.4 The nodes of a Leaf node of a

2. 3. 4. 5.

tree pointer

tree pointer

(b)

DBMS

X

<

<

pointer to next leaf node in tree

…. data pointer

data pointer

-tree. (a) Internal node of a -tree with q – 1 search values. (b) -tree with q – 1 search values and q – 1 data pointers.

where q ≤ p, each Pr is a data pointer, and P points to the next leaf node of the B tree. Within each leaf node, K < K < < K ; q ≤ p. Each Pr is a data pointer that points to the record whose search field value is K or to a file block containing the record (or to a block of record pointers that point to records whose search field value is K if the search field is not a key). Each leaf node has at least ⌈(p 2)⌉ values. All leaf nodes are at the same level.

The pointers in internal nodes are tree pointers to blocks that are tree nodes, whereas the pointers in leaf nodes are data pointers to the data file records or blocks––except for the P pointer, which is a tree pointer to the next leaf node. By starting at the leftmost leaf node, it is possible to traverse leaf nodes as a linked list, using the P pointers. This provides ordered access to the data records on the indexing field. A P pointer can also be included. For a B -tree on a nonkey field, an extra level of indirection is needed so the Pr pointers are block pointers to blocks that contain a set of record pointers to the actual records in the data file. Because entries in the internal nodes of a B -tree include search values and tree pointers without any data pointers, more entries can be packed into an internal node of a B -tree then for a similar B -tree. Thus, for the same block (node) size, the order p will be larger for the B -tree than for the B-tree. This can lead to fewer B -tree levels, improving search time. Because the structures for internal and for leaf nodes of a B -tree are different, the order p can be different. We will use p to denote the order for internal nodes and p to denote the order for leaf nodes, which we define as being the maximum number of data pointers in a leaf node. As with the B-tree, we may need additional information––to implement the insertion and deletion algorithms––in each node. This information can include the type of node (internal or leaf), the number of current entries q in the node, and pointers to the parent and sibling nodes. Hence, before we do the above calculations for p and p , we should reduce the block size by the amount of space needed for all such information. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 193


Theory of Computation

Part – 6: Theory of Computation 6.1: Introduction/ Preliminaries (string, alphabet, set, relation, equivalence of relation etc.) PRELIMINARIES: 1. String:A string is a finite sequence of symbols put together. Note: The length of empty string denoted by , is the string consisting of zero symbols. Thus I I = 0. 2. Alphabet:An alphabet is a finite set of symbols. It is denoted by 3. Formal language: A formal language is a set of strings of symbols from some 1 alphabet. Note: 1. The empty set, ∅, is a formal language. The cardinality (size) of this language is zero. 2. The set consisting of empty string, { } is a formal language. The cardinality (size) of this language is one. 4. Set: A set is a collection of objects (members of the set) without repetition. i. Finite Set: A set which contains finite number of elements is said to be finite set. ii. Countably Infinite Set:Sets that can be placed in one-to-one correspondence with the integers are said to be countably infinite or countable or denumerable. iii. Uncountable set: Sets that can't be placed in one-to-one correspondence with the integers are said to be uncountable sets. 5. Relations: A(binary) relation is a set of ordered tuple. The first component of each tuple is chosen from a set and the second component of each pair is chosen from a (possibly different) set.




6.2: Finite Automata Introduction: A finite automaton involves states and transitions among states in response to inputs. A Deterministic finite automaton is represented by a Quintuple (5-tuple): (Q, ,δ,q0,F) where Q : Finite set of states : Finite set of input symbols called the alphabet. δ : Q X ⇨ Q (δ is a transition function from Q X to Q) q0 : A start state, one of the states in Q F : A set of final states, such that F ⊆ Q.

Acceptance by an Automata : A string “X” is said to be accepted by a finite automaton M = (Q, , δ, q 0, F) if δ (q0, x) = P for some p in F. The language accepted by M, designated L (M), is the set {x | δ(q0,x) is in F}. A language is a regular set (or just regular) if it is the set accepted by some automaton. There are two preferred notations for describing Automata 1. Transition diagram 2. Transition table 1. Give DFA for accepting the set of all strings containing ‘111’ as substring. Transition diagram : 0/1

0 Start

1 q0

1 q2

q1 11

11 11

0 11 111 111 Transition Table: 1

q0 q1 q2 *q3

0

1

q0 q0 q0 1q3 1

q1 q2 q3 q3

111 111 1

111 0 111 1

111 111 1

1 q3 11 11 111 111 1

111 111 1

111 111 Extending transition function from single symbol to string: For the behavior of a finite automaton 1

on string, we must extend the transition function ‘δ’ to apply to a state and a string rather than a state and a symbol. We define a function δ̂ from Q X *⇨Q.




1. δ̂ (q, ) = q, and 2. for all strings “w” and input symbols ‘a’, δ̂ (q, wa) = δ( δ̂ (q,w),a)  In any DFA, for a given input string and state the transition path will always be unique.

Non-Deterministic Finite Automata: A nondeterministic finite state machine or nondeterministic finite automaton (NFA) is a finite state machine where for each pair of state and input symbol there may be several possible next states. An NFA is represented by a 5-tuple(Q, , δ, q0, F) where 1. Q is a finite set of states, 2. is a finite set of input symbols 3. δ : Q x ⇨ 2Q (‘δ’ is a transition function from Q x to power set of Q) 4. q0 , a member of Q, is the start state and 5. F, a subset of Q, is the set of final states. ⟶Non-deterministic finite automata can also be represented by transition diagram and transition table. Extending transition function from single symbol to string: The function ‘δ’ can be extended to function δ̂ mapping Q X * to 2Q and reflecting sequences of input as follows: 1. δ̂ (q, ) = {q} 2. Suppose ‘w’ is of the form w = xa, where a is the final symbol of w and x is the rest of w. Alsosuppose hatδ̂ (q, x)= {P1, P2 ……Pk}. Let ⋃ δ (pI, a) = { r1, r2, …… rm} Then δ̂ (q, w) ={r1, r2, ……. , rm}. Less formally, we compute δ̂ (q, w) by first computing δ̂ (q, x) and by then following any transition from any of these states that is labeled as ‘a’. Acceptance by NFA: An NFA accepts a string ‘w’ if it is possible to make any sequence of choices of next state, while reading the characters of w, and go form start state to any accepting state. if M = (Q, , δ, q0, F} is an NFA, then L(A) = {w / δ(q0, w) ∩ F ≠ ∅}. That is, L(A) is the set of strings w in * such that δ̂ (q0, w) containing at least one accepting state. 1. Construct an NFA for the set of all strings over the alphabet {0,1} containing 111 as substring. Transition diagram: 0/1 0/1 1

1

1

0 0 THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 196



Construction of DFA from NFA(Sub Set Construction): Let N = (QN, , δN q0, FN) be a NFA. We have to construct D = (QD, , δD{q0}, FD) such that L(D) = L(N)  QD is the set of subsets of QN, i.e..power set of QN has ‘n’ states, then QD will have 2n states. Often not all these states are accessible from the start state of QD. Inaccessible states can be “thrown away”. So effectively the number of states of D may be much smaller then 2n.  FD is the set of subsets S of QN such that S ∩ FN ≠ ∅. That is , FD is all sets of Ns states that include set S ⊆ QN and for each input symbol a in , δD(S,a) = ∪ δN (P,a) That is , to compute δD(S,a) We look at all states p in S, see what states N goes to from p on input ‘a’, and take the union of all those sets. NFA With -Transitions (Epsilon Transitions):  An NFA is allowed to make a transition spontaneously, without receiving an input symbol. These NFA’s can be converted to DFA’s accepting the same language.  We may represent an NFA exactly as we do an NFA with one exception Q δ: Q X U { } → 2 Closure: closure (q) is the set of all vertices p such that there is a path from q to p on alone. 1 ϵ

0

ϵ

ϵ q0 In this diagram

2

q1

q2

-closure of q0 = {q0, q1, q2} -closure of q1 = {q1, q2} -closure of q2 = {q2}

Extended transitions and language for -NFA’s: 1. δ̂(q, ) = -closure(q). That is, if the label of the path is , then we can follow only -labeled arcs extending from state q, 2. Suppose ‘w’ is of the form xa, where a is the last symbol of w. Note a is a member of ; It cannot be , which is not in . We compute δ̂(q,w)as follows: (A) Let {p1,p2,……..pk} be δ̂(q,x)). That is, the p’s are all and only the states that we can reach from q following a path labeled x. This path may end with one or more transitions labeled , and may have -transitions as well. (B) Let ∪ δ(pi,a) be the set {r1,r2…….rm}. That is, follow all transitions labeled ‘a’ from states i=1 we can reach from q along paths labeled x, The rj’s are some of the states we can reach from q along paths labeled w. The additional states we can reach are found from the r j’s followed by ϵ-labeled arcs in step(C), below. M




(C) Then δ̂(q,w) = ∪ ϵ-closure (rj). This additional closure step includes all the paths from j = 1. q labeled w, by considering the possibility that there are additional ϵ-labeled arcs that we can follow after making a transition on the final ‘real’ symbol a. Eliminating -Transitions (Construction of DFA from -NFA); Let E=(Q , , δ , q , F )be the given -NFA then the equivalent DFA D=(Q , δ , q , F )is defined as follows 1. q = closure (q ) 2. Q is the set of all subsets of Q more precisely ,we shall find that the only accessible states of D ate the -colsed subset of Q , that is those sets S≤Q such that S= -closure (S). 3. F is those sets of states that contain atleast one accepting state of E i.e., F ={S/S is in Q and S∩F ≠ ∅} 4. δ is computed ,for all a in and sets S in Q by (A) Let S={P , p … … , p } (B) Compute ∪δ(p , a) ; Let this set be {r , r … . . , r } m (C) Then δ (S, a) -closure (r ) ∪




6.3: Regular Expression This algebraic notation describes exactly the same languages as finite automata: the regular languages. The regular expression operators are union, concatenation (or “dot”), and closure (or “star”). A set of languages A ⊆ * is said to be a regular if A = L(M), for some finite automation M. where L(M) is the language accepted by M. Definitions: Union of two languages: L and M denoted by L U M is the set of strings that are either in L or M or both. For Example L = {0, 1} and M = {00, 11, 111} then L U M = {0, 1, 00, 11, 111} Concatenation of two languages: L and M denoted by LM is the set of strings that can be formed by taking any string of L and concatenating it with any string in M. Closure (Kleen closure or star) of a Language: It is denoted as L* and represents the set of those strings that can be formed by taking any number of strings from L, possibly with repetitions and concatenating all of them. That mean L* is the infinite union Ui>= 0 Li, where L0 = {ϵ}, L1 = L and Li for i > 1 is LLL…L (the concatenation of I copies of L) Regular Expression: Let

be a given alphabet. Then 1. ϕ, ϵ, and a ϵ are all regular expressions. These are called primitive regular expressions. 2. If r1 and r2 are regular expressions, so are r1+ r2, r1.r2, r1*, and (r1). 3. A string is a regular expression if and only if it can be derived from the primitive regular expressions by a finite number of applications of the rules in (2).

Languages Associated with Regular Expressions: Regular expressions can be used to describe some simple languages. If r is a regular expression, we will let L(r) denote the language associated with r. The language is defined as follows Definition: The language L(r) denoted by any regular expression r is defined by the following rules. 1. ∅ is a regular expression denoting the empty set(L(∅) = {}). 2. is a regular expression denoting the set L( ) = { }. 3. For every a , ‘a’ is a regular expression denoting the set {a}. If r1 and r2 are regular expressions, then 4. L(r1 + r2) = L(r1) ∪ L(r2) 5. L(r1 . r2) = L(r1) L(r2) 6. L(r1*) = (L(r1))*




Precedence of Regular Expression Operators: 1. Closure has higher precedence than concatenation. 2. Concatenation has higher precedence than union. Equivalence of Regular Expressions: Two regular expressions are said to be equivalent if they denote the same language Algebraic Laws For Regular Expressions: Let r1, r2 and r3 be three regular expressions 1. Commutative law for union: The commutative law for union, say that we may take the union of two languages in either order. r1 + r2 = r2 + r1 2. Associative law for union: The associative law for union says that we may take the union of three languages either by taking the union of the first two initially, or taking the union of the last two initially. (r1 + r2)+r3 = r1+(r2 + r3) 3. Associative law for concatenation: (r1r2)r3 = r1(r2r3) 4. Distributive Laws For Concatenation: → Concatenation is left distributive over union i.e., r1(r2 + r3) = r1r2 + r1r3 → Concatenation is right distributive over union i.e., (r1 + r2)r3 = r1r3 + r2r3 5.

Identities For Union And Concatenation: → ∅ is the identity for union operator i.e. r1 + ∅ = ∅ + r1 = r1 → is the identity for concatenation operator i.e. r1 = r1 = r1 6. Annihilators For Union And Concatenation:  An annihilators for an operator is a value such that when the operator is applies to the annihilator and some other value, the result is the annihilator. ∅ is the annihilator for concatenation. i.e., ∅r1 = r1∅ = ∅ There is no annihilator for union operator. 7. Idempotent Law For Union: This law states that if we take the union of two identical expressions, we can replace them by one copy of the expression. i.e. r1 + r1 = r1 8. Laws Involving Closure Let ‘r’ be a regular expression, then 1. (r*)* = r* 2. ∅* = THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 200



3. * = 4. r+ = r.r* = r*.r i.e r+ = rr* = r*r 5. r* = r+ + 6. r? = + r (Unary postfix operator? means zero or one instance) Converting Regular Expression To Automata ( -NFA): Basis: Automata for , ∅ and a are (a), (b) and (c) respectively start

a

start

start q0

q0

qf

q0

(a) r = (b) r = ∅ Induction: Automata for r + s, rs and r* are (p), (q) and (s) respectively. q1

(c) r = a

f1

M1

Є star t

qf

Є f0

q0 Є

Є q2

f2

M2 (p) r+s

start

Є q1

M1

f1

q2

M2

f2

(q)rs

Є Є

start q0

Є q1

M1

f1

ff0 0

Є (s)r*




Finite Automata with Output: Moore Machine: A Moore machine is a six-tuple (Q, , Δ, δ, λ, q0) Where Q: finite set of states : finite set of input symbols Δ: finite set of output alphabet δ: Q x → Q (transition function from Q x to Q λ: Q → Δ (λ is a function from Q to Δ) q0: start state In Moore machine output is associated with states Melay Machine: A Melay machine is sixtuple (Q, , Δ, δ, λ, q0) Q: finite set of states : finite set of input alphabet Δ: finite set of output alphabet δ: Q x → Q λ: Q x → Δ (i.e λ(q,a) gives the output associated with the transition from state q on input a) q0: starting state In Melay machine output is associated with each transition. Example: M = {{q0, p0, p1}, {0,1}, {y,n}, δ, λ, q0} 0/y p0 0/n 0/n 1/n q0 p1 1/n

1/y

Fig: Melay machine for (0+1)*(00 + 11) Equivalence of Moore and Melay machines: Melay machine equivalent to Moore machine: If M1 = (Q, , Δ, δ, λ, q0) is a Moore machine then there is a Melay machine M2 equivalent to M1. Construction of Moore machine: Let M = (Q, , Δ, δ, λ, q0) be the given Melay machine and M’ = (Q’, , Δ, δ’, λ’, [q0, b0]), Where B0 is an arbitrarily selected member of Δ THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 202



The states of Moore machine are [q0,y], [q0,n], [p0,y],[p0,n] [p1,y] and [p1,n]. As ‘q0’ is the start stae of Melay machine choose either [q0,n]. The start stae of Moore machine is [q0,n]

Regular Grammars Definition of a Grammar: A phrase-structure grammar (or simply a grammar) is ( V,T,P,S), where i. V is a finite nonempty set, whose elements are called variables. ii. T is a finite nonempty set, whose elements are called terminals. iii. V ∩ T = ϕ, iv. S is a special variable (i.e an element of V) called the start symbol, and v. P is a finite set whose elements are α → β, where α and β are strings on V ∪ T. α has at least one symbol from V. Elements of P are called productions or production rules or rewriting rules. Right-Linear Grammar: A grammar G = (V,T,S,P) is said to be right-linear if all productions are of the form A → xB A → x. Where A,B V and x T*. Left-Linear Grammar: A grammar G = (V,T,P,S) is said to be left-linear grammar if all productions are of the form A → Bx or A → x. Either right-linear or left-linear grammar is a Regular grammar. Example: The grammar G1 = ({s}, {a,b}, S, P1), with P1 given as S → abS/a is right-linear grammar. The grammar G2 = ({S, S1, S2}, {a,b}, S, P2} with productions. S → S1ab, S1 → S1ab|S2, S2 → a, is left-linear grammar. Both G1 and G2 are regular grammars. A language L is regular if and only if there exists a left-linear grammar G such that L = L(G). A language L is regular if and only if there exists a right-linear grammar G such that L = L(G). Construction of -NFA from right-linear grammar: Let G = (V,T,P,S) be a right-linear grammar. We construct an NFA with -moves, M = (Q,T,δ,[S],[ ]} that simulates deviation in ‘G’ Q consists of the symbols [α] such that α is S or a (not necessarily proper) suffix of some righthand side of a production in P. We define δ by: 1. If A is a variable, then δ([A], ) = { [α] | A → α is a production} 2. If a is in T and α in T* ∪ T*V, then δ([aα],a) = {[α]}




Construction of ϵ-NFA from a left-linear grammar: If ‘G’ is a left-linear grammar we will reverse all the right hand sides of the productions then we will get right-linear grammar from which we will construct ϵ-NFA. To get ϵ-NFA of given leftlinear we will exchange initial, final states and reverse the direction of all the edges. Construction of right-linear and left-linear grammars from a given Finite Automata: Right linear grammar: Let M = (Q, , δ, q0, F) be the given finite automata. First suppose that q0 is not a final state. Then L = L(G) for a right-linear grammar G = (Q, , P, q0), where P consists of production p → aq whenever δ(p,a) = q and also p → a whenever δ(p, a) is a final state. Now let q0 be final state, so ϵ is in L. So introduce a new start symbol S with productions S → q0 | ϵ. Right linear grammar A → 0A| B, B → aB|bB|a|b A → B can be written as A → B (since B = B) The resulting grammar is A → 0A|B, B → aB|bB|a|b As A → B is a unit production we can eliminate it by substituting and the resulting grammar is A → 0A | aB|bB|a|b, B → aB|bB|a|b Now substitute 10 for a and 11 for b A → 0A | 10B|11B|10|11, B → 10B|11B|10|11 Left linear grammar: This can be obtained by reversing all the right hand sides of the production in the right – linear grammars and the production set contains the following productions. A → A0 | B01|B11|01|11, B → B01|B11|01|11 Overview of Finite Automata and Regular Expressions: 1. Regular sets  DFA 2. NFA  DFA 3. NFA – ε – Moves  NFA 4. Regular  Regular grammar 5. Regular grammar Right linear grammar 6. Regular grammar Left linear grammar 7. Regular set  Unique minimal finite automata 8. Right linear  NFA 9. Left linear  NFA 10. Moore machine melay machine

Properties Of Regular Languages: Closure Properties of Regular Languages: 1. Regular languages are closed under union, concatenation and Kleene closure. 2. Regular languages are closed under complementation. That is, L is a regular language and L ⊆ *, then * - L is a regular languages. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 204



3. Regular languages are closed under intersection. That is if L1 and L2 are regular languages then L1 ∩ L2 and L1 ∪ L2 are also regular languages. 4. Regular languages are closed under difference. That is if L and M are regular languages, then so is L – M. 5. Regular languages are closed under string reversal. → The reversal of a string a1 a2 ……an is the string written backwards, that is an an-1 ….a1 we use WR for the reversal of a string w. The reversal of a language L, written LR, is the language consisting of the reversals of all its strings. Given a language L that is L (M) for some deterministic finite automata, we may construct an automata for LR by 1. Reverse all the arcs in the transition diagram for M 2. Make that start state of A be the only accepting states for new automata 3. Create a new start P0 with transitions on to all the accepting states of M. The result is an automata that simulates M “in reverse” and therefore accepts a string w if and only if A accepts wR. 6. Regular languages are closed under substitution. → Let R ⊆ * be a regular set and for each ‘a’ in , let Ra⊆ Δ* be a regular set Let f: → Δ* be the substitution defined by f (a) = Ra. Select regular expression denoting R and each Ra. Replace each occurrence of the symbol ‘a’ in the regular expression for R by the regular expression for Ra. Example: Let f (0) = a and f(1) = b* That is , f(0) is the language {a} and f(1) is the language of all strings of b’s then f(010) is the regular set ab*a. If L is the language 0*(0+1)1*, then f(L) is a*(a + b*) (b*)* =a*b*. 7. Regular languages are closed under homomorphism and inverse homomorphism. ⟶homomorphism: A homomorphism h is a substitution such that h(a) contains a string for each a. We generally take h (a) to be the string itself, rather than the set containing that string. Suppose and ∆ are alphabet, then a function h: ⟶∆* is called a homomorphism The domain of the function h is extended to strings in an obvious fashion: if W=a1a2 ….. an then h(w) = h(a1)h(a2) h(a3) …….h(an). If ‘L’ is a language on , then its homomorphic image is defined as h (L) = {h(w):w L}. 8. Inverse Homomorphism: The inverse homomorphic image of a language L is h-1(L) ={x / h (x) is in L} for string w,h-1(w) = {x / h (x) = w} 1. Regular languages are closed under quotient with arbitrary sets. Definition: The quotient of languages L1 and L2 written L1/L2 is {x | there exist y in L2 such that xy is in L1} 2. Regular languages are closed under INIT operation Definition: Let ‘L’ be a language. Then INIT(L) ={x/ for some y, xy is in L} THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 205



3. Regular languages are closed under Kleen closure. i.e if ‘L’ is a regular set then L* is also a regular set. Decision Algorithms for Regular languages: The set of sentences accepted by a finite automata M with n states is: 1. Non empty if and only if the finite automaton accepts a sentence of length less than ‘n’. 2. Infinite if and only if the finite automaton accepts some sentences of length l, where n ≤ l < 2n. Note: To test whether a DFA accepts the empty set, take its transition diagram and delete all states that are not reachable on any input from the start state. If one or more final states remain, the language is non empty. Then without changing the language accepted, we may delete “all states that are not final and from which one cannot reach a final state”. The DFA accepts an infinite language if and only if the resulting transition diagram has a cycle. The same method works for NFA’s also. Equivalence of Regular languages: There is an algorithm to determine if two finite automata are equivalent (i.e., if they accept the same language). ̅̅̅ ∩ L2) is accepted by →Let M1 and M2 be FA accepting L1 and L2 respectively. (L1∩ ̅̅̅ L ) ∪ (L some finite automaton M3. It is easy to see that M3 accepts a word if and only if L1 ≠ L2. Hence we can find whether L1 = L2 or not. Right invariant relation: A relation R such that xRy implies xzRyz is said to be right invariant (with respect to concatenation) Myhill-Nerode Theorem: The following three statements are equivalent. 1. The set L ⊆ * is accepted by some finite automaton. 2. L is the union of some of the equivalence classes of a right invariant equivalence relation of finite index. 3. Let equivalence relation RL be defined by :xRLy if and only if for all z in *, xz is in L exactly when yz is in L. Then RL is of finite index. Pumping Lemma for Regular languages: Pigeon Hole Principle: If we put ‘n’ objects into ‘m’ boxes (pigeon holes), and if n>m, then atleast one box must have more than one item in it. Pumping Lemma for Regular languages: Pumping Lemma uses pigeon hole principle to show that certain languages are not regular. This theorem state that all regular languages have a special property. There are three forms of Pumping Lemma.




1. Standard Form of Pumping Lemma: Let ‘L’ be an infinite regular language. Then there exists some positive integer ‘n’ such that any z L with |z| ≥ n can be decomposed as z = uvw with |uv| ≤ n, and |v| ≥ 1, such that z = uvw, is also in L for all i = 0,1,2……. To paraphrase this, every sufficiently long string L can be broken into three parts in such a way that an arbitrary number of repetitions of the middle part yields another string in L. We say that middle string is “pumped”, hence the term Pumping Lemma for this result. 2. Strong Form of Pumping Lemma: If L is an infinite regular then there exists a ‘n’, such that the following holds for every sufficiently long z L and every one of its decompositions z = z1z2z3, with z1,z2 *, |z2| ≥ n. The middle string z2 can be written as z2 = uvw with |uv| ≤ n, |v| ≥ 1, such that z1uvwz3 L for all i = 0,1,2…….. 3. Weak Form of Pumping Lemma: Suppose L is an infinite regular language, there are integers p and q, with q>0, so that for every n ≥ 0, L contains a string of length p + nq. In other words, the set of integers, Lengths (L) = { |z| |z L} contains the “arithmetic progression” of all integers p + nq (where n ≥ 0). L (an infinite language)

Regular Language no

Yes Satisfies Weak form no Yes Satisfies Standard form

no

L is not regular

Yes no Satisfies Strong form

Yes We cannot say anything regularity of L If a language satisfies Pumping Lemma it may or may not be regular. But a language which doesn’t satisfy Pumping Lemma is not regular.




Examples: Of pumping lemma 1. L = {0 | n prime} is not regular. 2. L = {a b | i > } is not regular. 3. L = {a b | n ≥ 0} is not regular 4. L = {ω ω ω ε } is not regular 5. L = {a b | n ≠ 1} is not regular 6. 7. 8. 9.

L = {a | n ≥ 1} is not regular. L = {0 | n ≥ 1} is not regular L = {∝ ω ∝ | ∝, ω in (0 + 1) } is not regular L = {∝ ∝ ω | ∝, ω in (0 + 1) } is not regular




6.4: Context free grammar CFG is a 4 – tuple G = (V, T, P,S) where V and T are disjoint finite sets, S is an element of V, and P is a finite set of formulas of the form A → ∝ where A V and ∝ (VUT)*. Context free language: Let G= (V, T, P, S) be a CFG. The language generated by G is L(G) = {x is a CFL if there is CFG G so that L = L(G).

T*/S*→ x}. A language L

Leftmost And Rightmost Derivations and Ambiguity  If at each step in a derivation a production is applied to the leftmost variable, then the derivation is said to be leftmost.  If at each step in a derivation a production is applied to the rightmost variable, then the derivation is said to be rightmost. Derivations Tree: Let G (V,T,P,S) be a CFG. A tree is a derivation (or parse) tree for G if: Every vertex has a label, which is a symbol of V ∪ T ∪ { } The label of the root is S. If a vertex is interior and has label A, then A must be in V. If Vertex n has label A and vertices n1, n2..nk are the sons of vertex n, in order from the left, with labels X1,X2….Xk, respectively, the A → X1,X2.Xk must be a production in P. If vertex n has label , then n is a leaf and is the only son of its father. If w is L (G) for a CFG G, then w has at least two leftmost derivations or at least two rightmost derivations or at least two derivation trees then that grammar is said to be ambiguous.

Simplication of Context – Free Grammars If L is a nonempty CFL then it can be generated by a CFG with the following properties. Each variable and Each terminal of G appears in the derivation of some word in L There are no productions of the form A→B where A and B are variables. Furthermore if is not in L, there need be no productions of the A→ . In fact, if is not in L, we can require that every production of G be of one of the forms A→BC and A→ a, where A, B and Care arbitrary variables and is an arbitrary terminal Alternatively, we could make every production of G be of the from A→ a∝ where ∝is a string of variables (possibly empty). Elimination of useless symbols Let G ≈ (V,T,P,S) be a grammar A symbol X is useful if there is a derivation → ∝XB→ w for some ∝, β and w, where w is in T , otherwise X is useless. There are two aspects to usefulness. First some terminal string must be derivable from X and second, X must occur in some string derivable from S. These two conditions are not however; sufficient to guarantee that X is useful since X may occur only in sentential forms that contain a variable from which on terminal string can be derived. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 209



(A→ B means Band be derived from A after zero or more number of intermediate steps) Lemmal 1: Given a CFG =(V,T,P,S) with L(G) ≠∅, we can effectively find an equivalent CFGG =(V ,T,p ,S) ,such that for each A in V there is some w in T for which s w. Lemmal 2: Given a CFGG=(V, T, P, S) with : (G) ≠ ∅, we can effectively find an equivalent CFG G = (V , T , P ,S) such that for each X in V ∪ T there exists ∝ andβ in (V ∪ T )* for which S∝ Xβ Procedure for Eliminating Useless Symbols: Let G(V,T,P,S) be given CFG then by Applying Lemmal. G =(V , T,P ,S) can be find as follows, Calculation of V begin OLDV:=∅; NEWV:={A/A→w for some win T }; while OLDV≠NEWV do begin OLDV:=NEWV; NEWV:=OLWV: ∪{A/A→∝o for some ∝ in (T∪OLDV) } End ; P is the set of all productions whose symbols are in V ∪T. Now by applying lemma2 on G we can fine G as follows Place Sin V lf A is placed inV and A→∝ /∝ …./∝ , then add all variables of ∝ ,∝ …..∝ to set V and all terminals of ∝ ,∝ ….∝ toT p is the set of productions of P containing only symbols ofV ∪T Notes:  Every Non empty CFL is generated by CFG with no useless symbols

Eliminating ϵ- productions : Productions of the form A →ϵare called -Productions Surely if ϵ is in L(G) we cannot eliminate all ϵproduction from G, but isϵ is not in L(G) it turns out that we can The method is to determine for each variable A whether A ϵ lf so we call A mullable we may replace each production B→X X …X by all productions formed by striking out some subset of those x ‘s that are nullable but we don ‘t include B → even if allX ‘s are nullable. →lf L = L(G) for some CFG G = (V,T,P,S) than L {ϵ} is L(G) for a C F GG with no useless symbols or ϵ- productions. Algorithm for finding nullable symbols of a CFG If A→ϵis a production, then A is nullable. Then if B→∝is a production and all symbols of ∝ have been found nullable, then B nullableWe repeat this process until no more nullable symbols can be found. The set of production P is constructed as follows. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 210



If A→X X .. ....X is in P, then add all productions A→∝ ∝ …..∝ isP then add all productions 1 If X is not nullable, then ∝ =X ; 2 If X is nullable, then ∝ is either X or ϵ; 3 not all ∝ s are ϵ lf G is CFG we can find an equivalent CFGG without nullable productions except S→ϵ when in L(G) lf S→ϵ is in the production set then ‘S’ dose appear on the right side of and other production Elimination of Unit Productions: Productions of the form A→B are called unit productions i.e., productions whose right hand side consists of single variable Procedure for eliminating unit productions: For every pair of different non terminals A and B, if a CFG with no ϵ- productions has unit production A→B of if there is a chain of unit productions leading from A to B, such as A=X =X =……. =B Where X , X are some non terminals, we then introduce new productions according the following rule lf the non –unit productions from B are B→∝ /∝ /∝ …. Create the productions A→∝ /∝ /∝ …. we do the same for all such pairs of A’s and B’s simultaneously We can then eliminate all unit productions. Note : Every CFL without is defined by a grammar with no useless symbols, -productions, or unit productions

Normal Forms Chomsky Normal Form Any CFL without έ is generated by a grammar in which all productions are of the form A BC or Aa. Here A,B and C are variables and ‘a’ is a terminal. Example: Consider the grammar ({S,A,B}, {a,b}, P,S} that has the productions SbA | aB A bAA | aS | a B aBB | bS | b And find an equivalent grammar in CNF. Solution: The only productions already in proper form are Aa and Bb. There are no unit productions, so we may begin by replacing terminals on the right by variables, except in the case of productions Aa and Bb. SbA is replaced by SCbA and Cb b Similarly AaS is replaced by ACaS and Caa AbAA is replaced by ACbAA SaB is replaced by SCaB BbS is replaced by BCbS, BaBB is replaced by BCaBB In the next stage THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 211



ACbAA is replaced by ACbD1 and D1AA BCaBB is replaced by BCaD2 and D2BB The complete lists of productions are SCbA|CaB ACaS|CbD1|a BCbS|CaD2|b D1AA D2BB Caa Cbb

Greibach Normal Form Every context-free language L without έ can be generated by a grammar for which every production is of the form Aaά, where A is a variable, a is a terminal and ά is a (possibly empty) string of variables. Lemma 3: Define an A-production to be a production with variable A on the left Let G = (V, T, P, S) be a CFG. Let Aά1 Bά2 be a production in P and Bβ1|β2|…..|βr be the set of all B-productions. Let G1 = (V, T, P1, S) be obtained from G by deleting the production Aά1Bά2 from P and adding the productions Aά1β1ά2|ά1β2ά2|ά1β3ά2|……..|ά1βrά2.Then L(G) = L(G1). Lemma 4: Let G = (V,T,P,S) be a CFG. Let AAά1|Aά2|. . . . |Aάr be the set of A-productions for which A is the leftmost symbol of the right-hand side. Let Aβ1|β2|. . . . |βs. Let Aβ1|β2|. . . . |βs be the remaining A-productions. Let G1 = (Y ∪ {B}, T, P1, S) be the CFG formed by adding the variable B to V and replacing all the A-productions by the productions. 1) Aβi 2) Bαi AβiB 1≤ i ≤ S BαiB 1≤ i ≤ r Then L (G1) = L (G). By applying Lemma1 and Lemma2 we can construct GNF form -free CFL. Inherently ambiguous context free language: A context free language for which we cannot an unambiguous grammar is inherently ambiguous CFL. Ex:- L = {an bncm dm|n≥ 1, m ≥ 1} ∪ {anbmemdn|n≥ 1, m ≥ 1}. An operator grammar is a CFG with no - productions such that no consecutive symbols on the right side of productions are variables. Every CFL without has an operator grammar If all productions of a CFG are of the form AωB or Aω, then L (G) is a regular set where ω is a terminal string.




Pushdown Automata Just as the regular sets have an equivalent automaton the finite automaton, the context free grammars has their machine counterpart - the pushdown automation. The deterministic version of PDA accepts only a subset of all CFL’s where as non-deterministic version allows all CFL’s. The PDA will have an input tape, a finite control, and a stack.

Q0 Z0 Where Qo is initial state and Zo is bottom stack symbol. The language accepted by a PDA can be defined in two ways. 1. The first is the language accepted to be the set of all inputs for which some sequences of moves causes the pushdown automaton to empty stack. Definition of PDA: A PDA M is a system (Q, ,F,δ,q0,Z0,F), where 1. Q is a finite set of states; 2. Ε is an alphabet called the input alphabet; 3. F is an alphabet called the stack alphabet; 4. Q0 in Q is the initial state; 5. Z0 in F is a particular stack symbol called the star symbol; 6. F ε Q is the set of final states; 7. δis a mapping from Q*( ∪ { }) x F to finite subsets of Q x F*. Instantaneous Descriptions: To formally describe the configuration of a PDA at a given instant we define an instantaneous description (ID), we define an ID to be trible (q, w, r), where q is a state, w is a string of input symbols, and ‘γ’ a string of stack symbols. If M = (Q,E,F,δ,qo,Z0,F) is a PDA, we say (q, aw, zά) (p, w, βά) if δ(q, a, z) contains (p, β) note that a may be ε or an input symbol. We use */M for the reflexive and transitive closure of /M. That is l */ I for each ID I, and I */M j and J*/M k imply I */M K. we write 1 I K if ID i can become k after exactly I moves the subscript is dropped from |M i|M and *|M whenever the particular PDA M is understood. We can define L(M) the language accepted by a PDA M = (Q, , , δ, q0 z0, ) find state to be, {w/(q0 w, z0) * (p, , γ) for some p in F and γ in F*}. We define N(M), the language accepted y empty stack (or null stack) to be {w/( q0 w, z0) * (p , ) for some in Q}. When acceptance is by empty stack, the set of final states is irrelevant, and in this case, we usually let the set of final states be the empty set. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 213



Deterministic Pda’s A PDA M = (Q, δ q0 z0, F), is deterministic if: For each q in Q and Z in , whenever δ (q, z) is nonempty, then δ (q, a z ) is empty for all a in : 2. For non q in q Z in and a in ∪{ } does δ (q, a, z) contain more than one element Note: for finite automata, the deterministic and nondeterministic models were equivalent respect to the languages accepted. The same is not true for PDA’s DPDA’s accepts only a subset languages accepted NPDAs. That is NPDA is more powerful tan DPDA.  If L is a CFL, then there exists a PDA, m that accepts L. 1.

Construction of CFGfrom A PDAwith Empty Stack If M = (Q, , , δ, q0, z0, ϕ) is a PDA with empty stack then there exist an equivalent CFG (V, ,P,S ). If Q is {q0,q1} then V consists of S, [q0,z0, q0], [q0,z0, q1], , [q0, A, q0], , [q0, A, q1], , [q1,z0, q0], , [q1,z0, q1] , [q1,A, q0], , [q1, A, q1]. Production set P is Rule 1: S → , [q0,z0, q],for each q is Q. Rule 2: Each move erasing a pushdown symbol given by (q1, ϵ) ϵ δ ( q, a, A) induces the production ,[q, A, q1] →a. Rule 3: Each move not erasing a push down symbol given by , [q1,B1B2.....Bm) ) ϵ δ (q, a, A) induces many production of the form ) [q, A qm+1] → a [q1, B1,q2], [q2,B2,q3]. . . . [qm, Bm,qm+1] Where each of the sates q, q1, q2…..qm+1 can be any one of the states in Q, each a in ⋃ { }.

Deterministric Pushdown Automata A PDA P = (Q , , ,δ, q0,z0,F) to be deterministic, if and only if the following condition, are met: 1. δ (q,a,x) has at most one member for any q in Q , a in , or a = , and x in . 2. If δ (q,a,x) is nonempty, for some a in , then δ (q, ,x) must be empty. {wcwR | w is in (0+1)*} can be recognized by a deterministic PDA. The strategy of the DPDA is to store 0’s and 1’s on its stack, until it sees the center mark c. It then goes to another state, in which it matches input symbols against stack symbols and pops the stack if they match. If it ever finds a non match, its input cannot be of the form wcw R. If it succeeds in popping its stack down to the initial symbol, which marks the bottom of the stack then it accepts its input. Important Points 1. A language L is accepted by some DPDA with empty stack if any only if L has the prefix property (A language L has the prefix property if there are no two different strings x and y in L such that x is a proper prefix of y) and L is accepted by some DPDA by final state. 2. The languages accepted by DPDA’s by final state properly include the regular languages, but are properly include the regular languages, but are properly included in the CFLS. 3. The two modes of acceptance by final state and empty stack are not the same for DPDA’s Rather, the languages accepted by empty stack are exactly those of the languages accepted by final state that have the prefix property. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 214



4. All the regular language are accepted (by final state) by DPDA’s and there are non regular languages accepted by DPDA’s. The DPDA languages are context free languages, and in fact are languages that have unambiguous CFG’s. The DPDA languages lie strictly between the regular languages and the context free languages. 5. Deterministic CFLs are closed under complements, inverse homomorphism, 6. Intersection with regular sets and regular difference (DCFL – regular). 7. Deterministic EFLS are closed under union, concatenation, kleene closer, homomorphism and intersection.

Properties of Context Free Languages Closure properties of context free languages. Context free languages are closed under union. i.e., union of two CFL’s is again a CFL. Context free languages are closed under concatenation (Product) i.e., when we concatenate two CFL’s we will get another CFL. Context free languages are closed under Kleene closure i.e., is a CFL then L* is also CFL. Context – free languages are closed under substitution. Let L be a CFL L ⊆ *, and for each a in let La be a CFL. Let L be L (G) and for each a in let La be L (Ga). Assume that the variables of G and the Ga’s are disjoint. Construct a grammar G1as follows The variables of G1 are all the variables of G Ga’s The terminals of G1 are the terminals of the Ga’s The stat symbol of G1 is the start symbol of G. The productions of G1 are all the productions of the Ga’s together with those productions formed by taking a production A → α of G and substituting Sa the start symbol of Ga for each instance of a in appearing in α. CFL’s are closed under homomorphism: A type of substitution that is of special interest is the homomorphism. A homomorphism h is a substitution such that h (a) contains a single string for each a. we generally take h(a) to be the string itself, rather than the set containing that string. As CFL’s are closed under substitution, we can say that CFL’s are closed under homomorphism. The context free languages are closed under inverse homomorphism. If ha is a homomorphism, and L is any language, then h-1 (L) is the set of strings w such that h(w) is in L. The PDA to accept the inverse homomorphism of what a given PDA accepts.



Input a

h


h(a) P DA state

Accept / Reject

The key idea in this diagram is that after input a is read, h (a) is placed in a “buffer”. The symbols of h(a) are used one at a time and fed to the PDA being simulated. Only when the buffer is empty does the constructed PDA read another of its input symbols and apply the homomorphism to it. If L is a CFL and R is a regular language, then L ∪ R is a CFL: A PDA and a FA can run in parallel to create a new PDA as shown below.

FA state

AND

Accept/Reject

PDA state

stack

If L is a CFL’s and R is a regular language then L - R is a context – free language If L is a CFL’s and R is a regular language then Intersection between these two is a context – free language If ‘L’ is a CFL then LR (Reversal of L) is a CFL. Let L = L (G) for some CFG G = (V,T,P,S). Construct. GR = (V,T.PR,S), where PR is the “reverse” of each production in P. That is, if A → α is a production of G, then A→αR is a production of GR An easy induction on the lengths of derivations of sentential forms of G, and vice versa.




CFL’s are closed under INIT operation. INIT (L) = {w | for some x, wx is in L}. ‘L1’ and ‘L2’ are two CFL’S then L1 ∩L2 may or may not CFL . That is CFL’s are not closed under intersection. May CFG All regular languages are context- free. The intersection of two regular languages is regular therefore, if L1and L2 are regular and context – free, then L1 ∩L2 is both regular and context free. May not CFG Let L1 = {anbnam| where n, m = ,1,2,3……., but n is not necessarily the same as m} Let L2 = {anbmam| where n, m = ,1,2,3……., but n is not necessarily the same as m} Both the languages are context – free, but their intersection is the language L3 = L1 ∩ L2 {anbnan for n = 1,2,3……} Which is not a CFL (we will prove it later). Therefore, the intersection of two context – free languages can be non – context- free The Complement of a context-free language may or not be context-free.

Decision Algorithms for CFL’s: 1. Given a CF G G = (V,T,P,S) there exists an algorithm for deciding whether or not L(G) is empty, finite or infinite.  Assume the given language does not contain . Find the reduced grammar (i.e., eliminate useless productions, - productions and unit productions) of the given grammar. 1. If the reduced grammar vanishes then the given language is empty. 2. Draw a graph with the productions of the reduced grammar. If the graph contains cycle the given grammar generates infinite language, otherwise it generates finite language.

Non-Context-Free Languages Important points: 1. Let G be a CFG in CNF. Let us call the productions of the form Non-terminal → Nonterminal Nonterminal; Live and the productions of the form Nonterminal → terminal; Dead. 2. If G is a CFG in CNF that has p live productions and q dead productions, and if w is a word generated by G that has more than 2p letters in it, then somewhere in every derivation tree for w there is some non-terminal being used twice where the second z is descended from the first z. 3. In a given derivation of a word in a given CFG, a non-terminal is said to be self-embedded if it occurs as a tree descendent of itself.




The Pumping Lemma for CFLs Pumping Lemma for CFL’s is used to show that certain languages are non context free. There are three forms of pumping lemma. Standard form of pumping lemma:- Let “L” be any infinite CFL. Then there is a constant such that i) |vx| ≥ 1, ii) |vwx| ≤ n, and iii) for all i ≥ 0, uv’wx’y is in L. Strong form of pumping lemma (Ogden’s Lemma): Let L be an infinite CFL. Then there is a constant n such that if z is any word in L, and we mark any n or more positions of z “distinguished”, then we can write Z = uvwxy such that v and x together have atleast one distinguished position, vwx has atmost n distinguished positions, and for all i ≥ 0, uv’wx’y is in L. Weak form of pumping Lemma: Let L be on infinite CFL. When we pump the length of strings are |uvwxy| = |uwy| + |vx| |uv2wx2y| = |uwy| + 2|vx| ………………………… |uv’wx’y| = |uwy| + i|vx| when we pump the lengths are in Arithmetic progression

Membership Algorithm For Context – Free Grammars Assume that we have a grammar G = (V, T, S, P) in Chomsky normal form and a string w = a 1 a2 …an We define sub strings wij = ai……aj and subsets of V Vij = { A V : A

Wij}

Clearly w ϵ L (G) if and only if s ϵ V1n. To compute Vij observe that A ϵ Vii can be computed for all 1 ≤ i≤ n by inspection of w and the productions of the grammar. To continue, notice that for j >i, A derives wij if and only if there is a production A → BC, with B

wik and C

W(k + 1)j for

some k with i ≤ k < j. In other words, Vij* = ∪ {A : A → BC, with B ϵVik, C ϵ V(k + 1)j}. kϵ { i,i + 1 , ….j – 1} All the Vij ‘s can be easily computed if we follow the following sequence. Compute V11, V22, …….Vnn Compute V12, V23, …….Vn -1, n Compute V13, V24, …….Vn – 2, n And so on.




6.5: Turing Machines A tuning machine is an automaton whose temperature storage is a tape. This tape is divided into cells, each of which is capable of holding one symbol. Associated with the tape is a read-write read that can be travel right or left on the tape and that can be read and write a single symbol on each move A diagram giving an intuitive visualization of turning is … … .

… … .

an

B

… … ….

Finite Control Definition: A Turing machine M is defined by M = (Q,ε, δ q B, F) where Q is the inite set of internal states is the finite set of allowable tape symbols B a symbol of is the blank εis a subset if not including B is the set of input symbols, δis the next move function a napping from Q x to Q x x {L,R} (δ may however undefined for some arugements) Q0 in Q is the start state, F Q is the head of final states We denote instantaneous description (ID) of the Turing machine M by α qα Here q the current state of M is in Q, α α is the string in we assume that Q and are disjoint to avoid confusion Finally the tape head is assumed to be scanning the left most symbol of α or if α = ,the head is scanning a blank We define a move of M as follows Let x1,x2,x3……xi ….qx….xn be an ID suppose δ(q, x) = (P, , L) where if i-l=1=n then x is taken to be B if i=1 then there is no next ID as the tape head is not allowed to fall off the left end of the tape if i>1 then we write x1,x2…….xi-1qx1….xn However if any suffix is completely blank that suffix is deleted in (1) Note that in the case i-1=n the string x1…..xn is empty and the right side of (2) is longer than the left side if two ID’s are related by M say that the second results from another by some finite number of moves including zero moves they are related by the symbol M The language accepted by M denoted L (M) is the set of those words in that cause M to enter a final state when placed justified at the left on the tape of M with M is state q0 and the tape head of M at the left most cell .Given a TM recognizing a language L we assume without loss of generality that the TM halts i.e., has no next move whenever the input is a accepted However for words not accepted it is possible that the TM will never halt.




Modification of Turing Machines 1.

Two – Way infinite TM: L is recognized by a turing machine with a two-way infinite tape if and only if it is recognized by a TM with one-way infinite tape

……. .

a

a2

………… ……….

a1

B

…… ….

…… …

Finite control 2. Multitape – TM: If L is a language, accepted by a multi tape turing machine it is accepted by a single-tape machine Finite control

……… ……… ……… ………

…… …… …… …… …… ……

……… ……… ……… Fig: Multitape TM 3. Non – deterministic TM:If L is accepted by a non – deterministic turing machine M1,then L is accepted by some deterministic turing machine M2. 4. Multi Dimensional TM:In K-dimensional TM the tape consists of K- dimensional array cells infinite in all 2K direction for some fixed K. If L is accepted by a K-dimensional turing machine M1, then L is accepted by some Single tape turing machine M2. 5. Multi Head Turing Machine: A K- head TM has some fixed numer K of heads The heads are numbered I through K and a move of TM depends on the state and on the symbol scannd by each head In one move the haeds may move independently left right or remain stationery If L is accepted by some K-head by some K-head TM M1, it is accepted by a one head TM M2,




…………… …

…………… … Finite control

Fig: K – head TM 6. MULTI TRACK TURING MACHINE: we can imagine that the rope of the TM is divied into k tracks, for any finite k.

k-tracks

Finite control 7. TURING MACHINE WITH STAY OPTION: In these TMs the read-writehead can stay atthe current position upon reading an input symbol (possibly changing) without moving left to right. 8. OFF-LINE TURING MACHINE: An off-line TM is a multitape TM whose input tape isread-only. Usually we surround the input by end makers’ c on the left and s on the right. The turing machine is not allowed to move the input tape head off the region between c and s it should be obvious that the off-line TM is just a special case of the Multiple TM. An off-line TM can simulate any TM M by using one more tape than M. The first thing the off-line TM does is copy its own input onto the extra tape and it then simulates M as if extra tape were M’s input. All these Modifications does not add any language accepting power and all these are equivalent to the Basic model. POST MACHINE: A post machine denoted PM, is a collection of five things: 1. The alphabet of input letters plus the special symbol 2. A linear storage location (a place where a string of symbols is kept) called the STORE or QUEUE which initially contains the input string. We allow for the possibility that characters not in can be used in the STORE, characters from an alphabets called the store alphabet. 3. Read states, for example which remove the left most character from the STORE and branches accordingly? The only branching in the machine takes place at the Read states. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 221


There may be a branch for every character in from the READ have the same label.


or PMs are deterministic so on two edges

#

a

read

b

4. ADD states

ADD

a

ADD

b

ADD

#

Which concatenate a character onto the right end of the string in the STORE. This is different Form PDA pushes state .No branching can take place at an ADD state.It is possible to have an ADD state for every letter in and 5.

A start state (unutterable) and some halt states called Accept and REJECT

start

ACCEPT

REJECT

We could also define a Non deterministic post machine NPM. This would allow for more than one edge with the same label to come from a READ state. In their strength NPM-PM. Two - Stack Machine: A two-push down stack machine a 2PDA is like a PDA except that it has two push down STACKS,STACKS1,STACKS2.When we wish to push a character x into a stack, we have to specify which stack either PUSH1 x or PUSH2 x . When we POP a STACK for the purpose of branching we must specify which STACK either POP1 or POP2 (Read a character from read-only input tape) the functions of start, Read, Accept and Reject are same as in the post machine.




Counter-Machines: A counter machine may be thought of in one or two ways: 1. The counter machine has the same structure as the multistack machine but in place of each stack is a counter .Counters hold any nonnegative integer but we can only distinguish between zero and nonzero counters. That is the move of the counter machine depends on its state input symbol and which if any of the counter are zero . In one or more the counter machine can a. Change state b. Add or subtract 1 form any of its counters, independently. However a counter is not allowed to become negative, so it cannot subtract from a counter that is currently 0 2. A counter machine may also be regarded as a restricted multistack machine. The restrictions are as follows. a. There are only two stack symbols. Which we shall refer to as z (the bottom of stack marker), and X. c. Zo is initially on each state d. We may replace Z0 only by a string of the form XZ0 for some i ≥ 0. e. We may replace X only by X’ for some i ≥ 0That is Z0appears only on the bottom of each stack and all other stack symbols if any are X. The two definitions clearly define machines of equivalent power 1. Every recursively enumerable language is accepted by a three counter machine( we can simulate two stacks by 3-counters only) 2. Every recursively enumerable language is accepted by a two-counter machine(we can simulate two stacks by 2-counters only) 3. Every recursively enumerable language is accepted by a 3-pebble machine (3-pebble machine are sufficient to simulate two counters).

….

….

….

….

….

….

…… .

3-PEBBLES

Storage in the State: We can use the finite control not only to represent a position in the ‘program’ of the turing machine but to hold a finite amount of data. The technique requires no extension to the TM model we merely think of the state as a tuple.



State

q A


B

C

Storage

……… … The transition function of above TM is as follows: 1. δ([q , B], a = ([q , a], a, R) For a=0 or a=1initially is the control state , and the data in state is B. The symbol scanned is copied into second component of the state and M moves right entering control state as it does so. 2. δ([q , a], a = ([q , a], a, R) wherea is the “complement” of a that is 0 if a=1 and 1 if a=0 in state M skips over each symbol 0 or 1 that is different from the one it has stored in its state, and continues moving right 3. δ([q , a], B = ([q , B], B, R)for a=0 or a=1 if M reaches the first blank it enters the accepting state [ , ]. Notice that M has no definition for δ([q , a], a) for a=0 or a=1 Thus if M encounters a second occurrence of the symbol stored initially in its finite control it halts without having entered the accepting state,

Multiple Tracks: Another useful “ trick” is to think of the tape of TM as composed of several tracks each track can hold one symbol and tape alphabet of the TH consists of tuples, eith one component for each track. Like the technique of storage in the finite control, using multiple tracks does not extend what the TM cando. A common use of the multiple track is to treat one track as holding the data and a second track as holding a mark. We can check of each symbol as we use it or we keep track of a small number of positions within the data marking those positions. Finite control

X

Track 1 Track 2 Track 3

Y X




Universal Turing Machine: An universal TM Mo is an automation that given as input the description of any M. and a string w. can simulate the computation of M on we to occur construct such an Mo. We first choose a standard way of description turing machines we maywithout of generality, assume that. Q = {q ,q q } With q the initial state q the single final state and ={a. , a , … … … … . . a } Where a represents the blank. We then select an encoding in which q is represented 1 q is represented by 11 and so on. Similarly a is encoded as 1 a as 11 etc. the symbol ‘O’ is used as a separator between the 1’ s wit the initial and final stare and the blank defied by this conversion any turingmachine can be descried withδ only. The transition function is encoded according to this scheme, with the arguments and result in some prescribed sequence For example δ(q , a ) = (q a , L)might apper as ……………… 10110110111010………………… It follows from this that any Turing machine has finite encoding as a string on {0,1} +, and that any encoding of M, we can decode it uniquely. Some strings will not represent any Turing machine (e.g) the string 00011), but we can easily spot these, spot these, so they are of no concern. A universal TM M , then has an input alphabet that includes {0, 1} and the structure of a multitape machine as shown below . Control Unit of

Description of M

Internal states of M

Tape contents of M For any input M and W tape will keep an encoded definition of M, tape 2 will contain the tape contents of M and tape 3 the internal state of M Mu looks first at the contents of tape 2 and 3 to determine the configuration of M. it then consults tape1 to see what M would do in this configuration. Finally tapes 2 and tapes 3 will be modified to reflect the result of the move This implementation clearly can be done using some programming languages. There, we expect that it can also be done by a standard Turing machine Context sensitive grammar: A grammar G=(V,T,P,S) is said to be context-sensitive if all production are of the form α → β Where α, β ε (VUT)+and |α|≤ |β| THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 225



This definition shows clearly one aspects of this type of grammar. It is non contracting in the sense that the length of successive sentential forms can never decrease. Context sensitive language: A language is said to be context sensitive if there exists a context sensitive grammar G, such that L = L (G) or L=L(G) U {ε} Context sensitive grammar does not contain productions of the form αε so, that a contextsensitive grammar can never generate a language containing the empty strong By including the empty string in the definition of a context-sensitive language, we can claim that the family of context-free language is a subset of the family of context sensitive language

Linear Bounded Automata A linear bounded automata (LBA) is a non deterministicTuring machine satisfying the following two conditions. 1. Its input alphabet includes two special symbols C and $, the left and right end markers, respectively 2. The LBA has no moves left from C or right from $, nor it may print another symbol over C or $. The linear bounded automaton is simply a Turing machine which, instead of having potentially infinite tape on which to compute, is restricted to the portion of the tape containing the input & plus the two tape squares holding the end markers . Restricting the Turing machine to an amount of tape that, on each input is bounded by some linear function of the length of the input would result in the identical computational ability as restricting the turing machine to the portion of the tape containing the input- hence the name “linear bounded automation” An LBA will be denoted M = (Q. , ,δ, q , C, $, F ), where (Q. , ,δ, q , C, $, F ). Were Q. , ,δ, q , and F are as for a Nodeterministic TM; C and $ are symbols in the left and right end markers L (M) , the language accepted by M, is {w|w is in ( - {C, $ })* and q c w $

∝ qβ for some q in F}

M 1. If L is a CSL, then L is accepted y some LBA. 2. IF L = (M) for LBAM = ((Q. , ,δ, q , C, $, F ) then L – { } is a CSL 3. Every CSL is recursive but converse is not true.  A string is accepted a LBA if there is a possible sequence of moves q C w$ C ∝ q ∝ $ For some q F, ∝ , ∝ *.The language accepted by the LBA is the set of all such accepted strings. Hierarchy of formal languages (Chomsky Hierarchy ) 1. Unrestricted grammars (or Type 0 of grammars ): A grammar G = (V, T, P, S) is called unrestricted if all the production are of the form ∝β + Where ∝ is in (VUT) and β is in (VUT)* Any language generated by an unrestricted grammar is recursively enumerable.




2. Content sensitive grammars (or type 1 grammars ): A grammar G = (V, T, P, S ) is said to be context sensitive if all productions are of the form ∝β Where ∝ B, (VUT) + and |∝| ≤|β| (Except for S and start symbol S does not appear on the right hand side of any production) 3. Context free grammars (Tupe2 grammars): A grammar G = (V,T,PS) is said to be context – free if all production in P the form A∝, Where A V and ∝ (VUT)* 4. Regular grammar: A right linear grammar consists of production of the form AxB, Ax where A,B ε V and x T*. A left linear grammar consists of productions of the form AB x Ax where A,B ε V and x ε T*. Type 0 Type 1

Type 2

TYPE3 OR REGULAR GRAMMAR

OR CONTEXT FREE GRAMMAR OR CONTEXT SENSITIVE GRAMMAR

OR UNRESTRICTED GRAMMAR

TYPE3 TYPE2 TYPE1 TYPE 0




UNDESIDABILITY: Recursive language: A language L over the alphabet Σ is called recursively enumerable if there is a TM T that access every word in L Recursive enumerable language: A language L over the alphabet Σ is called recursively enumerable if there is a TM T that accepts Every word in L the either rejects or loops for every word in the language L , the complement L. Non recursively enumerable language : For any nonempty Σ there exists languages there exist language that are not recursively enumerable Suppose we have a list of (0+1)* in canonical order(if = {0,1}, The canonical order is 0,1,00,01,10,11,000,000…}Where W is the ith word, and M is the TM whose code is the integer j written in binary, imagine an infinite table that tells for all I and j whether W is in L(M ) J

i

1

2

3

4

…………..

1

0

1

1

0

……………

2

1

1

0

0

…………..

3

0

0

1

0

…………..

4

0

1

0

1 Diagonal

We construct language L by using the diagonal entries of the table to determine membership inL To guarantee that no TM accepts L .We insist that W is in L if and only if if the (i.i) entry is 0. that is , if M doesnot accept W Suppose that some TM MJ accepted L . Then we are faced with the following contradiction if W .is in L then (j.j) entry is 0, implying that w is tin L(M ) and contradiction if L = L(M ) or the other hand if W is not in L the (j, j) entry is 1, implying that W is I l(M ) , which again contradicts L = L(M ) , as W is either in or not in Ld we conclude that our assumption L = L(M ) is false. Thus no TM in the list accepts L . The universal language: Define L , the “universal language “ to be { / M accepts w} we call L “ universal since the question of whether any particular string W in ( 0+1)* is accepted by any particular TM.. M THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 228



is equivalent to the question of whether is in Lu. Where M’ is the one tape TM with tape alphabet {0,1,B} equivalent to M. Lu is recursively enumerable but not recursive Ld is recursively enumerable but not recursive Relationship between recursive, recursively enumerable and non enumerable languages

Non recursive enumerable

Recursively enumerable

Recursive

But not recursive enumerable

Reductions

If there is a reduction from p1 to

then:

a) If p1 is un decidable then so is b) If p1 is non-RE, then so is

yes

yes N0 o

no

P1

P2




Turing machines that accept empty language: (L∅ ) L = { all encoded TM’s whose language is empty } or { M/L(M)=ø} (L ∅ ) L ={ all encoded TM’s whose language is not empty} or {M/L(M)≠ø} i.eL is the language of all codes for TM’s that accept at least one input string . L is recursively enumerable. We have only to exhibit a TM that accepts L Accept

guessed W U

M

Accept

Accept

M for

Fig: construction of NTM to accept The operation of M is as follows 1. M takes as input a TMcode M 2. Using it’s non determine capability. M guess an input W that M might accept 3. M tests whether Mi accepts W for this part M can simulate the universal TM U that accepts Lu 4. If Mi accepts wi, then W accepts it’s own input , which isM in this manner, If M1accepts even one string M will guess that string and accept Mi. however, if L(M )=ø then no guess w leads to acceptance by M . So M does not accept M thus L(M)=L . L is non recursively enumerable Let A be a hypothetical algorthim accepting L . There is an algorithm B that given Constructs a TM M that accepts ø if M does not accept W and accepts (0+1)* if M accepts w. the plan of the construction ofM is as shown below W

accept

accept

M X

M’




M ignores it’s own input x and instead simulates M on input W, accepting note that M is not B. Rather B is like a compiler that takes as “ source program” and produces M’ as “ object program “ the construction of B is simple, it takes and isolates W say W = a a …….. a is of length n, B creates n+3 states q , q ,-----------q +3. With moves δ(q X) = (q ,$, R) for any X (print maker), δ(q ,X)=(q +1, a , R) for any X and 2≤ 1 ≤ n+1 (print w) δ(q ,X) = (q ,B,R) for X≠B (erase tape) δ(q ,B) = (q ,B,L) δ(q ,X) = δ(q ,,B,L) for X ≠$ (find maker) having produced the code for these moves B then adds n+3 to the indices of the states of M and includes the move δ(q , $) = δ(q , $,R) /* state up M*/ and all the moves of M it is generating TM Now suppose algorithm A accepting L shown below.

exists. Then we construct an algorithm C for L

as

yes yes (M,w)

A

B

No No

Fig: algorithm constructed for

assuming that algorithm A for

exists.

As L is RE it doesn’t have algorithm . thusL is not recursive. If L were recursive, L would also. Thus L is R.E but not recursive, If L were R.E then L and L would be recursive . ThenL is not recursively enumerable .

Yes start w

M

yes

X

Fig: TM




As in the previous example, we have described the output of A. we leave the construction of A to the reader. ̅̅̅ as shown below Given A and M we could construct a Tm acceptingL

yes yes

A

Which behaves as follows. On input the TM uses A to produce M , uses M to determine if the set accepted by M is recursive, and accepts if and only if L(M ) is recursive iff L(M ) = ∅, which mean M does not accept w. thus the above hypothetical TM accepts off is in L Now let us turn to L . Suppose we have TM M accepting L and an algorithm B, to be ̅̅̅ B takes as input and produces as output a TM constructed by the reader, to accept L M such that L(M ) = if M accepts w, L if M does not accept w.

M

yes

yes

yes

X

yes

B

Thus M accepts a recursive language iff M accepts w. M which B must produce, is shown in fig (a) and TM to accept ̅̅̅ L given B and M is shown in fig (b) The TM of fig (b) accepts iffL(M ) is not recursive or equivalently, if and only if M does ̅̅̅ since we have already shown that not accept w. i.e., the TM accepts iff is in L




no such Tm exists , the assumption that M exists is false. We conclude that L is not recursively enumerable. Rice theorem for recursive index sets: We can’t decide if he set accepted by a turingmachine is empty, recursive, finite, infinite regular, context free or has an even number of strings, or satisfies many or satisfies many other predicates. We can decide only trivial predicates,“ does a TM accepts a R.E which are either true for all Turing machines or false for all TM’s. Rice’s theorem: any nontrivial property of the r.e languages is undecidable. Rice’s theorem for recursively enumerable index sets The condition under which a set L is R.E is for more complicated. We shall show that L is R.E iff satisfies the following three conditions. 1. If L is in and L L , for some R.EL then L is in 2. If L is an infinite language in , then there is a finite subset of L in 3. The set of finite languages in is enumerable, in the sense that there is a Tm that generates the (possibly) infinite string code1# code2#........., where code1 is a code for the ith finite language in (in any order). The code for finite languages { w , w , … w } is just w ,w ,…w . IMP Points: Point 1: If does not have the containment property then, L is not R.E Point2: If has an infinite language L such that no finite subset of L is in , than L is not R.E Point3: If L is R.E, then the of binary codes for the finite sets in L is enumerable L isR.Eiff 1. If L in and L L for some R.E, L then L , is in . 2. If L is an infinite set in then there is some finite subset L of L that is in . 3. The set of finite languages in is enumerable The following properties of R.E sets are not R.E (A) L = ∅ (B) L = * (C) L is recursive (D) L is not recursive (E) L is a singleton (has exactly one member) (F) L is a regular set (G) L – Lu = ∅ Proof: in each case condition 1 is violated, except for (B) where 2 is violated and (G), where 3 is violated. The following properties of R.E sets are R.E (A) L ≠ ∅ (B) L contains at least 10 members. (C) W is in L for some fixed word w. (D) L ∩ Lu ≠ ∅




Post correspondence problem: It is a valuable tool in establishing other problems to undecidable. An instance of post correspondence problem (PCP) consist of two lists, A = w , . . w and B = x , … x of strings over some alphabet . This instance of PCP has a solution if there is any sequence of integers i , i , … i with m ≥1, such that w w ….w = x x …x The sequence i , i , … i is a solution to this instance of PCP . In the following table D means decidable,U means undecidable, T means Trivially decidable and ? means open question( answer is not known).

Question

Regular Sets D

DCFL's

CFL's

CSL's

D

D

D

D

D

2. is L=∅? (Emptiness problem)

D

D

D

U

U

U

3.is L = *? (completeness problem)

D

D

D

U

U

U

4. is L1 = L2? (equality problem)

D

?

U

U

U

U

5.is L1⊑ L2 ? (subset problem)

D

U

U

U

U

U

6. is L1 ∩ L2 = ∅?

D

U

U

U

U

U

7.i s L - R, where R is a given regular set

D

D

U

U

U

U

8.is L regular?

T

U

U

U

U

U

9.i s the intersection of two languages, a language of the same type?

T

U

U

T

T

T

1. is w in L? (membership problem)

Recursive Sets r.e Sets




10. is the complement of a language also a language of the same type

T

T

u

?

T

U

11.is L finite or infinite?

D

D

D

U

U

U

DCFL's

CFL's

Closure properties of formal languages Regular Sets

CSL's

Recursive Sets ✓

R.E Sets

1. Union

✓

X

✓

✓

2. Concatenation

✓

X

✓

✓

✓

✓

3. Kleene closure

✓

X

✓

X

X

✓

4. Intersection

✓

X

X

✓

✓

✓

5. complementation.

✓

✓

X

?

✓

X

6. Homomorphism

✓

X

✓

X

X

✓

7. Inverse Homomorphism

✓

✓

✓

✓

✓

✓

8. Substitution

✓

X

✓

✓

X

✓

9. Reversal

✓

X

✓

✓

✓

✓

10. Intersection with regular sets

✓

✓

✓

✓

✓

✓

11. Quotient regular sets

✓

✓

✓

X

X

✓

with

✓

The Classes P &NP  P consists of all those languages or problems accepted by some deterministic Turing Machine that runs in some polynomial amount of time, as a function of its input length  NP is the class of languages or problems that are accepted by non deterministic TM's with a polynominal bound on the time taken along any sequence of nondeterministic choices.  The P » NP question : it is un known whether or not P and NP are really the same classes of languages, although we suspect strongly that there are languages in NP that are not in P.




 Polynominal-Time Reductions: If we can transform instances of one problem in deterministic polynominal time into instances of a second problem that has the same answer -yes or no - then we say the first problem is polynomial time reducible to the second.  NP Complete problems; A language is NP-Complete if it is in NP, and there is a Polynominaltime reduction from each language in NP to the language in question. We believe strongly that none of the NP-complete problems are in P, and the fact that no one has ever found a polynominal time algorithm for .any of the thousands of known NP-complete problems is mutually re-enforcing evidence that none are in P.  If P| is NP~complete, and there is a. poly nominal-time reduction of P1 to P2then P2 is NPcomplete.  If some NP-complete problem is in P then P = NP. NP~ Hardproblem: If there is a language L such that every language L1 in NP can be polynominally reducible to L and we cannot prove that L is in NP then L is said to be NP-hard problem. (Note;If we can prove that L is in NP and every NP problem can be poly nominally reducible to L then L is said to be NP-complete problem) Some of the NP - Complete problems: 1. (cook's Theorem) Boolean satisfiability problem simply SA T is NP-complete. 2. CSAT,3SAT Problems are NP-Complete. 3. Traveling sales man problem (TSP) is NP-complete. 4. Hamilton circuit and directed Hamilton circuit problems are NP-complete. 5. Vertex cover problem: which asks us to find the smallest set of nodes that cover all the edges, in the sense that at least one end of every edge is in the selected set., is NP-complete. 6. Independent set problem :Given a graph, find a maximal independent set. This problem is NP-complete. 7. Chromatic Number Problem is NP-complete. 8. The partition problem: Given a list of integers i1 , i2, …..ik, does there exist a subset whose sum is exactly ½ (i1 , i2, …..ik). Note that this problem appears to be in P until we remember that the length of an instance is not i1 , i2, …..ik, but sum of the length of the i1’s written in binary or some other fixed base. 9. A K-Clique in a graph G is a set of k nodes of G such that there is an edge between every two nodes in the clique. CLIQUE isNP-Complete. 10. The coloring problem: Given a graph G and an integer k, is G "K- colorable", that is can we assign one of k-colors to each node of G in such a way that no edge has both of its ends colored with the same color. The coloring problem is NP-Complete. 11. The subgraph - isomorphism problem :given graphs G1 and G2does G1 contain a copy of G2as a subgraph ? That is, we can find a subset of the nodes of G1 that, together with the edges among them in G1, forms an exact copy of G2when we choose the correspondence between nodes of G2and nodes of the subgraph of G1 properly? The subgraph - isomorphism problem is NP-complete. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 236



12. The edge cove problem : Given a graph G and an integer k, does G have an "edge cover" of k edges, that is, a set of k edges such that node of G is an end of at least one edge is the edge cover. The edge cover problem is NP-complete. 13. The linear integer programming problem is NP-complete. 14. The dominating set problem .-.Given a graph G and an integer k, does there exist a subset S of k nodes of G such that each node is either in S or adjacent a node of S? This problem is NP complete. 15. The half-clique problem :Given a graph G with an even number of vertices, does there exist a clique of G consisting of exactly half the nodes of G? The half-clique problem is NP-complete. 16. The unit-execution time scheduling problem is NP - complete. 17. Exact cover problem: Given a set S and a set of subsets S1,S2, ....Sn of S, is there a set of subsets T ⊆ {S1, S2,...Sn} such that each element x of S is inexactly one member of T? Exact cover problem is NP-Complete. 18. The knapsack problem is NP-Complete. 19. Given graphG and an integer k, does G have a spanning tree with at most k leaf vertices. 20. Given graph G and an integer d, does G have a spanning tree with no node of degree greater than d ?. (The degree of a node n in the spanning tree is the number of edges of the tree that have n as an end).This problem is NP-Complete. 21. Do two FA's with the same input alphabet recognize different languages isNP-Complete. 22. Do two R.E E1on E2over the operators (+,.,*) represent "different languages is NP- Complete. 23. Do two regular grammars G1and G2generate different languages is NP-Complete. 24. Does a given CFG generates a given string x is NP-Complete. 25. Satisfiability, CNF-satisfiability problems are NP- complete. Some of the NP-Hard Problems: 1. Halting problem is to determine for an arbitrary deterministic algorithm A and an input I whether algorithm A with input I ever terminates. It is well known that this problem is undecidable. Hence, there exist no algorithm (of any complexity) to solve this problem. So, it clearly can't be in NP. Two problem L1 and L2 are said to be polynomially equivalent if and only if L1 is polynomilally reducible to L2 and viceversa. Only a decision problem can be NP - complete. However, an optimization problem may be NP-hard. Further more if L1 is a decision problem and L2 an optimization, it is quite possible that L1 and L2 (Li is polynomially reducible to L2). Knapsack decision → Optimization Clique decision → Optimization Yet optimization problems can't be NP-complete where as decision problems can. There also exist NP - Hard decision problems that are not NP-complete. → To show that a problem L2 is NP-hard, it is adequate to show L1 and L2, where L1 is some problem already known to be NP-hard. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 237



Intractable Problems: *The problems solvable in polynomial time on a deterministic TM are tractable. * The problems which require more than polynomial time on a deterministic TM are intractable. The Class of languages Co-NP P is closed under complementation but its is not known whether NP is closed under complementation .A suspected relationship between Co-NP and other classes of languages is shown below NP-Complete Problems

NP P Co-NP

Complements of NPComplete Problems * If P = NP then P, NP and Co-NP all are same. *NP = Co-NP if and only if there is some NP-Complete problem whose complement is in NP.



Computer Organization

Part – 7: Computer Organization 7.1: Introduction to Computer Organization

 Computer architecture deals with the structure and behavior of the computer system.  It includes the information formats, the instruction set and the hardware units that implement the instructions alongwith the techniques for addressing memory.  Computer organization deals with the way the various hardware components operate and the way they are connected together to form the computer system.  It also deals with the units of the computer that receive information from external sources and send computed results to external destinations.  Computer design is concerned with the hardware design of the computer. This aspect of computer hardware is sometimes referred to as computer implementation. Basic blocks of a Computer System  Input Unit: It is a medium of communication between the user and the computer. With the help of input unit, it is possible to give programs and data to the computer. Examples: Keyboard, floppy disk drive, hard disk drive, mouse, Magnetic Ink Character Recognition (MICR), Optical Character Recognition (OCR), paper tape reader, Magnetic tape reader, Scanner etc.  Output Unit: It is a medium of communication between the computer and the user. With the help of output unit only it is possible to take results from the computer. Example: Printers, Video Display Unit (VDU), Floppy disk drive, Hard disk drive, Magnetic tape drive, punched cards, paper tape, plotter, digitizer etc.  Memory: The memory unit is responsible for storing the user programs and data as well as system programs. The digital computer memory unit consists of two types of memories: Read Only Memory (ROM) and Read Write Memory (R/WM) or Random Access Memory (RAM).  ALU: All arithmetic and logical operations are performed within this unit.

 Control unit: It is used to generate necessary timing and control signals to activate different blocks in the computer to perform the given task.  Central Processing Unite (CPU): The ALU and Control Unit together are called CPU. It is the heart of any digital computer. Byte Ordering or Endianness When computers try to read or store multiple bytes. Where does the biggest byte appear? THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 239





Big endian machine: Stores data big-end (MSB) first. When looking at multiple bytes, the first byte (lowest address) is the biggest.



Little endian machine: Stores data little-end (LSB) first. When looking at multiple bytes, the first byte is smallest.

Memory Unit  ROM:ROM is used to store permanent programs or system programs. It does not have write capability. Types: PROM, EPROM, EEPROM  RAM: It is also called user memory because the user programs or application programs are stored in this memory. The CPU is able to write or read information into or from this type of memory. Types: static, dynamic, scratch pad etc.

ALU (Arithmetic Logic Unit) The data processing part of CPU is responsible for executing arithmetic and logical instructions on various operand types including fixed point and floating point numbers. 1. Combinational ALU’s: The simple ALU’s combine the functions of 2’s complement adder-subtracter with those of a circuit that generates word based logic functions of the form f(x, y). 2. Sequential ALU’s:  Both multiplication and division can be implemented by combinational logic. It is generally impractical to merge these operations with addition and subtraction into a single, combinational ALU.  The combinational multipliers and dividers are costly in terms of hardware. They are also much slower than addition and subtraction. Circuits, a consequence of their many logic levels.  The bus lines carry data, address and control signals.Since, this bus can be used only for single transfer at a time multiplebuses are introduced so as to achieve more concurrency in operations by allowing two or more transfers to be carried out at the same time. Hereby, increasing performance but at an increased cost.  The internal organization of a digital system is defined by the sequence of primitiveoperations; it performs on the data stored in its registers.  In a special purpose digital system, the sequence of micro-operation is fixed by the hardware and the system performs the same specific task over and over again.  The data for the digital computer can be represented most frequently in two different ways a) Fixed point representation, and b) Floating point representation  While storing the signed binary numbers in the internal registers of a digital computer, most significant bit position is always reserved for sign bit and the remaining bits are used for the magnitude representation.




 When the binary number is positive, the sign bit is represented by ‘0’. When the binary number is negative, the sign bit is represented by ‘1’.  The representation of the decimal point (or binary point) in a register is complicated by the fact that it is characterized by a position between two flip- flops in the register. There are two ways of specifying the position of the decimal point in a register. 1. Fixed Point

2. Floating Point

 The fixed point method assumes that the decimal point (or binary point) is always fixed in one position. The two positions most widely used are (1) a decimal point in the extreme left of the register to make the stored number a fraction, and (2) a decimal point in the extreme right of the register to make the stored number an integer.  Negative number can be represented in one of three possible ways Signed- magnitude representation Signed -1’s complement representation Signed -2’s complement representation  The 2’s complement of a given binary number can be formed by leaving all least significant zeros and the first non-zero digit unchanged, and then replacing 1’s by 0’s and 0’s by 1’s in all other higher significant digits.  Subtraction using 2’s complement represent the negative number in signed 2’s complement form. Add the two numbers, including their sign bit and discard any carry out of the most significant bit.  Since negative number are represented in 2’s complement form, negative results also obtained in signed 2’s complement form.  2’s complement form is usually chosen over 1’s complement to avoid the occurrence of a negative zero.  The 1’s complement of 1’s complement of a given number is same number.  The general form of floating point number is Smr .Where S= sign bit, M= Mantissa, r = base, e = exponent.  The Mantissa can be a fixed point fraction or fixed point integer  Normalization: Getting non-zero digit in the most significant bit or digit position of the mantissa is called Normalization.  It is possible to store more number of significant digits as a result accuracy can be improved, if the floating point number is normalized.  A zero can not be normalized because it does not contain a non- zero digit.  The hexadecimal code is widely used in digital systems because it is very convenient to enter binary data in a digital system using hexcode. There are mainly two types of numbering systems: a) Non positional number systems

b) Positional number systems

Roman numeral system is an example of a Non positional number system. The most widely used number system is Positional number system: Positional number system has a ‘radix’ or ‘base’. A number with radix ‘r’ is represented as an an-1 an-2……………………………a0. a a ………………….a and can be interpreted as anr +a r +a r + an-3 r +……………..+ a0r +a 0r a r +……..+a r For example a number system has a radix 10. A number in this system is represented as a 10 +an-1 10 +an-2 10 + …………..a010 + a 10 + a 10 + ….+a 10 THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 241



This number system is known as Decimal number system. A Decimal number system of counting having ten different digits or symbols namely 0…………..9. This number system said to have a base of ten as it has ten different digits. The commonly used number systems with their symbols and bases. Number system Binary Octal Decimal Hexadecimal

Radix 2

Essential Digits 0, 1

8

0, 1, 2, 3, 4, 5, 6, 7

10

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

16

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Binary Number System: All digital computers and systems are based on binary number system.Binary means two. Thebinary number system uses only two digits, i.e. 0 and 1. Hence the radix is 2. Binary number are strings of 0’s and 1’s. The electronic devices are most reliable when designed for two-state (binary) operation. All input and output voltages are either low or high. Where low voltage represents binary 0, and high voltage represents binary 1. Binary to Decimal Conversion: We can summarise the binary –to – decimal conversion by the following procedure:  Write the binary number.  The binary number write2 , 2 ,2 2 2 ……………..working from right to left.  Add the decimal weights to obtain the decimal equivalent. Decimal to Binary conversion: To convert decimal numbers to binary number is the ‘double dabble’ method.  In this method, divide the decimal number by 2  Write down the remainder after each division.  The remainders, take in reverse order (down to up) to form the binary number. Octal Numbers: The number system with base (or radix) eight is known as ‘Octal number system’. The eight digits 0, 1,2,3,4,5,6 and 7 are used to represent numbers.




Octal – to – Decimal Conversion:  Octal to decimal is similar to binary to decimal conversion.  In the Octal number system, each digit position corresponds to a power of 8.  To convert from octal to decimal, multiply each octal digit by its weight and add the resulting products Decimal –to – Octal Conversion:  Decimal to octal conversion is similar to decimal to binary conversion  “Octal dabble” is the method used with octal numbers, while converting the decimalnumbers into octal numbers  Instead of dividing by 2, divide by 8 writing down theremainders from down to up, after each division.

Data Representation In digital computer system, the information is represented by means of binary sequences, which are organized in words. A word is a unit of information of a fixed length.The binary information in digital computers is stored in memory or in processor registers. This binary information may be in the form of either data or control information. Types of Information: 

Information o Instructions o Data  Numerical  Non – numerical  Fixed – point  Binary  Decimal  Floating point  Binary  Decimal

Fixed – Point Representation: The fixed point method assumes that the decimal or binary point is always fixed in one position. Thereare two ways of specifying the position of the decimal point or binary point in a register, i.e. by a ‘fixed’ position or by employing a ‘floating point’ representation. The two positions most widely used are i) A decimal point in the extreme left of the register to make the stored number as a fraction. ii) A decimal point in the extreme right of the register to make the stored number as an integer.




Binary Fixed –Point Representation: A fixed point binary number is positive the sign is represented by’0’ and the magnitude by a positive binary number. When the number is negative the sign is represented by ‘1’. Decimal Fixed –Point Representation: The representation of the decimal number in registers is a function of the binary code usedto represent a decimal digit. A 4 bit decimal code required four flip- flops for each decimaldigit. requires 16 flip flops. This is the wastage of storage space since the number of bits needed to store a decimal number in a binary code is greater than the number of bits needed for its equivalent binary representation.

Floating –Point Representation: The floating point representation of a number consists of three parts. These are Mantissa(M), Exponent (E) and the Base (B) . These components together represent the number, as MX . The mantissa has a 0 in the left most position to denote a plus sign. The mantissa is considered to be a fixed point fraction. The exponent contains the decimal number +04 and the base is decimal number 10. MIPS(Million of Instruction Per Second) : MIPS is calculated by dividing the number of instruction executed in a running program by the time required to run the program and typically expressed in Million of Instruction Per Second(MIPS). CPI (Clock Per Instruction): The CPI of a given program on a given system is calculated by dividing the number of clock cycles required to execute the program by the number of instruction in running the program.  IPC (Instruction Per Clock): IPC is calculated by dividing the number of instruction executed in running a program by the number of clock cycles required to execute the program, and is the reciprocal of CPI. IPC do not contain any information about a system’s clock rate and therefore less useful to measure the performance of a actual system. Speedup: Speedup is simply the ratio of the execution times before and after a change is made, so: Speedup = Amdahl’s Law: Amdahl’s law states that the performance improvement to be gained from using some faster mode of execution is limited by the function of the time the faster mode can be used. Overall speed up =

(

)

f: fraction of program that is enhanced s: speed up of the enhanced portion.




7.2: Memory Hierarchy The objective of memory hierarchy is to provide the matching of the data transfer rate of faster processor to that of lowest level memory at a reasonable cost. Memory Unit: It is an essential unit in any digital computer since it is needed for storing the programs and data that are executed by the CPU. Main Memory: The memory unit that communicates directly with the CPU is called the mainmemory. Example; ROM AND RAM.  Main memory is also called primary memory  The principal technology used for the main memory is based on semiconductor integrated circuits.  The static RAM consists essentially of internal flip-flops that store the binary information.  The stored information remains valid as long as power is applied to the unit.  The dynamic RAM stores the binary information in the form of electric charges that are applied to capacitors.  The stored charge on the capacitors tends to discharge with time and the capacitors must be periodically recharged by refreshing the dynamic memory. Refreshing is done by cycling through the words every few milliseconds to restore the decaying charge.  The dynamic RAM offers reduced power consumption and larger storage capacity in a single memory chip. The static RAM is easier to use, has shorter read and write cycles. Memory Hierarchy: Total memory capacity of a computer can be visualized as being a hierarchy of components. In memory hierarchy from top to down. 0. 1. 2. 3.

Size increases Access time increases Cost/bit decreases Decrease in frequency of access

Ci>Ci +1 (Cost per Bit) Si < Si + 1 (Size) Ti < Ti+ 1 (Access Time)  The memory hierarchy system consists of all storage devices employed by a computer system from the slow but high –capacity auxiliary memory devices, to a relatively faster main memory, to an even smaller and very fast buffer memory accessible to the high speed processing logic.  At the bottom of the hierarchy, the relatively slow magnetic tapes used to store removal files.  The main memory occupies a central position in the memory hierarchy and can communicate directly with the CPU and with auxiliary devices through an I/O processor.  The high speed small memory called cache memory occupies top position in the memory hierarchy. Cache Memory: A special very-high-speed memory called a cache is sometimes used to increase the speed of processing by making current programs data available to the CPU at a rapid rate. The cache memory is employed in computer systems to compensate for the speed difference between main memory access time and processor logic.




Capacity: The capacity of a cache is simply the amount of data that can be stored in the cache, so a cache with a capacity of 32KB can store 32 Kilobytes of data. What is Cache? High speed memory module connected to the processor for it’s private use. Contains copies of recently referenced material. Copies between cache and memory in lines or blocks. By using cache memory the speed of operation will be increased and execution time will be reduced.  Cache memory is also called high speed buffer memory    

Processor

Cache

Main Memory

Fig.7.2.2 Example of Cache Memory

Line length: The line length of a cache is the cache’s block size. Associativity: The associativity of a cache determines how many locations within the cache may pertain to a given memory address. The speed of the main memory is very low in comparison with the speed of modern processors. An efficient solution is to use a fast cache memory which essentially makes the main memory appears to the processor to be faster than it really is. The effect of the cache mechanism is based on a property of computer programs called locality of reference. It manifests itself in two ways. i) temporal and ii) spatial. Temporal: Definition: Recently accessed items are likely to be accessed in future. The temporal aspect of the locality of reference suggests that whenever an information item (instruction or data) is first needed, this item should be brought into the cache, where it will hopefully remain until it is needed again. Spatial: Definition: Items whose addresses are near one another are tend to be referenced close together in time. The spatial aspect suggests that instead of fetching just one item from the main memory to the cache, it is useful to fetch several items that reside at adjacent addresses as well. We will use the term block to refer to a set of contiguous address locations of some size. Another term that is often used to refer to a cache block is cache line. The correspondence between the main memory blocks and those in the cache is specified by a mapping function. When the cache is full and a memory word (instruction or data) that is not in the cache is referenced, the cache control hardware must decide which block should be removed to create space for the new block that contains the referenced word. The collection of rules for making this decision constitutes the replacement algorithm. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 246



In a Read operation, the main memory is not involved. For a Write operation, the system can proceed in two ways. In the first technique, called the write-through protocol. The cache location and the main memory location are updated simultaneously. The second technique is to update only the cache location and to mark it as updated with an associated flag bit, often called the dirty or modified bit. The main memory location of the word is updated later, when the block containing this marked word is to be removed from the cache to make room for a new block. This technique is known as the write-back, or copy-back, protocol. When the addressed word in a Read operation is not in the cache, a read miss occurs. The block of words that contains the requested word is copied from the main memory into the cache. After the entire block is loaded into the cache, the particular word requested is forwarded to the processor. Alternatively, this word may be sent to the processor as soon as it is read from the main memory. The latter approach, which is called load-through or early restart reduces the processor’s waiting period. The performance of virtual memory or cache memory is measured with hit ratio. Hit ratio is defined as the number of hits divided by the total number of CPU references to the memory. (hits plus misses) Tavg = HC + (1 – H) M Where H = Hit ratio of cache memory C = time to access information in cache memory M = miss penalty + Main Memory access time + Cache Memory access time The average memory access time of a computer system can be improved considerably by use of a cache. Cache Coherence Problem:   

Will have multiple copies of the “same” data in different caches Data in the caches is modified locally. Cache become inconsistent between caches, and between cache and main memory from which the cached data was copied

The transformation of data from main memory to cache memory is referred to as a mapping process. Multilevel Cache Hierarchy One of the fundamental issues is tradeoff between cache latency and hit rate. Larger caches have better hit rates but longer latency. To address this tradeoff, many computers use multiple levels of cache, with small fast caches backed up by larger slower caches. Multi-level caches generally operate by checking the smallest Level 1 (L1) cache first; if it hits, the processor proceeds at high speed. If the smaller cache misses, the next larger cache (L2) is checked, and so on, before external memory is checked. Multi-level caches introduce new design decisions. For instance, in some processors, all data in the L1 cache must also be somewhere in the L2 cache. These caches are called strictly inclusive. Other processors have exclusive caches — data is guaranteed to be in at most one of the L1 and L2 caches, never in both Address Mapping: a) Associative Mapping: The tag bits of an address received from the processor are compared to the tag bits of each block of the cache to see if the desired block is present. This is called to associative-mapping technique. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 247



An associative cache employs a tag, that is, a block address, as the key. At the start of a memory access, the incoming tag is compared simultaneously to all the tags stored in the cache’s tag memory. If a match (cache hit) occurs a match indicating signal triggers the cache to service the requested memory access. A no match signal identifies a cache miss, and the memory access requested is forwarded to the main memory for service. Main memory Block0 Tag

Block 0 Block 0

Tag

Block 1

Block i Block 127

Tag

Tag

Block Block 4095

Fig.7.2.3

Associative Mapped cache

b) Direct Mapping: An alternative, and simpler, address mapping technique for cache is known as Direct Mapping. Main memory Block 0 Block 1 Cache tag tag

tag

Block 127

Block 0

Block 128

Block 1

Block 129

Block 255 Block 256

Block 127

Tag

Block

Word

5

7

4

Main memory address Block 4095

Fig. 7.2.4 Direct Mapping Cache




 The direct- mapping technique is easy to implement, but it is not very flexible.  The main drawback of direct mapping is that the cache’s hit ratio drops sharply if two or more frequently used blocks happen to map onto the same region in the cache. c) Set Associative Mapping: Here, combination of the direct and associative mapping techniques can be used. Blocks of the Cache are grouped into sets, and the mapping allows a block of the mainmemory to reside in any block of a specific set. At the same time, the hardware cost is reduced by decreasing by the size of the associative search. The tag field of the address must then be associatively compared to the tags of the two blocks of the set to check if the desired block is present. This two-way associative search is simple to implement. The number of blocks per set is a parameter that can be selected to suit the requirement of a particular computer. A cache that has k-blocks per set is referred to as a k-way setassociative cache. One more control bit, called the valid bit, must be provided for each block. This bit indicates whether the block contains valid data or not. Main memory

Block 0 Block 1 Set 0

Set 1

Cache Block 0 Block 1 Block 2

tag tag tag

Block 3

Set 63

Block 63 Block 64 Block 65 Block 127

tag tag

Block 126 Block 127

Block 128 Block 129

Block 4095 Tag 6

Set 6

Word 4 Fig. 7.2.5 Main memory address

The time required to find an item stored in memory can be reduced considerably if stored data can be identified for access by the content of the data itself rather than by an address. Match logic is used in the associative memory to identify data item.




Page replacement policies The page replacement policy decides which frame entry information will be removed from the page table in case of conflict. 1. FIFO policy: This policy simply removes pages in the order they arrived in the main memory. Using this policy we simply remove a page based on the time of its arrival in the memory. 2. LRU policy: LRU expands to least recently used. This policy suggests that we remove a page whose last usage is farthest from current time. 3. NFU policy: NFU expands to not frequently used. This policy suggests using the criterion of the count of usage of page over the interval T. Let us briefly discuss the merits of choices that one is offered. FIFO is a very simple policy and it is relatively easy to implement. All it needs is the time of arrival. However, in following such a policy we may end up replacing a page frame that is referred often during the lifetime of a process LRU and NFU policies are certainly better in that regard but as is obvious we need to keep the information about the usage of the pages by the process. A more advanced technique of page replacement policy may look-up the likely future references to pages. Such a policy frame would require use of some form of predictive techniques. Virtual memory: The term virtual memory refers to the abstraction of separating LOGICAL memory--memory as seen by the process--from PHYSICAL memory--memory as seen by the processor. Because of this separation, the programmer needs to be aware of only the logical memory space while the operating system maintains two or more levels of physical memory space. The virtual memory abstraction is implemented by using secondary storage to augment the processor's main memory. Data is transferred from secondary to main storage as and when necessary and the data replaced is written back to the secondary storage according to a predetermined replacement algorithm. If the data swapped is designated a fixed size, this swapping is called paging, if variable sizes are permitted and the data is split along logical lines such as subroutines or matrices, it is called segmentation. TLB (Translation Look aside Buffer): The basic idea in the translation look-aside buffer access is quite simple. TLB is very effective in improving the performance of page frame access. This is a cache buffer to keep copies of some of the page frames in a cache buffer.. Note that a cache buffer is implemented in a technology which is faster than the main memory technology. So, a retrieval from the cache buffer is faster than that from the main memory. The hardware signal which looks up the page table is also used to look up (with address translation) to check if the cache buffer on a side has the desired page. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 250



This nature of look-up explains why this scheme is called Translation Look-aside Buffer (TLB) scheme. The basic TLB buffering scheme is shown in Figure. Suppose we wish to access page frame p. The following three possibilities may arise: 1. Cache presence: There is a copy of the page frame p. In this case it is procured from the lookaside buffer which is the cache. 2. Page table presence: The cache does not have a copy of the page frame p, but page table access results in a page hit. The page is accessed from the main memory. 3. Not in page table: This is a case when the copy of the page frame is neither in the cache buffer nor does it have an entry in the page table. Clearly, this is a case of page-fault. It is handled exactly as the page-fault is normally handled.

Offset CPU

offset

Main Memory

Page Table

TLB

The dotted lines shows the translation look aside buffer operation

Fig. 7.2.6 With 32-bit addresses and 1kB page, the VPN and PPN are 22 bits each. With 128 entries and 4 ways set associativity, there are 32 sets in the TLB; so 5 bits of the VPN are used to select a set. Therefore, we only have to store 17 bits of the VPN in order to determine if a hit has occurred, but we need all 22 bits of the PPN to determine the physical address of a virtual address. This gives a total of 41 bits per TLB entry. 41× 128 = 5,125 kB Memory mapping table is used to translate virtual address to physical address The virtual memory is divided into pages and the main memory is divided into blocks or frames. The size of the page must be equal to the size of the block or frames. Random access memory page table techniques or Associate memory page table techniques can be used to translate virtual address into main memory address. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 251



In Associative memory page table technique, the number of memory locations required to store memory mapping table is equal to the number of block available in the main memory. The wastage of memory is minimum in the case of associative memory page table technique. The most commonly used page replacement algorithms in virtual memory are (a) first-in-first-out and (b) the least recently used (LRU).




7.3: Pipeline The speed of execution of programs is influenced by many factors. One way to improve performance is to use faster circuit technology to build the processor and the main memory. Another possibility, is to arrange the hardware in a manner so that more than one operation can be performed at the same time. Pipelining is particularly effective way of organizing concurrent activity in a computer system. Pipeline is commonly known as an assembly-line operation.

A pipeline can be visualized as a collection of processing segments through which binary information flows.

There are two area of computer design where the pipeline organization is applicable. 1. An Arithmetic Pipeline: It divides an arithmetic operation into sub-operations for execution in the pipeline segments.  Pipeline arithmetic units are usually found in very fast speed computer. They are used to implement floating-point operations, multiplication of fixed-point number, and similar computations encountered in scientific problem. 2. An Instruction Pipeline: It operates on a stream of instructions by overlapping the fetch, decode and execute phases of the Instruction Cycle. Example of Instruction Pipeline

IF

ID

EX

OF

Totaltimewithpipe

T

WB

[ + (n 1)]T n

Where ‘n’ is the number of instructions ‘ ’ is the number of stages in pipe ‘T’ is the time for a stage which is maximum of all stages. [ +(

1)]

Cyclic Time of pipelined processors: Cycle Time Pipelined =

+ Pipeline Latch Latency




Speed Up: The Speed Up of a pipeline processing over equivalent non pipeline processing is defined by the ratio of S=

(

)

Where ‘tn’ is the time taken to complete an instruction without pipeline. ‘tp’ is the time taken for a stage which is maximum of all stages.

Speed Up factor can also given by S= For ideal case, Pipeline stall cycles/instruction is zero. S = Pipeline depth= no. of stages Stall Cycles: Stall cycles are cycles which are used if one stage operation is delayed, then the previous stages for next instruction are waited for completion by using some empty cycles, these empty cycles are called as stall cycles.

a)

Sequential Execution: I1

F1

I3

I2

E1

F2

E2

F3

E3

…………

Interstate buffer B1 Instruction fetch unit

Execution unit

Fig. 7.3.1



b)


Hardware organization Time Clock cycle

1

2

F1

E1

3

4

Instruction I1

F2

I2

E2 F3

I3

E3

Fig. 7.3.2 Pipelined execution Consider how the idea of pipeline can be used in a computer. The processor executed a program by fetching and executing instructions. Execution of a program consists of a sequence of fetch and execute steps, as shown in figure7.3.1. Time Clock cycle

1

2

3

F1

D1

E1

F2

D2

4

5

6

7

Instruction I1 I2

F3

I3 I4

W1 E2

W2

D3

E3

W3

F4

D4

E4

W4

Fig. 7.3.3 Instruction execution divided into four steps Inter-stage buffers

D: Decode instruction and fetch operands

F : Fetch Instruction B1

E:Execute operation

B2

W : Write results B3

Fig. 7.3.4 Four-segment pipeline THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 255



The instruction fetched by the fetch unit is deposited in an intermediate storage buffer. This buffer is needed to enable the execution unit to execute the instruction while the fetch unit is fetching next instruction. The computer is controlled by a clock whose period is such that the fetch and execute steps of any instruction can each be completed in one clock cycle. An interchange storage buffer, B1, is needed to hold the information being passed from one stage to next. New information is loaded into this buffer at the end of each clock cycle. A pipelined processor may process each instruction in 4 steps: F

Fetch: Read the instruction from the memory.

D

Decode: decode the instruction and fetch the source operand.

E

Execute: perform the operation specified by the instruction.

W

Write: store the result in the destination location.

The sequence of events for this case is shown in Figure 7.3.3. Four instructions are in progress at any given time. This means that four distinct hardware units are needed, as shown in Figure 7.3.4. These units must be capable of performing their tasks simultaneously and without interfering with one another. Information is passed from one unit to the next through a storage buffer. As an instruction progresses through the pipeline all the information needed by the stages downstream must be passed along. For example, during clock cycle 4, the information in the buffers is as follows:  



Buffer B1 holds instruction , which was fetched in cycle 3 and is being decoded by the instruction-decoding unit. Buffer B2 holds both the source operands for instruction and the specification of the operation to be performed. This is the information produced by the decoding hardware in cycle 3. The buffer also holds the information needed for the write step of instruction (step ). Even though it is not needed by stage E, this information must be passed on to stage W in the following clock cycle to enable that stage to perform the required Write operation. Buffer B3 holds the results produced by the execution unit and the destination information for instruction .

Pipeline Performance: In above figure, processor completes the processing of one instruction in each clock cycle, which means that the rate of instruction processing is four times as compared to sequential operation. For a variety of reasons, one of the pipeline stage may not be able to complete its processing task for a given instruction in the time allotted. Some operations, such as divide, may require more time to complete. Idle periods are called stalls. They are also referred to as bubbles in the pipeline. Once created as a result of a delay in one of the pipeline stages, a bubble moves down stream until it reaches the unit pipelined operations on the above situation, is said to have been stalled for two clock cycles. Any condition that cause the pipeline to stall is called a hazard.




A data hazard is any condition in which either the source or the destination operands of an instruction are not available at the time expected in the pipeline. As a result some operations has to be delayed, and the pipeline stalls. The pipeline may also be stalled because of a delay in the availability of an instruction. For example, this may be a result of a miss in the cache, requiring the instruction to be fetched from the main memory. Such hazards are often called control hazards (or instruction hazards). A third type of hazards that may be encountered in pipelined operation is known as a structural hazard. This is the situation when two instructions require the use of a given hardware resource at the same time. The most common case in which this hazard may arise is in access to memory, or structure hazard occurs when the processor’s hardware is not capable of executing all instruction in the pipeline simultaneously. If instructions and data reside in the same cache unit, only one instruction can proceed and the other instruction is delayed. Many processors use separate instruction and data caches to avoid this delay. Data Hazards: A data hazard is a situation in which the pipeline is stalled hold because the data to be operated on are delayed for some reason. We must ensure that the results obtained when instructions are executed in a pipelined processor are identical to those obtained when the same instruction are executed sequentially. Pipeline Conflicts: In general, there are three major difficulties that cause the Instruction Pipeline to deviate from its normal operation. 1. Resource Conflicts caused by access to memory by two segments at the same time. Most of these conflicts can be resolved by using separate instruction on data memories. 2. Data Independency Conflicts arise when an instruction depends on the result of previous instruction, but this result is not yet available. 3. Branch Difficulties arise from branch and other instruction that change the value of PC. Data Hazards avoiding techniques: 1. Hardware Interlock or delayed load or bubble: The hardware interlock preserves the correct execution pattern of an instruction sequence. In general a hardware interlock detects the data hazard and stalls the pipeline until the hazard is cleared. 2. Operand Forwarding: The operand forwarding is a hardware technique to minimize the stalls in a pipelined execution of sequence of instructions. The key insight the forwarding is that the result of a previous instruction is directly fed to the next instruction through the pipelined registers without waiting to be written in WB stage to the register file. Branch Hazards avoiding techniques 1. Pipeline flushing : Simplest solution to the handle branches is to freeze or flush the pipeline, deleting any instruction after the branch until the target of the branch is known 2. Branch prediction: Continuing with the sequence of instructions as if branch were taken or not taken. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 257



3. Delayed Branching: A branch delay slot is introduced after the branch instruction and filled with the instruction will be executed irrespective of branch is taken or not. There are two types of Branch Prediction techniques. They are

(1) Static Branch Prediction

– done by compiler

(2) Dynamic Branch Prediction – done by processor H/w. Throughput of Pipeline Computer: Throughput = is cache miss penalty

is branch penalty.

Anti Dependence Write After Read (WAR) Hazard: Dependence resulting from reuse of a name. Data Dependence Read After Write (RAW) Hazard or True Dependence: True dependence resulting from use of a data value produced by an earlier statement. O/P Dependence Write After Write (WAW) Hazard: Dependence resulting by writing a value before a preceding write has completed.




7.4: Instruction Types An instruction is a binary code, which specifies a basic operation (eg. add, subtract, and, or) for the computer. Types of Instructions Instruction varying accordingly in terms of word length, number and type of register between computers, are listed below: Load: Copy the contents of a specific location to the contents of a register. Add: Adds the contents of a specific location to the contents of a register. Subtract: Subtract the contents of a specific location to the contents of a register. Store: Copy the contents of a register into a specific memory location. Branch: Switch the CPU control to another instruction address other than the next in sequence. Register-register: Moves contents from one register to another. Shift: Moves bits in a memory location to the left or to the right for arithmetic purposes or for pattern manipulation. Input/output: Effects data transfer between a peripheral devices and memory. Logical Operator: (AND, OR, NOT, NAND, and so on)- combine the contents of a register and a specified memory location or register. The range of instructions available to any particular computer varies from system to system. Branch Instruction: Branch instruction cover a series of instruction that divert the program flow from one series of memory locations to a second non-contiguous series. This can be readily achieved by having an instruction that alters the contents of the Program Counter (PC) register. Branch instructions may be either conditional or non-conditional. In the former case some test is performed and if the out come of test is true then the contents of the PC are altered and hence the program flow jumps to the new point in memory. In non-conditional jump is always made. Branch instructions enable complex programs to be developed in which:  

certain pieces of code are only executed if a condition is met (involved conditional branches) and loops are possible. Such that pieces of code can be executed over and over again.

 Micro-operation: It is the elementary operation performed on the binary data stored in the internal registers of a digital computer during one clock period. The result of the operation THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 259



may replace the previous binary information of a register or may be transferred to another register.  Macro-operation or operation: A set or micro operations is called a macro-operation.  Instruction Code: It is a group of bits that tell the computer to perform a specific operation. It is divided into parts called fields, each having its own particular interpretation. The most basic part of an instruction code is its operation part.  The basic instruction code format consists of two parts or two fields. Opcode

Address

 OpCode: It is also called Operation Code field. It is a group of bits that define the operation performed by the instruction.  The number of bits required for the operation part of an instruction code is a function of the total number of operations used. It must consist of at least n-bits for given 2n (or less) distinct operations.  The address part of the instruction tells the control, where to find an operand in the memory.  Operand: It is the data on which the given operation is to be performed.  Stored Program Connect: It is the most important concept used in all the digital computers. In this concept the instructions are stored in one section of the memory and data is stored in another section of the memory. The CPU fetches one by one instructions from the memory and they will be decoded and executed.  Computers that have a processor register usually assign to it name ‘’accumulator’’ and label it AC.  In general, the length of Accumulator register or processor register must be equal to the length of each memory location.  In general, the basic registers available in the digital computer are, Program Counter (PC), Memory Buffer Register (MBR),Memory Address Register (MAR), Operation Register (OPR), Accumulator register(AC), Mode bit flip- flop (1), Extension (E) flip-flop etc.  PC: The Program Counter always holds the address of the next instruction to be fetched from the memory location.  The length of the PC always depends on the addressing capacity of the CPU or the number of memory locations available in the memory. If the memory consist of 2n memory location then n bit PC is required.  OPR: The operation register is used to hold the operation part of the instruction. The length of this register depends on the length of the operation part of the instruction.  The length of MAR must be equal to the length of PC.  The length of MBR is equal to the length of AC and length of each memory location.  The use of mode flip-flop (l) is to hold mode bit. With the help of this mode bit the CPU can distinguish direct address instruction and indirect address instruction.  The E flip-flop is an extension of the AC. It is used during shift operations. It receives the end –carry during addition, etc.  The basic instruction code formats can be of three types: Immediate operand instructions, Direct address instructions, and Indirect address instructions.  Immediate operand instructions: If the second field of an instruction code specifies an operand, the instruction is said to have an immediate operand. Op Code

Operand




 Direct address instructions: It is second field of an instruction code specifies the address of an operand, then it is called direct address instruction. Op Code

Address of operand

 Indirect address instructions: If the second field of an instruction code designates an address of a memory word in which the address of the operand is found, is called the indirectaddressinstruction. Op Code

Address of address operand

 Todistinguish direct and indirect address instructions a third field (one bit field) called Mode field is used in the instruction code format. I

Op Code Op Code

Address of operand

I = 0 Direct address instruction I = 1 Indirect address instruction

 The most common fields found in the instruction code formats are: (a) mode bit field (b) Op Code field and (c) address field.  The CPU requires zero memory references to complete the execution of the immediate types of instruction, once by instruction code is transferred from memory into CPU.  The CPU requires one memory reference to complete the execution of the direct address type of instruction, once the instruction code is transferred from memory into the CPU.  The CPU requires two memory references to complete the execution of the indirect address type of instruction, once the instruction code is transferred into the CPU.  Depending on the way the CPU refers the memory, Input-Output and registers, the instructions of the basic computer can be of three types: Memory reference instructions, Registers reference instructions and Input-Output reference instructions.  Memory reference instructions: The CPU is supposed to refer to the memory to get the operand for the completion of the execution of the instruction.  Register reference instruction: The CPU is supposed to refer to the internal registers of the CPU to get the operand for the execution of the instruction.  Input-Output reference instructions: The CPU is supposed to refer the input or output devices to complete the execution of the instruction.  Most computers fall in three types of CPU organizations: Single AccumulatorOrganization, General Register Organization and Stack Organization.  Based on the number of address fields available in the address part of the instruction, there are four different types of instructions. 1. Three address instructions 2. Two address instructions 3. One address instructions 4. Zero address instructions  Three address instructions: The address field of the instruction code is divided into three sub fields. Example: ADD R 1, R2, R3 (R1) (R2) + (R3) The advantage of the three- address format is that it result in short program when evaluating arithmetic expression. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 261



 Two address instructions: In this type of instruction the address field of the instruction code is divided into two sub fields. Example: ADD R1, R2 (R1) (R1) + (R2)  The example of a computer which supports two address instructions is PDP -11  One Address Instructions: In this type of instructions, an implied accumulator (AC) register are used for all data manipulation. Example: LOAD A, AC M [A] ADD B, AC M [B] + AC The example of a computer which supports one – address instruction is PDP-8  Zero–address instructions: Some operational instructions do not require address field, such instructions are called zero address instructions. Example: computer which supports zero address instructions is Burroughs - 6700  Single accumulator organized computers uses one address instruction.  General register organized computer uses two and three address instructions.  Stack organized computer uses zero address instructions.  Stack: It is a set of memory locations of the RAM which are used to store information in such a manner that the item stored last, is the first item retrieved.  A very useful feature that is included in the CPU of many computers is a stack or Last In – First Out (LIFO) list. Stack Pointer (SP): It is a register, which always holds the address of the top of the stack. The length of the stack pointer register is equal to the length of PC register.  PUSH and POP instructions are used to communicate with stack memory.  FULL and EMPTY flip- flops are used to indicate status of the stack memory.  The arithmetic expressions can be written in one of three ways 1. Infix notation 2.Prefix notation and 3.Postfix notation  If the operator is placed between the operands then it is called infix notation. Ex: A+B  The prefix notation is also called polish notation. In this notation the operator is placed before the operands. Ex: +AB  The postfix notation is also called as reverse polish notation. In this notation the operator is placed after the operands. Ex: AB+  The reverse polish notation is suitable for stack organized computers.  The expression A* B +C* D can be represented in reverse polish notation as AB*CD*+

RISC and CISC Architecture Complex Instruction Set Computer (CISC): A computer with large number of instructions is classified as a complex instruction set computer. Reduce Instruction Set Computer (RISC): A computer with fewer number of instructions is classified as a reduce instruction set computer. Example: • CISC –VAX, Intel X86, IBM 360/370, etc. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 262



• RISC – MIPS, DEC Alpha, SUN Sparc, IBM 801 RISC Vs CISC Characteristics of ISA (Instruction set architecture) CISC

RISC

Variable length instruction

Fixed length instruction

Variable format

Fixed field decoding

Memory Operands

Load/Store Architecture

Complex instruction

Simple Operations

Features CISC

RISC

Small chip size

Large chip size

Small code size

Large code size

less burden on software

Greater burden on software

Large number of instruction

Less number of instructions

Less register count

Large register count

Multiple clocks are required to complete a The instruction usually take a single clock instruction Difficult to take advantage of pipelining

More suitable for pipelining




7.5: Addressing Modes The different ways in which the location of an operand is specified in an instruction are referred to as Addressing modes or the addressing mode specifies a rule for interpreting or modifying the address field of the instruction before the operand actually referenced.

General Adressing Modes: Name

Assembler Syntax

Addressing function

Implied

NIL

Implied

Immediate

# value

Operand =Value

Register

Ri

EA= Ri

Absolute ( Direct )

LOC

EA= LOC

Indirect

Ri

EA= [Ri]

Index

X (Ri)

EA = [Ri] +X

Base with index

(Ri, Rj)

EA= [Ri] + [Rj]

Base with index and offset

X (Ri,Rj)

EA= [Ri] + [Rj] + X

Relative

X (PC)

EA = [PC] + X

Auto increment

(Ri) +

EA = [Ri]; Increment Ri

Auto decrement

-(Ri)

Decrement Ri; EA = [Ri]

Fig7.5.1 Generic addressing modes Here, EA = Effective Address, Value = a signed number.  Addressing Mode: The way in which the operand is specified in the address part of the instruction is called addressing mode. The addressing mode specifies a rule for interpreting or modifying the address field if the instruction before the operand is actually referenced.  Computers use addressing mode techniques to give programming versatility to the user by providing such facilities such as a pointer, counters, indexing, program relocation and to reduce the number of bits in the address field of instruction.  Different types of addressing modes used in computers are implied, immediate, Register, Register indirect, Auto increment or Auto decrement, direct address mode, Base register addressing mode, indirect address mode, relative addressing mode and indexed addressing mode.




 Effective Address: The effective address is defined to be the memory address obtained from the computation dictated by the given addressing mode. The effective address is the address of the operand in a computational type instruction.  Implied mode: In this mode the operands are specified implicitly in the definition of the instruction. Zero – address instruction in a stack – organized computer are implied-mode instruction. EX: PUSH  Immediate Mode: In this mode the operands are specified in the instruction itself.  Register Mode: In this mode the operands are in register that reside within the CPU.  Register Indirect Mode: In this mode the instruction specifies a register in the CPU whose contents give the address of operands in memory.  Auto increment or Auto decrement: This is similar to the register indirect mode except that the register is incremented or decremented after (or before) its value is used to access memory.  Direct Address Mode: In this mode the effective address is equal to the address part of the instruction.  Indirect Address mode: In this mode the address field of the instruction gives the address where the effective address is stored in the memory.  Relative Address Mode: In this mode the content of the program counter (PC) is added to the address part of the instruction in order to obtain the effective address. Relative addressing is often used with branch type instruction when the branch address is in the area surrounding the instruction word itself.  Indexed Addressing Mode: In this mode the content of a index register is added to the address part of the instruction to obtain the effective address. The index register is a special CPU register that contains an index value.  Base Register Addressing Mode: In this mode the content of a base register is added to the address part of the instruction to obtain the effective address.  The addressing modes supported by one processor may differ from the addressing modes supported by the other processors. (A) Direct = 400 (B) Immediate = 301 (C) Relative = 302+400=702 (D) Register indirect = 200 (E) Indexed = 200+400 = 600  Computer cycle: Digital computers provide timing sequence of 8 to 16 repetitive timing signals. The time of one repetitive sequence is called a computer cycle. Types of computer cycles are Fetch cycle, Indirect cycle, Executive cycle and Interrupt cycle.  Fetch cycle: When an instruction is read from memory the computer is said to be in an instruction fetch cycle. The first cycle of any instruction cycle must be always a fetch cycle.  Indirect cycle: When the word read from the memory is an address of operand, the computer is said to be in an indirect cycle.  Execute cycle: When the word read from memory is an operand, the computer is said to be in an execute cycle.  The execute cycle can come after fetch cycle or indirect cycle. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 265



 The indirect cycle must come after fetch cycle.  The CPU will enter into interrupt cycle only after completion of execute cycle.  The interrupts are generally used to transfer the program control from one place to another place.  There are three types of interrupts that cause a break in the normal execution of the program. 1. External interrupts 2. Internal interrupts 3. Software interrupts  External interrupts come from input-output devices, from a timing device, from a circuit monitoring the power supply, or from any other external source. Example: I/O device requesting transfer of data.  Internal interrupts arise from illegal or erroneous use of an instruction or data. Internal interrupts are also called as traps. Example: attempt to divide by zero, stack overflow, invalid operation code etc.  External and Internal interrupts are initiated from signals that occur in the hardware of the CPU.  A software interrupt is initiated by executing an instruction.  Internal interrupts are synchronous with the program while external interrupts are asynchronous. If the program returns. The internal interrupts will occur in the same place each time.  The programmer to initiate an interrupt procedure at any desired point in the program can use software interrupts.  The collection of all status bit conditions in the CPU is sometimes called a program status word or PSW.  Program interrupt refers to the transfer of the program control from a currently running program to another service program as a result of an external or internal generated request.




7.6: I/O Data Transfer  Interfacing: Writing software instructions and designing hardware circuit to enable the central processing unit to communicate with peripheral devices is called interfacing, and the hardware circuit used is called interfacing circuit. Need for interface: 1. Most of the peripherals devices are electromechanical and electromagnetic devices and their manner of operation is different from the operation of the CPU and memory, which are electronic devices. 2. The data transfer rate of peripherals is much slower than the transfer rate in the central computer. 3. The operation of the peripheral must be synchronized with the operation of the CPU and memory unit. 4. Data format in peripherals differ from the data format in the central processor.  The I/O processor is sometimes called a data channel controller.  There are two ways to connect peripheral devices to CPU. (a) Memory mapped I/O and (b) Isolated I/O or I/O mapped I/O. Programmed I/O  CPU has direct control over I/O Sending status Read/Write Commands Transferring data  CPU waits for I/O module to complete operation  Wastage CPU time

Details      

CPU requests I/O operation. I/O module performs operation. I/O module sets status bits. CPU checks status bits periodically. I/O module does not interrupt CPU. CPU may wait or come back later.

Interrupt Driven I/O  Overcome CPU waiting.  No repeated CPU checking of device.  I/O module interrupts when ready.




I/O Mapping  Memory mapped I/O.  Devices and memory share an address space.  I/O looks just like memory read/write.  No special commands for I/O.  Large selection of memory access commands available. Isolated I/O  Separate address space.  Need I/O or memory select lines.  Special commands for I/O.  Limited set. Basic Operation  CPU issues read command.  I/O module gets data from peripheral devices while CPU does other work.  I/O module interrupts CPU.  CPU requests data.  I/O module transfers data.  In memory mapped I/O techniques, the I/O devices are also treated as memory locations, under that assumption that they will be given addresses. Same control lines are used to activate memory locations as well as I/O devices.  In I/O or Isolated I/O technique, the I/O devices are given separate addressed and separate control signals are used to activate memory locations and I/O devices.  Data transfer between CPU and peripherals is handled in one of three possible modes:  Data transfer under program controlled.  Interrupt initiated data transfer.  Direct memory Access (DMA) transfer.  Program controlled operations are the result of I/O instruction written in the computer  Each data item transfer is initiated by an instruction in the program.  The disadvantage of program controlled data transfer is that, the processor stays in a program loop until the I/O unit indicates that it is ready. This is a time-consuming process since it keeps the processor busy needlessly.  In interrupt initiated data transfer, when the peripheral is ready for data transfer, it generates an interrupt request to the processor. Then the processor stops momentarily the task it is doing, branches to a service routine to process the data transfer and then returns to the task it was performing.  In Direct Memory Access (DMA) the interface transfers data into and out of memory unit through the memory bus generally to transfer bulk amount of data from memory to peripheral or from peripheral to CPU, DMA techniques is used.  There are basically two formats of data transfer: parallel and serial.  In parallel mode data bits (usually a byte) are transferred parallely over the communication lines referred to as buses. Thus all the bits of a byte are transferred simultaneously within the time frame allotted for the transmission.  In serial data transfer, each data bit is sent sequentially over a single data line.  In order to implement serial data transfer, the sender and receiver must divide the timeframe allotted for the transmission of a byte into subintervals during which each bit is sent and received.  In serial transmission the information is transferred in the form of frames. The frame consists of three parts. Start bit, character code and stop bits.  Start bit is always logic ‘0’ and stop bits are always at logic ‘1’ level. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 268



 The data transfer in both the parallel and serial mode of operation can be either synchronous.  In synchronous mode both source unit and destination unit work in synchronous mode with same control signal.  In asynchronous mode the source unit and destination units have their own independent control signals.  Asynchronous parallel data transfer can be of three types. (a) Strobe control (b) Two wire hand shaking method and (c) Three wire handshaking method.  Asynchronous parallel transfer o Strobe control  Source initiated  Destination initiated o Two wire handshaking  Source initiate  Destination initiate o Three wire handshaking  Asynchronous data transfer between two independent unit requires the control signals to be transmitted between the communication units to indicate the time at which data is being transmitted. One way of achieving this is by means of a strobe pulse supplied by one of the unit to indicate to the other unit when the transfer has to occur.  Exchange of control signals between source and destination units during the data transfer operation is called as handshaking.  The disadvantage of two-wire handshaking method is, it is not possible to connect more than one destination unit to a single source unit. Transmitter is used to interface serial I/O device to a parallel bus structure.  In DMA transfer, the CPU initializes the DMA by sending a memory address and the number of words to be transferred.  During DMA transfer, the CPU is idle and has no control of the memory buses.  A DMA controller takes over the buses to manage the transfer directly between the I/O device and memory.

DBUS

Bus request BR

ABUS

Address bus bus Data bus

CPU

Bus grant

BG

RD

Read

WR

Write

High- impedance (disable) When BG is enabled

Fig 7.6.1 CPU bus signal for DMA transfer  DMA is preferred data transfer for high speed peripheral devices such as magnetic disks.  There are mainly three different modes of DMA operation. 1. Continuous DMA or burst mode of DMA 2. Cycle stealing and 3. Interleaved DMA THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 269



 In burst mode a sequence of arbitrary length of data is transferred in a single continuous burst during which the DMA controller is the master of the system bus.  In cycle stealing technique, the transfer is done by first checking if the memory unit is not used by the CPU and then the DMA steals one memory cycle to access a word in memory.  Interleaved (multiplexed) DMA: In this Technique the processor and DMA access of memory can be interleaved on alternate half cycle of the clock. The DMA access the memory during first half of the cycle while CPU accesses during second half.  Interrupt: It is the exceptional event, which causes the CPU to temporarily suspend the current program being executed. The control is subsequently transferred to some other programreferredto as interrupt service routine, which specifies the actions to be taken if the exceptional event occurs.  Priority interrupt: It is an interrupt system that establishes a priority over the various sources to determine which condition is to be serviced first when two or more requests arrive simultaneously.  Software or hardware can establish the priority of simultaneous interrupts.  A polling procedure is used to identify the highest priority source by the software.  In polling technique, there is one common branch address for all interrupts. The common service program begins at the branch address and polls the interrupt source in sequence.  The disadvantages of the software method is that if there are many interrupts, the time required to poll them can exceed the time available to service the I/O device.  The hardware priority function can be established either by a serial or a parallel connection of the interrupt links.  The daisy-chain method of priority selection consists of all devices that request an interrupt from the processor. The device with the highest priority is placed in the first position, followed by lower priority device, up till the device with the lowest priority which is placed last in the chain.

VAD 1

VAD 2

VAD 3

Device 1 O I

Device 2 I O

Device 3 I O

Interrupt request Interrupt Acknowledge

To Next Device

I T C U I T AC

Fig 7.6.2 Daisy-Chain Priority Interrupt




 The parallel priority interrupt method uses a register whose bits are set separately by the interrupt request from each device priority is established according to the position of the bits in the register. Interrupts 1 Introduction  An interrupt is signal either sent by an external device or internally by the software, indicating the need for attention or change in execution.  Interrupts arising from an external events, is called asynchronous  Interrupts arising from an internal even are generated by the software and generally called synchronous Following are the examples are both type of interrupts  Externalities: Interrupts come typically from I/O devices which have completed a task, have run out of some resources or have run into some problems, such as an attempt to write to a write-disabled disk.  Page faults: Running under virtual memory implies that many parts of a program will reside on disk. If references to those resident pages cannot be answered in reasonable time, a fault interrupt is issued which calls for a context switch to a task that can make use of the CPU while the missing pages are brought in from disk.  Address translation errors: Address translation errors occur if events such as trying to write to read-only-memory or doing any operation in a space, that is not open to the running program.  Illegal instructions fault: This type of faults includes undefined opcodes and instructions reserved for higher level of privilege.  Arithmetic errors: This class includes divide-by-zero, word and half-word overflow. Direct Memory Access (DMA) 1. Overview Direct memory Access (DMA) is a operational transfer mode which allows data transfer within memory or between memory and I O device without processor’s intervention. A special DMA controller manages that data transfer. A DMA controller can be implemented as separated controller from the processor or integrated controller in the processor. In the following integrated DMA controller is considered. The DMA mechanism provides two unique methods for performing DMA transfer:  Demand-mode transfer (synchronized to external hardware): Typically used for transfers between an external device and memory. In this mode, external hardware signals are provided to synchronize DMA transfers with external requesting devices.  Block-Mode transfer (non-synchronized): Typically used to move block of data within memory. To perform a DMA operation the DMA controller uses microcode the core’s multi-process resources, the bus controller and internal hardware dedicated to the DMA controller. Loads and stores are executed in DMA microcode to perform each DMA transfer. Multi-process resources THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 271



are used to enable DMA operations to be executed concurrently with user’s program. The bus controller, directed by the DMA microcode, handles data transaction in external memory. External bus access is shared equally between the user and the DMA process. The bus controller executes bus requests by each process in alternating fashion. The DMA controller hardware synchronizes transfers with external devices or memory, provides the programmers interfaces to the DMA controller itself, and manages the priority for servicing DMA requests. 2. Data transfers Different DMA transfer modes are explained in the following paragraph. Multi-cycle Transfer Multi-cycle Transfer comprises of two or more bus requests: loads from source address are followed by stores to a destination address. To execute the transfer, DMA microcode issues the proper combination of bus requests. The processor effectively buffers the data for each transfer. When the DMA is configured for destination synchronization, the DMA controller buffers source data, waiting for the request from the destination requestor. The initial DMA request still requires the source data to be loaded before the request is acknowledged. 32bit memory Data source

Integrated DMA controller

32bit I/O device DREQ

Buffer load data

DACK

Data Destination

External system bus

Fig 7.6.3 Source data buffering for destination synchronized DMA’s Fly-By Single-Cycle Transfer Fly-by transfers are only executed with only a single load or store request. Source data is not buffered internally. Instead of this, the data is passed directly between sourced and destination via the external data bus. Fly-by transfers are common used for high-performance peripheral to memory transfers which can be described in the following example of a source-synchronized demand DMA mode. The source data is a peripheral device at a fixed address and the destination data is the system memory. Each data transfer is synchronized with source. The source requests a transfer by asserting the request pin *DRQ’s. As soon as the request is serviced, a store is issued to the destination memory while the requesting device is selected by the DMA acknowledged pin *DAC ’s. When selected the source data device must drive the data bus for the store instead of the processor. In this case, the processor must floats the data bus. If the destination is the requestor, i.e., destination synchronization, a load is issued to the source while the destination is selected with the acknowledge pin. The destination then reads the load data; the processor ignores the data from the load.



32bit I/O device Data source

DREQ DACK

Integrated DMA controller

32bit memory Data Destination

Buffer load data

Source driven data


One word step

External system bus

Processor floats bus during store operation

Fig 7.6.4 Source-synchronized fly-by DMA operation A Fly-by DMA in block mode is started by software and can be carried out in the same way like any block mode operation multi-cycle. Fly by DMA in block mode can be used to implement highperformance, memory-to-memory transfers where source and destination addresses are fixed at the block boundaries. In this case, the acknowledge pin must be used in conjunction with external hardware to unique address, the sources and destination for the transfer. With DMA, the processor is free to work on other tasks, except when initiating each DMA and responding to the interrupt at the end of each transfer. This takes 2500 cycles/transfer, or a total of 6,250,000 cycles spent handling DMAs each second. Since the processor operates at 200 MHz this means that 3.125 percent of each second or 3.125 percent of the processor’s time is spent handling DMAs, less than one-third of the overhead without DMA. 3. DMA Controller implementation Integrated DMA controller’s functions including data chaining data alignment and so on are implemented in microcode. Processor clock cycles are required to setup and execute a DMA operation. When considering whether to use the DMA controller, two questions generally arise: 1. When a DMA transfer is executing, how many internal processor clock cycles does the DMA operation consume? 2. When a DMA transfer is executing, how much of the total bus bandwidth is consumed by the DMA bus operation? A process switch from user process to DMA process occurs as a result of a DMA event. A DMA event is signaled when a DMA channel requires services or is in the process of setting up a channel. Signaling the DMA event is controlled by DMA logic. A DMA rarely uses the maximum available cycles for the DMA process. Actual cycle allocation between user process and DMA process depends on the type of DMA operation performed, DMA channel activity external bus loading and performance. 4. DMA Transfer Performances DMA transfer performance is characterized by two values: throughput and latency. Throughput measurement is needed as a measure of the DMA transfer bandwidth. Worst-case latency is required to determine if the DMA is fast enough in responding to transfer requests from DMA devices.




Throughput describes how fast data is moved by a DMA operation. It is defined as the number of the controller clock cycles per DMA request. This value is denoted as The established measure of throughput, in units of byte/second, is derived by the following equation: Throughput (Bytes/Second) = (

f )

where:

: Number of controller clock cycles per DMA request Ng :Bytes per DMA request f : DMA Controller clock frequency In general, the DMA throughput for a particular system depends on the following factors:  DMA transfer type.  Memory system configuration and  Bus activity generated by the user process. Latency is defined as the maximal time delay measured between the assertion of the request pin *DREQ and the assertion of the acknowledgement pin *DACK. The latency here is derived in number of the controller clock cycles, which is denoted as . The established measure of DMA latency, in units of seconds, is derived by the following equation: DMA latency (Second) =

f , where:

: Number of controller clock cycles, f : DMA Controller clock frequency Address bus

Data bus

Data bus buffer

Address bus buffer

Address register DMA Select

DS

Register Select

RS

Read

RD

Write

WR

Bus Request

BR

Word count register

Control Logic

Control register DMA request

Bus Grant

BG

Interrupt

Interrupt

DMA acknowledge to I/O Device

Fig 7.6.5 Block Diagram of DMA Controller




The DMA latency in a system depends on the following factors:  DMA transfer type and subsequently the worst-case throughput value calculated for that transfer;  Number of channels enabled and the priority of the requesting channel;  Status of the suspend DMA on interrupt bit in the DMA control register.

INPUT/OUTPUT:  The devices that are connected to the periphery of the CPU are called peripheral devices. Example: input and output devices.  Input devices: It is a medium of communication between user to the computer. Example: keyboard, Floppy disk drive, hard disk drive, Mouse, Magnetic tape drive, paper tape Deader, card reader, VDU etc.,  Output device: It is a medium of communication between computer to the user. Example: VDU printers, Floppy disk drive, punched cards, potters etc.  On – line devices: Devices that are under the direct control of the processor are said to be connected on line.  Off –line devices: When a device is offline then it is operated independently of the computer.  All peripheral devices are electromechanical & electromagnetic devices.  The I/O organization of a computer is a function of the size of the computer and the devices connected to it. Auxiliary Memory: The device that provide backup storage are called auxiliary or secondary memory. The secondary memory is not directly assessable to the CPU. Example: Hard disk, floppy disk, magnetic tape etc. The important characteristics of any device are its access mode, access time, transfer rate, capacity, and cost. Seek Time: It is the time required, to move the read/write head to the proper track. This depends on the initial position of the head relative to the track specified in the address. Rotational delay (latency time): This is the amount of time that elapses after the head is positioned over the correct track until the starting position of the addressed sector passes under the read/write head. On average, this is the time for half a rotation of the disk. Access Time: The sum of these two delays is called the disk access time.




Disk Controller: Operation of a disk drive is controlled by disk controller, which also provides an interface between the disk drive and the bus that connects it to the rest of the computer System. The disk controller may be used to control more than one drive. The disk controller keeps track of such sector ad substitutes other sector instead. Main memory Address: The address of the first main memory location of the block of words involved in the transfer. Disk Address: The location of the sector containing the beginning of the desired block of words. Word Count: The number of words in the block to be transferred. Magnetic Hard Disks

Tracks

Sector

Read And Write Head

Fig 7.6.6 Magnetic Disk The disk system consists of three key parts. One part is the assembly of disk platters, which is usually referred to as the disk. The second part comprises the electromechanical mechanism that spins the disk and moves the read/write heads; it is called the disk drive. The third part is the electronic circuitry that controls the operation of the system which is called the disk controller. Each surface is divided into concentric tracks, and each track is divided into sectors. The set of corresponding tracks on all surfaces of a stack of disks forms a logical cylinder. The data on all tracks of a cylinder can be accessed without moving the read/write heads. The data are accessed by specifying the surface number, the track number, and the sector number. Floppy disks: Floppy disks are smaller, simpler, and cheaper disk units that consist of a flexible, removable, plastic diskette coated with magnetic material. One of the simplest schemes used in the first floppy disk for recording data is phase or Manchester encoding. Disks, encoded in this way are said to have single density. A more complicated variant of this scheme, called double density. The main feature of floppy disk is their low cost and shipping convenience.




RAID Disk Arrays: RAID stands for, Redundant Array of Inexpensive Disks. Using multiple disks also makes it to improve the reliability of the overall system. Six different configurations were proposed. They are known as RAID levels even though there is no hierarchy involved. RAID 0 is the basic configuration intended to enhance performance. A single large file is stored in a several separate disk units by breaking the file up into a number of smaller pieces and storing these pieces on different disks. This is called data striping. RAID 1 is intended to provide better reliability by storing identical copies of data on two disks rather than just one. The two disks are said to be the mirrors of each other. RAID 2, RAID 3 and RAID 4 levels achieve increased reliability through various parity checking schemes without requiring a full duplication of disks. All of the parity information is kept on one disk. RAID 5 also makes use of parity based error recovery scheme. However, the parity information is distributed among all disks, rather than being stored on one disk. Indeed, the term RAID has been redefined by the industry to refer to “independent” disks. ATA/EIDE Disks: EIDE (Enhanced Integrated Drive Electronics) or as ATA (Advanced Technology Attachment). Many disk manufactures have a range of disks that have EIDE/ATA interfaces. In fact Intel’ s entium chip sets include on controller that allows EIDE ATA disks to be connected to the motherboard one of the main drawback is that separate controller is needed for each drive if two drives are to be used concurrently to improve performance. RAID Disks: RAID disks offer excellent performance and provide a large and reliable storage. They are used either in high-performance computers. Optical Disks & CD Technology: The optical technology that is used for CD systems is based on a laser light source. A laser beam is directed onto the surface of the spinning disk. Physical indentation in the surface are arranged along the tracks of the disk. They reflect the focused beam toward a photo detector, which detects the stored binary patterns. The laser emits a coherent light beam that is sharply focused on the surface of the disk. Coherent light consists of synchronized waves that have the same wavelength. If a coherentlight beam iscombined with another beam of the same kind, and the two beams are in phase, then the result will be a brighter beam. But, if the waves of the two beams are 180 degrees out of phase, they will cancel each other. Thus, if a photo detector is used to detect the beams, it will detect a bright spot in the first case and a dark spot in the second case. The bottom layer is polycarbonate plastic, which function as a clear glass base. The surface of this plastic is programmed to store data by indenting it with pit. The unindented parts are called lands. The laser source and the photo detector are positioned below the polycarbonate plastic. The emitted beam travels through this plastic reflects off the aluminum layer, and travels back toward the photo detector.




CD-ROM: Science information is stored in binary form in CD’s they are suitable for use as a storage medium in computer system. The biggest problem is to ensure the integrity of stored data. Because pits are very small, it is difficult to implement all of pits perfectly. Stored data are organized on CD-ROM tracks in the form of blocks that are called sectors. Error handling Possible errors on a disk subsystem are: Programming error: For example the driver requests the controller to seek to a nonexistent sector. Most disk controllers check the parameter given to them and complain if they are invalid. Transient checksum error: That are caused by dust on the head. Most of the time they are eliminated by just repeating the operation few times. If error persists, the block has to be remarked as a bad block and avoided. Permanent checksum error: In this case, the disk blocks are assumed to be physically damaged. These errors are unrecoverable errors and these blocks are remarked as bad block and avoided. Seek error: For example, the arm was sent to cylinder 6, but it went to cylinder 7. Normally, it keeps track of the arm position internally. To perform a seek, it issues a series of pulses to the arm motor, one pulse per cylinder to move the arm to the destination cylinder. Then, the controller reads the actual cylinder number to check whether the seek operation is correct or not. If the seek error occurs, the controller moves the arm as far as it will go out, resets the internal current cylinder to 0 and tries it again. If it does not help, the drive must be repaired. Disk controller error: The controller refuses to accept command from the connected computer. It has to be replaced. Disk Structure: Disk drives are addressed as large 1-dimensional arrays of logical blocks, where the logical block is the smallest unit of transfer.   

The 1-dimensional array of logical blocks is mapped into the sectors of the disk sequentially. Sector 0 is the first sector of the first track on the outermost cylinder. Mapping proceeds in order through that track, then the rest of the tracks in that cylinder, and then through the rest of the cylinders from outermost to innermost.



Digital Logic

Part – 8: Digital Logic 8.1: Number Systems & Code Conversions Characteristics of any number system are: 1. Base or radix is equal to the number of possible symbols in the system, 2. The largest value of digit is one (1) less than the radix Decimal to Binary Conversion: (a) Integer number: Divide the given decimal integer number repeatedly by 2 and collect the remainders. This must continue until the integer quotient becomes zero. (b) Fractional Number: Multiply by 2 to give an integer and a fraction. The new fraction is multiplied by 2 to give a new integer and a new fraction. This process is continued until the fraction becomes 0 or until the numbers of digits have sufficient accuracy. Note: To convert a decimal fraction to a number expressed in base r, a similar procedure is used. Multiplication is by r instead of 2 and the coefficients found from the integers any range in value from 0 to (r-1). The conversion of decimal number with both integer and fraction parts separately and then combining the answers together.    

Don’t care values or unused states in BCD code are 1010, 1011, 1100, 1101, 1110, 1111. Don’t care values or unused state in excess – 3 codes are 0000, 0001, 0010, 1101, 1110, 1111. The binary equivalent of a given decimal number is not equivalent to its BCD value. Eg. Binary equivalent of 2510 is equal to 110012 while BCD equivalent is 00100101. In signed binary numbers,MSB is always sign bit and the remaining bits are used for magnitude. A7 A6 A5 A4 A3 A2 A1 A0

Sign Bit  

Magnitude

For positive and negative binary number, the sign is respectively ‘0’ and ‘1’. Negative numbers can be represented in one of three possible ways. 1. Signed – magnitude representation. 2. Signed – 1’s complement representation. 3. Signed – 2’s complement representation.



Example: Signed – magnitude

+9 0 0001001

Digital Logic

-9 (a) 1 000 1001 signed – magnitude (b) 1 111 0110 signed – 1’s complement (c) 1 111 0111 signed – 2’s complement

     

Subtraction using 2’s complement: Represent the negative numbers in signed 2’s complement form, add the two numbers, including their sign bit and discard any carry out of the most significant bit. Since negative numbers are represented in 2’s complement form, negative results also obtained in signed 2’s complement form. The range of binary integer number of n-bits using signed 1’s complement form is given by +(2 – 1) to –(2 – 1),which includes both types of zero’s i.e., +0 and -0. The range of integer binary numbers of n-bits length by using signed 2’s complement representation is given by + (2 – 1) to – 2n-1 which includes only one type of zero i.e. + 0. In weighted codes, each position of the number has specific weight. The decimal value of a weighted code number is the algebraic sum of the weights of those positions in which 1‘s appears. Most frequently used weighted codes are 8421, 2421 code, 5211 code and 84 2’1’ code.



Digital Logic

8.2: Boolean Algebra & Karnaugh Maps 1. Boolean properties: a) Properties of AND function 1. X . 0 = 0

2. 0 . X = 0

3. X . 1 = X

4 .1.X = X

b) Properties of OR function 5. X + 0 = X

6. 0 + X = X

7. X + 1 = 1

8. 1 + X = 1

c) Combining a variable with itself or its complement 9. X .X’ = 0

10. X . X = X

11. X + X = X

12. X + X’ = 1

13. (X’)’ = X d) e) f) g)

Commutative laws: Distributive laws: Associative laws: Absorption laws:

h) Demorgan’s laws:

14. 16. 18. 20.

x. y = y. x x(y +z) = x.y + x.z x(y.z) = (x. y) z x + xy= x

15. 17. 19. 21.

x+y=y+x x + y. z = ( x+y) (x + z) x + ( y + z) = (x + y) +z x(x + y) = x

22. x + x’y = x+ y

23. x(x’ + y) = xy

24. (x + y)’ = x’ .y’

25. (x . y)’ = x’ + y’

 Duality principle: It states that every algebraic expression deducible from theorems of Boolean algebra remains valid if the operators and identify elements are interchanged.  To get dual of an algebraic function, we simply exchange AND with OR and exchange 1 with 0.  The dual of the exclusive – OR is equal to its complement.  To find the complement of a function is take the dual of the function and complement each literal.  Maxterm is the compliment of its corresponding minterm and vice versa.  Sum of all the minterms of a given Boolean function is equal to 1.  Product of all the maxterms of a given Boolean function is equal to 0 Boolean Algebraic Theorems Theorem No. Theorem ̅) = ( + B). ( + B 1. ̅ 2. B + C = ( + C)(̅ + B) ( + B)(̅ + C) = C + ̅ B 3. 4. B + ̅ C + BC = B + ̅ C ̅ ( + B)( + C)(B + C) = ( + B)(̅ + C) 5. ̅̅̅̅̅̅̅̅ ̅ + C̅ + 6. . B. C. = ̅ + B ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅. C̅ 7. + B + C + = ̅. B



Digital Logic

Karnaugh Maps (K – maps):  A map is a diagram made up of squares. Each square represents either a minterm or a maxterms.  The number of squares in the karnaugh map is given by 2 where n = number of variable.  Gray code sequence is used in K – map so that any two adjacent cells will differ by only one bit. No. of cells Number of No. of variables No. of literals present containing 1’s variables eliminated in the resulting term grouped 4 2 0 2 1 1 2 1 0 2 8 3 0 4 2 1 2 1 2 3 1 0 3 16 4 0 8 3 1 4 2 2 4 2 1 3 1 0 4



Digital Logic

8.3: Logic Gates  OR, AND, NOT are basic gates  NAND and NOR gates are called Universal gates because, by using only NAND gates or by using only NOR gates we can realize any gate or any circuit.  EXOR, EXNOR are arithmetic gates.  There are two types of logic systems: 1) Positive level logic system (PLLS) : Out of the given two voltage levels, the more positive value is assumed as logic ‘1’ and the other as logic ‘0’. 2) Negative level logic system (NLLS):out of the given two voltage levels, the more negative value is assumed as logic ‘1’ and the other as logic ‘0’.  NOT gate:Truth Table A Y 0

1

1

0

+VCC

Symbol A Y=̅

 AND gate: Truth Table A B Y 0 0 0 0 1 0 1 0 0 1 1 1

VCC A B

Y = AB

A B

 OR gate: A 0 0 1 1

B 0 1 0 1

Y 0 1 1 1

Y=̅

A

Y

A Y = A+B

B

A Y B

 NAND gate: A 0 0 1 1

B 0 1 0 1

Y 1 1 1 0

A B

Y = ̅̅̅̅ B



Digital Logic

 NOR gate: A 0 0 1 1

B 0 1 0 1

Y 1 0 0 0

A Y = ̅̅̅̅̅̅̅ +B

B

 The circuit, which is working as AND gate with positive level logic system, will work as OR gate with negative level logic system and vice-versa.  The circuit which is behaving as NAND gate with positive level logic system will behave as NOR gate with negative level logic system and vice – versa.  Exclusive OR inputs”. A B 0 0 0 1 1 0 1 1

gate (X– OR): “The output of an X – OR gate is high for odd number of high A

Y 0 1 1 0

Y = A⊕B= B’ + ’B

B

 Exclusive NOR gate (X–NOR): The output is high for odd number of low inputs”. (OR) “The output is high for even number of high inputs”. A B Y A 0 0 1 Y = A⨀B= B + ’B’ 0 1 0 B 1 0 0 1 1 1  Realization of Basic gates using NAND and NOR gates:  NOT gate A

NAND Y=̅

A A 1

NOR Y = ( . )’ A = Y = ( .1)’ A 0 = ’

( + )’ = ’ Y = ( + 0)’ =

 AND gate A A B

A Y =AB

B

Y =AB

Y =AB

B

 OR gate: A A B

A Y =A+B B

Y = A+B B

Y = A+ B



Digital Logic

 Realization of NAND gate using NOR gates: A

A Y = ( B)’

Y = ( B)’ B  Realization of NOR gate using NAND gates: A

A Y = ( + B)’

B

B

Y = ( + B)’

 Realization of X – OR gate using NAND and NOR gates: A Y = B’+ ’B

B A

Y = B’ + ’B B A `

Y = B’ +

B

B  The minimum number of NAND gates required to realize X – OR gate is four.  The minimum number of NOR gates required to realize X – OR gate is five.  Equivalence Properties: 1. (X ⊕Y)’ = X’Y’ + XY = X 2. X 0 = X’ 3. X 1 = X 4. X X = 1 5. X X’= 0 6. X Y = Y X 7. (X Y)’ = X ⊕ Y

Y



Digital Logic

Alternate Logic Gate Symbols: A bubbled NAND gate is equivalent to OR gate A `

A Y=( B)

B `

=A+B

Y = A+B

B

 A bubbled NOR gate is equivalent to AND gate A Y=(

B

A `

+ B ) =AB

B `

Y= B

 A bubbled AND gate is equivalent to NOR gate A ` B `

A Y=

B = ( + B)

Y = ( + B)

B

 A bubbled OR gate is equivalent to NAND gate A B

Y=

+ B =( B)

A ` B `

Y = ( B)



Digital Logic

8.4: Combinational Digital Circuits Digital circuits can be classified into two types: o Combinational digital circuits and o Sequential digital circuits. Combinational Digital Circuits: In these circuits “the outputs at any instant of time depends on the inputs present at that instant only.” For the design of Combinational digital circuits, basic gates (AND, OR, NOT) or universal gates (NAND, NOR) are used. Examples for combinational digital circuits are adder, decoder etc. Sequential Digital Circuits: The outputs at any instant of time not only depend on the present inputs but also on the previous inputs or outputs. For the design of these circuits in addition to gates we need one more element called flip-flop. Examples for sequential digital circuits are Registers, Shift register, Counters etc. Half Adder: A combinational circuit that performs the addition of two bits is called a halfadder. Sum = X ⊕ Y = XY’ + X’ Y

Carry = XY

Half Subtractor: It is a Combinational circuit that subtracts two bits and produces their difference. Diff. = X ⊕ Y = XY’ + X’Y Borrow = X’ Y Half adder can be converted into half subtractor with an additional inverter. Full Adder: It performs sum of three bits (two significant bits and a previous carry) and generates sum and carry. Sum=X⊕ ⊕Z Carry = XY + YZ + ZX Full adder can be implemented by using two half adders and an OR gate. X Y

H.A.

H.A.

Sum

Z

Carry

Full subtractor: It subtracts one bit from the other by taking pervious borrow into account and generates difference and borrow. Diff.=X⊕ ⊕Z Borrow = X’Y + YZ + ZX’ 

Full subtractor can be implemented by using two half- subtractors and an OR gate. X Y Z

H.S.

H.S.

Diff.

Borr.



Digital Logic

Multiplexers (MOX)   

It selects binary information from one of many input lines and directs it to a single output line The selection of a particular input line is controlled by a set of selection lines There are 2 input lines where ‘n’ is the select lines i/p then n = log 2 : 1 MUX I 2:1 MUX

I

Y=S̅I + SI

Y

S 4 : 1 MUX I I I I

4:1 MUX

S1

S1 0 0 1 1

Y

S0 0 1 0 1

Y I I I I

S0

Y=S̅ S̅ I + S̅ S I + S S̅ I + S S I  Decoder: Decoder is a combinational circuit that converts binary information from ‘n’ input lines to a maximum of 2 unique output lines. Truth table of active high output type of decoder.

X

Y

D

D

D

D

0

0

1

0

0

0

0

1

0

1

0

0

1

0

0

0

1

0

1

1

0

0

0

1

X 2

4

Y



Digital Logic

 Encoder  Encoder is a combinational circuit which has many inputs and many outputs  It is used to convert other codes to binary such as octal to binary, hexadecimal to binary etc.  Clocked S-R Flip-flop: It is called set reset flip-flop. No change Reset set Forbidden

0

0

0

1

0

1

0

1

1

1

*

Pr S Clk R

Q

Cr = S +R

PRESET S

Q

Clk

Q’

R CLEAR

 S and R inputs are called synchronous inputs. Preset (pr) and Clear (Cr) inputs are called direct inputs or asynchronous inputs.  The output of the flip-flop changes only during the clock pulse. In between clock pulses the output of the flip flop does not change.  During normal operation of the flip flop, preset and clear inputs must be always high.  The disadvantage of S-R flip-flop is S=1, R=1 output cannotbe determined. This can be eliminated in J-K flip-flop.  S-R flip flop can be converted to J-K flip-flop by using the two equation S=J ’ and R= K .

J Q’

Pr

S

Q

J

Clk

Clk Q’

R Q K

Q

Cr

=J

K

Q’

+K



Digital Logic

Truth table

0

0

0

1

0

1

0

1

1

1

 Race around problem is present in the J-K flip flop, when both J=K=1.  Toggling the output more than one time during the clock pulse is called Race around Problem.  The race around problem in J-K flip-flop can be eliminated by using edge triggered flip-flop or master slave J-K flip flop or by the clock signal whose pulse width is less than or equal to the propagation delay of flip-flop.  Master-slave flip-flop is a cascading of two J-K flip-flops Positive or direct clock pulses are applied to master and these are inverted and applied to the slave flip-flop.  D-Flip-Flop: It is also called a Delay flip-flop. By connecting an inverter in between J and K input terminals. D flip-flop is obtained. Truth table J

D 0

0

1

1

D

Q Clk

K

Q’

T Flip-flop: J K flip-flop can be converted into T- Flip-flop by connecting J and K input terminals to a common point. If T=1, then Q n+1 = . This unit changes state of the output with each clock pulse and hence it acts as a toggle switch. Truth table T 0 1

T

J

Q Clk

K

’

 Ring Counter: Shift register can be used as ring counter when Q0 output terminal is connected to serial input terminal.  An n-bit ring counter can have “n” different output states. It can count n-clock pulses.  Twisted Ring counter: It is also called Johnson’s Ring counter. It is formed when output terminal is connected to the serial input terminal of the shift register.  An n-bit twisted ring counter can have maximum of 2n different output states. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 290


Digital Logic

Counters: The counter is driven by a clock signal and can be used to count the number of clock cycles counter is nothing but a frequency divider circuit.  Two types of counters are there: (i) Synchronous (ii) Asynchronous  Synchronous counters are also called parallel counters. In this type clock pulses are applied simultaneously to all the flip – flops  Asynchronous counters are also called ripple or serial counter. In this type of counters the output of one flip – flop is connected to the clock input of next flip – flop and soon.



Digital Logic

8.5: Semiconductor Memory The capacity of a memory IC is represented by 2 xm, where ‘2 ’ represents number of memory locations available and ‘m’ represents number of bits stored in each memory location. Example:- 2 8 = 1024 8 To increase the bit capacity or length of each memory location, the memory ICs are connected in parallel and the corresponding memory location of each IC must be selected simultaneously. Eg. 1024 × 8 memory capacity can be obtained by using 4 ICs of memory capacity 1024×2. Types of Memories:

Memories

Semiconductor Memories

Magnetic Memories

Drum

Read/Write Memory (RAM or user memory)

Disk

Bubble

Core

Read Only Memory (ROM)

PROM Static RAM

Tape

EPROM

EEPROM

Dynamic RAM

 Volatile Memory: The stores information is dependent on power supply i.e., the stored information will remain as long as power is applied. Eg. RAM  Non- Volatile Memory: The stored information is independent of power supply i.e., the stored information will present even if the power fails. Eg: ROM, PROM, EPROM, EEPROM etc.  Static RAM (SRAM): The binary information is stored in terms of voltage. SRAMs stores ones and zeros using conventional Flip-flops.  Dynamic RAM (DRAM): The binary information is stored in terms of charge on the capacitor. The memory cells of DRAMs are basically charge storage capacitors with driver transistors. Because of the leakage property of the capacitor, DRAMs require periodic charge refreshing to maintain data storage.  The package density is more in the case of DRAMs. But additional hardware is required for memory refresh operation.  SRAMs consume more power when compared to DRAMs. SRAMS are faster than DRAMs.



Complier Design

Part – 9: Complier Design 9.1 Introduction to Compilers Translator: A translator is a program that takes as input a program written in one language and produces as output a program in another language. Beside program translation, the translator performs another role, the error-detection. Any violation of the HLL specification would be detected and reported to the programmers. Important role of translator are: translating the HLL program input into an equivalent ML program and providing diagnostic messages wherever the programmer violates specification of the HLL. Type of Translators: - Interpreter, Compiler, Preprossessor, Etc Source program Preprocessor Source program Compiler Target assembly program Assembler Relocatable machine code Loader/link editor

library, relocatable object file

Absolute machine code Fig. 9.1.1.A language-processing system. In addition to a compiler, several other programs may be required to create an executable target program. Figure 9.1.1 shows a typical language processing process along with the compilation”. Interpreter: Unlike compiler, interpreter takes single instruction & executes. Advantage of Interpreter: 

Certain language features supported by Interpreter rather than compiler. “Portability”



Complier Design

 COMPILERS 

A compiler is a program that reads a program written in one language – the source language – and translates it into an equivalent program in another language – the target language As an important part of this translation process, the compiler reports to its user the presence of errors in the source program.



Source code

Applications:   

Compiler

Machine code

Errors

Design of Interfaces Design of language migration tools Design of Re – engineering Tools

Two-Pass Assembly: The simplest form of assembler makes two passes over the input, where a pass consists of reading an input file once. In the first pass, all the identifiers that denote storage locations are found and stored in a symbol table. In the second pass, the assembler scans the input again. This time, it translates each operation code into the sequence of bits representing that operation in machine language, and it translates each identifier representing a location into the address given for that identifier in the symbol table. The output of the second pass is usually relocatable machine code, meaning that it can be loaded starting at any location L in memory; i.e., If L is added to all addresses in the code, then all references will be correct. Thus, the output of the assembler must distinguish those portions of instructions that refer to addresses that can be relocated. Loaders and Link-Editors:   

A program called loader performs the two functions of loading and link-editing. The process of loading consists of taking relocatable machine code, altering the relocatable addresses and placing the altered instructions and data in memory at the proper locations. The link-editor makes a single program from several files of relocatable machine code.

The Phases of a Compiler A compiler includes the different phases, each of which transforms the source program from one representation to another. From figure 9.1.2 the compiler structure has following:      

Lexical analysis Syntax Analysis Semantic analysis Intermediate code generation Code optimization Target code generation



Source code

Front end

Intermediate Language

Back end

Complier Design

Machine code

Errors

Front end is 0(n) or 0(n logn) Back end is NP-complete The six phases divided into 2 Groups 1. Front End: Depends on stream of tokens and parse tree ( also called analysis phase) 2. Back End: Dependent on Target, Independent of source code ( also called synthesis phase) The Compilation Model There are two parts to compilation: analysis and synthesis. High level program

Lexical Analyzer Stream of tokens Syntax Analyzer Parse tree Semantic Analyzer Annotates Parse tree Symbol Table Management

Intermediate code Generation

Error Handling Table

Intermediate form Code Optimization Optimized intermediate form Code Generatin

Assembly Program Fig. 9.1.2. Compiler structure



Complier Design

Symbol-Table Management  

A symbol table is a data structure containing a record for each identifier, with fields for the attributes of the identifier. Symbol table is a data Structure in a compiler used for managing information about variables & their attributes.

Error Detection and Reporting   

The syntax and semantic analysis phases usually handle a large fraction of the errors detectable by the compiler. The lexical phase can detect errors where the characters remaining in the input do not form any token of the language. Errors where the token stream violates the structure rules (syntax) of the language are determined by the syntax analysis phase.

ANALYSIS PHASE OF THE SOURCE PROGRAM 1. Linear or Lexical analysis, in which stream of characters making up the source program is read from left-to-right and grouped into tokens that are sequences of characters having a collective meaning. 2. Hierarchical or Syntax analysis, in which characters or tokens are grouped hierarchically into nested collections with collective meaning. 3. Semantic analysis, in which certain checks are performed to ensure that the components of a program fit together meaningfully. Lexical Analysis:   

The lexical analyzer is the first phase of a compiler. Its main task is to read the input characters and produce as output a sequence of tokens that the parser uses for syntax analysis. Sometimes, lexical analyzers are divided into a cascade of two phases, the first called “scanning” and the second "lexical analysis." The scanner is responsible for doing simple tasks, while the lexical analyzer does the more complex operations. Consider the expression t=t t where t,t t are floats Lexical analyzer will generate id

id

id

Syntax Analysis:   

Hierarchical analysis is called parsing or syntax analysis. It involves grouping the tokens of the source program into grammatical phrases that are used by the compiler to synthesize output. The hierarchical structure of a program is usually expressed by recursive rules. For example, we might have the following rules as part of the definition of expressions: 1. Any identifier is an expression. 2. Any number is an expression. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 296


Complier Design

3. If expression1 and expression2 are expression, then so are expression1 + expression 2 expression1 * expression 2 (expression1) Ex. Parser will generate

= id

+ ×

id

12

id Semantic Analysis:   

The semantic analysis phase checks the source program for semantic errors and gathers type information for the subsequent code-generation phase. It uses the hierarchical structure determined by the syntax-analysis phase to identify the operators and operands of expressions and statements. An important component of semantic analysis is type checking. Ex. Now as t +1 &t are float. 12 is also converted to float

= id

+ ×

id id

Int to float 12

Intermediate Code Generation (or) ICG:  

After syntax and semantic analysis compiler generate an explicit intermediate representation of the source program. This intermediate representation should have two important properties; easy to produce, and easy to translate into the target program. Ex. Intermediate code will be te = id . te id

id te

te



Complier Design

Code Optimization: The code optimization phase attempts to improve the intermediate code, so that faster-running machine code will result. Some optimizations are trivial. Advantages of Code Optimization:Improves Efficiency Occupies less memory Executes fast

-

Ex. Optimized code will be te id

id id

. te

Code Generation: The final phase of the compiler is the generation of target code, consisting normally of relocatable machine code or assembly code. Memory locations are selected for each of the variables used by the program. Then, intermediate instructions are each translated into a sequence of machine instructions that perform the same task. A crucial aspect is the assignment of variables to registers. Machine code will look like MUL ADD MOV id Where

. contains id &

contains id .

 Lexical Analysis 

A token is a string of characters, categorized according to the rules as a symbol (e.g. IDENTIFIERS, KEYWORDS, OPERATORS, CONSTANTS, LITERAL STRINGS and PUNCTUATION SYMBOLS such as parenthesis, commas, and semicolons.).



The process of forming tokens from an input stream of characters is called tokenization and the lexer categorizes them according to a symbol type.



A lexeme is a sequence of characters in the source program that is matched by the pattern for a token. For example, the following statement



Int number ; The substring ‘nu ber’ is a lexe e for the token “identifier” or “ID” ‘int’ is a lexe e for the token “keyword” and ‘;’ is a lexe e for the token”;” THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 298


Complier Design

For example, in lexical analysis the characters in the assignment statement position : = initial + rate * 60, would be grouped into the following tokens: 1. 2. 3. 4. 5. 6. 7.

The identifier position. The assignment symbol : = The identifier initial. The plus sign. The identifier rate. The multiplication sign. The number 60.

The main purpose of a lexical analyzer in a compiler application is to translate the input stream into a form that is more manageable by the parser. However the tasks of a lexical analyzer can be divided into two phases. They are: Scanning & Lexical analysis. Lexical analyzer can also detect some lexical errors. a) Scanning: In the scanning phase it scans the input file and eliminates comments and white spaces in the form of blank, tab and new-line characters. So the parser will have not to consider it. The alternative is to incorporate white space into the syntax which is not nearly as easy to implement. This is why most compilers do such tasks at scanning phase. b) Lexical Analysis: At the second phase it matches pattern for each lexeme to generate token. In some compilers, the lexical analyzer is in charge of making a copy of the source program with the error message marked in it. It may also implements preprocessor functions if necessary. Issues in Lexical Analysis : There are several reasons for separating the analysis phase of compiling into lexical analysis and parsing. 1. Simpler design is perhaps the most important consideration. 2. Compiler efficiency is improved. A separate lexical analyzer allows us to construct a specialized and potentially more efficient processor for the task. 3. Compiler portability is enhanced. Input alphabet peculiarities and other device-specific anomalies can be restricted to the lexical analyzer. Tokens, Patterns, Lexemes (Important Point)   

When talking about lexical analysis, we use the terms "token," "pattern," and "lexeme" with specific meanings. There is a set of strings in the input for which the same token is produced as output. This set of strings is described by a rule called a pattern associated with the token. A lexeme is a sequence of characters in the source program that is matched by the pattern for a token.



Complier Design

TOKEN

SAMPLES LEXEMES

INFORMAL DESCRIPTION OF PATTERN

const

const

const

If

If

If

relation

<, < =, =, <>, >, > =

< or < = or <> or > = or >

Id

pi, count, D2

letter followed by letters and digits

Num

3.1416, 0, 6.02E23

any numeric constant

literal

“core du

any characters between “and” exce t”

ed”

Fig. 9.1.2 Examples of tokens Lexical Errors: Few errors are discernible at the lexical level alone. But, suppose a situation does arise in which the lexical analyzer is unable to proceed because none of the patterns for tokens matches a prefix of the remaining input. Perhaps the simplest recovery strategy is “ anic ode” recovery. We delete successive characters from the remaining input until the lexical analyzer can find a well-formed token. This recovery technique may occasionally confuse the parser, but in an interactive computing environment it may be quite adequate. Other possible error-recovery actions are: 1. 2. 3. 4.

Deleting an extraneous character Inserting a missing character Replacing an incorrect character by a correct character Transposing two adjacent characters.



9.2

Complier Design

Syntax Analysis

The syntax of programming language constructs can be described by context-free grammars or BNF (Backus-Naur Form) notation. Grammars offer significant advantages to both language designers and compiler writers.   



A grammar gives a precise, yet easy-to-understand syntactic specification of a programming language. From certain classes of grammars we can automatically construct an efficient parser that determines if a source program is syntactically well formed. A properly designed grammar imparts a structure to a programming language that is useful for the translation of source programs into correct object code and for the detection of errors. Tools are available for converting grammar-based descriptions of translations into working programs. Languages evolve over a period of time, acquiring new constructs and performing additional tasks. These new constructs can be added to a language more easily when there is an existing implementation based on a grammatical description of the language.

The Role of The Parser In our compiler model, the parser obtains a string of tokens from the lexical analyzer, as shown in Fig. 9.2.1 and verifies that the string can be generated by the grammar for the source language. token source progra m

lexical analyze r

Parser get next token

parse tree

rest of front end

intermediate representation

symbol table

Fig. 9.2.1 Position of parser in compiler model Syntax Error Handling: Planning the error handling right from the start can both simplify the structure of a compiler and improve its response to errors. Error-Recovery Strategies: panic mode, phrase level, error productions, global correction

Context-Free Grammars Grammars: It is a set of finite rules that may define infinite sentence. A context- free grammar (grammar for short) consists of terminals, non terminals, a start symbol, and productions.



Complier Design

A context-free grammar G is a four-type (T,NT,S,P)  

 

T is the set of terminals Non terminals are syntactic variables that denote sets of strings. The non terminals define sets of strings that help to define the language generated by the grammar. They also impose a hierarchical structure on the language that is useful for both syntax analysis and translation. In a grammar, one non terminal is distinguished as the start symbol, and the set of strings it denotes is the language defined by the grammar. The productions of a grammar specify the manner in which the terminals and non terminals can be combined to form strings. Each production consists of a non terminal, followed by an arrow (sometimes the symbol: = is used in place of the arrow), followed by a string of non terminals and terminals. i.e., Productions of the form NT (T+NT)*

Derivations: The central idea here is that a production is treated as a rewriting rule in which the non terminal on the left is replaced by the string on the right side of the production. We can take a single Non terminal and repeatedly apply productions in any order to obtain a sequence of replacements. We call such a sequence of replacements as derivation. Likewise, we use +  to mean “derives in one or more steps. To ean “derives in zero or ore ste s”. "Given a grammar G with start symbol S, we can use the S

relation to define L (G); the

language generated by G. Strings in L (G) can contain only terminal symbols of G. We say a string of terminals w is in L (G) if and only if S

w. The string w is called a sentence of G. A language

that can be generated by a CFG is said to be a context-free language, If two grammars generate the same language, the grammars are said to be equivalent. If S

α where α

ay contain non ter inals then we say that α is a sentential form of G.

Note: A sentence is a sentential form with no non terminals. Parse Trees and Derivations: A parse tree may be viewed as a graphical representation for a derivation that filters out the choice regarding replacement order. The leaves of the parse tree are labeled by non terminals or terminals and, read from left to right; they constitute a sentential form, called the yield or frontier of the tree. Note : Let G = {V, T, P, S} is a grammar. The parse trees for G are trees with the following conditions: 1. Each interior node is labeled by a variable in V. 2. Each leaf is labeled by terminal or epsilon. However, if the leaf is labeled epsilon, then it must be the only child of its parent.



Complier Design

Parsing means construct a parse tree. Using this parse tree we determine whether a string can be generated by a grammar. We can construct a parse tree in the following two ways: Top-Down parsing: When we construct a parse tree expanding the root, then expand all the non terminals until we get the leaves. Bottom-up parsing: When we construct a parse tree from bottom i.e., from leaf and get the root this parsing process is known as bottom-up parsing. A parse tree ignores variations in the order in which symbols in sentential forms are replaced. These variations in the order in which productions are applied can also be eliminated by considering only leftmost (or rightmost) derivations. It is not hard to see that every parse tree has associated with it a unique leftmost and/or a unique rightmost derivation. Grammar could be ambiguous or unambiguous. Ambiguous Grammars

Unambiguous Grammars

1. There exist more than one “LMD/ given string

MD ” for a

2. More than one parse tree for a given string

1. Unique “LMD/ MD” for a given string 2.

Unique parse tree for a given string

Ambiguity: A grammar that produces more than one parse tree for some sentence is said to be ambiguous. Eliminating Ambiguity Example : S S+S|S S|a Expression :a+a*a S S a

+ S a

S S+T|T T T*F|T

S S

S *

S

S

a

a

* S

S a

+ a

Two different parse trees

F a This grammar is equivalent to S S+S|S*S|a But it is unambiguous



Complier Design

Importance of Left Recursion: Consider the example, one left-recursive: S –>Sx | x and one right-recursive: S –>xS | x There are the same number of states in the LR(1) parsers for the two grammars so there is no advantage in the runtime memory requirements for either table. Consider parsing the input xxx...x with a thousand x’s. In the left-recursive form, the parser shifts the first x, immediately reduces it to S, shifts another x, and then reduces Sx to S. It does this again and again for each subsequent x. The parse stack grows and shrinks, only getting three deep at its maximum. This left recursive will give infinite loop to LL (1) parser but LR parser will not get any problem. Hence elimination left recursive form is required for LL (1) parsers or Predictive parsers or Top-Down parsers. For the right-recursive grammar, the parser shifts the first x, and then shifts the second x, and so on. The arser doesn’t reduce until it gets to the $ at the end of the in ut. It then reduces the last x on top of the stack to S, then reduces the xS on the stack to S, again and again until the stack is empty. The stack had a thousand states on it by the time we got to the end of the input Although the LR parsing can handle either, the left-recursive grammar is handled more efficiently.

Elimination of Left Recursion: 

A grammar is left recursive if it has a non terminal A such that there is a derivation A



Non-recursive predictive parser LL (1) methods cannot handle left-recursive grammars, so a transformation that eliminates left recursion is needed. We can eliminate immediate left recursion from them by the following technique. First, we group the A- productions as



Aα

A→Aα1|Aα2|………..|Aαm|β1|β2|……..|βn Where no β begins with an A. Then we re lace the A-productions by A →β1A’ | β2A’|………..|βnA’ A’→α1A’|α2A’|………..|αmA’|ϵ   

Left recursion may appear either immediate or indirect left recursion in the grammar. If productions contain immediate left recursion then the above rule can be applied individually to the A-productions. If productions contain indirect left recursion then substitution procedure applied in the grammar. SAa|b AAc|Sd|∈ After removing left recursion for A, we will get SAa|b ASdA’ |A’ A’cA’ | ∈



But still there left recursion. Consider ASdA’ substitution first SAa | b A Ac | Aad | bd | ∈ After removing left recursion, SAa | b AbdA’ | A’ AcA’ | adA’ | ∈

Complier Design

S Aa, So it will be AAadA’. So we use

Left Factoring: Left factoring is a grammar transformation that is useful for producing a grammar suitable for predictive parsing. Left factoring is useful to avoid backtracking nature for parsers. The basic idea is that when it is not clear which of two alternative productions to use to expand a non terminal A, we may be able to rewrite the A-productions to defer the decision until we have seen enough of the input to make the right choice. For example, if we have the two productions st t → if ex r then st t else st t | if ex r then st t on seeing the input token if, we cannot immediately tell which production to choose to expand statement. In general if A →αβ1| αβ2 are two A-productions, and the input begins with a nonempty string derived fro A we do not know whether to ex and A to αβ1or to αβ2by seeing α. However, we ay defer the decision by ex anding A to αA'. Then after seeing the in ut derived fro we ex and A' to β1 or to β2. That is, left-factored, the original productions become A →α A’ A’→β1| β2 Example:Sabc | abd | ae | f Removing left factoring, SabS’ | ae | f S’c|d Once, again, repeat, the same procedure: SaS’’ | f S’ c | d S’’bS’ | e

Top-Down Parsing Recursive-Descent Parsing / Predictive Parsing: 

Top-down parsing, called recursive descent, that may involve backtracking, that is, making repeated scans of the input.



Complier Design

Transition Diagrams for Predictive Parsers: To construct the transition diagram of a predictive parser from a grammar, first eliminate left recursion from the grammar, and then left factor the grammar. Then for each non-terminal A do the following: 1. Create an initial and final (return) state. 2. For each production A→XR1 , XR2 . . . . XRn , create a path from the initial to the final slate, with edges labeled XR1 , XR2 . . . .XRn  A predictive parsing program based on a transition diagram attempts to match terminal symbols against the input, and makes a potentially recursive procedure call whenever it has to follow an edge labeled by a non-terminal.  A non-recursive implementation can be obtained by stacking the states  The above approach works if the given transition diagram does not have non determinism. Non Recursive Predictive Parsing: 

It is possible to build a non-recursive predictive parser by maintaining a stack explicitly, rather than implicitly via recursive calls.  The key problem during predictive parsing is that of determining the production to be applied for a non-terminal.  A table-driven predictive parser has an input buffer, a stack, a parsing table, and an output stream. a. The input buffer contains the string to be parsed, followed by $, a symbol used as a right end marker to indicate the end of the input string. The stack contains a sequence of grammar symbol with a $ on the bottom, indicating the bottom of the stack. Initially the stack contains the start symbol of the grammar on top of $. b. The parsing table is a two-dimensional array M [X, a], where X is a non-terminal, and a is a terminal or the symbol $. c. The key problem during predictive parsing is that of determining the production to be applied for a non-terminal. The non-recursive parse looks up the production to be applied in the parsing table  The parser is controlled by a program that behaves as follows. The program considers X, the symbol on top of the stack, and a, the current input symbol. These two symbols determine the action of the parser. There are three possibilities. a. If X = a = $, the parser halts and announces successful completion of parsing. b. If X = a = $, the parser pops X off the stack and advances the input pointer to the next input symbol. c. If X is a non-terminal, the program consults entry M [X, a], a| of the parsing table M. This entry will be either an X-production of the grammar or an error entry.



Complier Design

FIRST and FOLLOW   

The construction of a predictive parser is aided by two functions associated with a grammar G. These functions, FIRST and FOLLOW, allow us to fill in the entries of a predictive parsing table for G, whenever possible. If a is any string of grammar symbols let FI ST (α ) be the set of ter inals that begin the strings derived from a. If ϵ , then e is also in FI ST(α). Define FOLLOW(A), for non-terminal A, to be the set of terminals a that can appear immediately to the right of A in some sentential form, that is, the set of terminals a such that there exists a derivation of the form S αAaβ for so e α and β Note that there ay at so e ti e during the derivation have been sy bols between A and α but if so they derived ϵ and disappeared. It A can be the rightmost symbol in some sentential form, then $ is in FOLLOW {A). To build FIRST (X) 1. 2. 3. 4.

If X is a terminal, FIRST (X) is {X} If X ϵ then ϵ ∈ FIRST (X) If X … . then put FIRST ( ) in FIRST (X) If X is a non-terminal and X …. then a ∈ FIRST (X) if a FIRST ( ) and ϵ ∈ FIRST ( ) for all ≤ j
To compute FOLLOW(A) for all non-terminals A, apply the following rules until nothing can be added to any FOLLOW set. To build follow(X): 1. 2. 3. 4.

Place $ in follow (〈goal〉) If A β then ut {first(β)-ϵ} in follow(B) If A then put follow (A) in follow (B) If A β and ϵ ∈ first (β) then ut follow (A) in follow( )



Example: S aBDh B cC C bC D EF E gl∈ F fl∈

Complier Design

Follow S | {$} Follow B | {g,f,h} Follow C | {g,f,h} Follow D | {h} Follow E | {h,f} Follow F | {h}

Construction of Predictive Parsing Tables: The following algorithm can be used to construct a predictive parsing table for a grammar G. The idea behind the algorithm is the following. Suppose A → α is a roduction with a in FIRST (α). Then, the parser will expand A by a when the current input symbol is α. The only co lication occurs when a ϵor α ϵ, In this case, we should again expand A by a if the current input symbol is in FOLLOW(A), or if the $ on the input has been reached and $ is in FOLLOW(A). Algorithm for Construction of a predictive parsing table. Input: Grammar G. Output: Parsing table M. Method: 1. ∀ roduction A perform steps 2- 4 2. ∀ ter inal a in first ( ) add A to M[A,O] 3. If ϵ ∈ first ( ) add A to M[A,b] ∀ terminal b∈ follow (A) 4. If ϵ ∈ first ( ) and eof ∈ follow (A), add A to M[A,eof] Example (i) Consider the grammar: E E+T|T TT×F|F F(E)|id First remove the left recursion. E TE’ E’ TE’ |E TFT’ T FT’ |E F(E) | id, (i) first (E) = first(T) = first(F) = {∁, id} First (E’) { E} first (T’) {×,E} Then follow (E) {$} follow (E’) {$ )} follow (T) (F) = {×,+,$,)}

{ $ )} follow (T’)

{ $ )} follow



Complier Design

(ii) id E

(

)

$

E’∈

E’∈

T’∈

T’∈

ETE’ E’ TE’

TFT’

TFT’ T’∈

T’ F

×

ETE’

E’ T

+

T’×FT’

Fid

(iii) consider the input to id+id Stack Input id+id$ $E id+id$ $ET id+id$ $E’T’F id+id$ $E’T’ID +id$ $E’T’ +id$ $E’ +id$ $E’T’ id$ $E’T id$ $E’T’F id$ $E’T’id $ $E’T’ $ $E’ $ $

F(E)

Action ETE’ TFT’ Fid Pop id T’ε E’ TE’ Pop+ TFT’ FID Pop id T’ε E’ε Success

 LL (1) Grammars  

A grammar whose parsing table has no multiply-defined entries is said to be LL (1) Grammar. The first "L" in LL(1) stands for scanning the input from left to right the second "L” for producing a leftmost derivation and the “1” or using one input symbol of look ahead at each step to make parsing action decisions. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 309


 

Complier Design

No ambiguous or left recursive grammar can be LL (1). A grammar G is LL(1) if and only if whenever A → α | β are two distinct roductions of G the following conditions hold: 1. For no ter inal α do both α and β derive strings beginning with a 2. At ost one of α and β can derive the e ty string. 3. If β ϵ, then a does not derive any string beginning with a terminal in FOLLOW (A).

Check whether Grammar is LL (1) or not: Follow the order to check the given grammar is LL (1) or not: 1) Unambiguous Suppose given grammar is Ambiguous Grammar then it is not LR (0), LL (1), LR (1) (or) CLR (1), SLR (1), LALR (1) 2) Left factored and No left recursion: Grammar may be LL (1), if grammar is left factored and contains no left recursion 3)LL (1) Grammar (Validations): .A→

|

If first (

|

), first (

……. | ) are mutually disjoint then

First ( ) ∩ First ( ) = ∅ ,for all i, j are in between 1 and n 2. A → /∈ First ( ) ∩ Follow (A) = ∅ Note: For given Grammar, if LL(1) parsing table is Constructed without any multiple entries, then Grammar is LL(1). Example : SAa | b | cB | d AaA | d BbB | d It is not LL(1), because let S 1 | | | Where Aa b c d First ( 1) ∩ first ( ) = {a,d} ∩ {d}≠ ∅

Bottom-up parsing 

Shift-reduce parsing attempts to construct a parse tree for an input string beginning at the leaves (the bottom) and working up towards the root (the top).



Complier Design

Handles       

Informally, a "handle" of a string is a substring that matches the right side of a production, and whose reduction to the nonterminal on the left side of the production represents one step along the reverse of a rightmost derivation. In any cases the left ost substring β that atches the right side of so e roduction A→β is not a handle, because a reduction by the production A →β yields a string that cannot be reduced to the start symbol. A handle of a right-sentential form is a production A β and a position in where β ay be found. If (A β k) is a handle then re lacing the β in at osition k with a roduces the revious right-sentential for in a right ost derivation of . If G is unambiguous, then every right-sentential form has a unique handle. The process we use to construct a bottom-up parse is called Handle-Pruning. To construct a rightmost derivation S =   …  We set i to n and apply the following simple algorithm Do i = n to 1by -1 (1) Find the handle (A β k ) in (2) Replace β with A to generate

Stack Implementation of Shift-Reduce Parsing: There are two problems that must be solved if we are to parse by handle pruning.     



The first is to locate the substring to be reduced in a right-sentential form. Second is to determine what production to choose in case there is more than one production with that substring on the right side. A convenient way to implement a shift-reduce parser is to use a stack to hold grammar symbols and an input buffer to hold the string w to be parsed. We use $ to mark the bottom of the stack and also the right end of the input. Initially, the stack is empty, and the string w is on the input. The arser o erates by shifting zero or ore in ut sy bols onto the stack until a handle β is on to of the stack. The arser then reduces β to the left side of the a ro riate roduction. The parser repeats this cycle until it has detected аn error or until the stack contains the start symbol After entering this configuration, the parser halts and announces successful completion of parsing.

While the primary operations of the parser arc shift and reduce, there are actually four possible actions a shift-reduce parser can make: (1) shift, (2) reduce. (3) accept and (4) error. 1. In a shift action, the next input symbol is shifted onto the top of the stack. 2. In a reduce action, the parser knows the right end of the handle is at the top of the stack. It must then locate the left end of the handle within the stack and decide with what nonterminal will be used to replace the handle. 3. In an except action, the parser announces successful completion of pursing. 4. In an error action, the parser discovers a syntax error that has occurred and calls an error recovery routine.



Complier Design

LR parsers: 

    

The technique is called LR(k) parsing; the "L" is for left-to-right scanning of the input, the "R" is for constructing a rightmost derivation in reverse, and the * for the number of input symbols of look ahead that are used in making parsing decisions. When (k) is omitted, k is assumed to be 1. LR parsing is attractive for a variety of reasons. LR parsers can be constructed to recognize virtually all programming- language constructs for which context-free grammars can be written. The LR parsing method is the most general non-backtracking shift-reduce parsing method known, yet it can be implemented as efficiently as other shift-reduce methods. The class of grammars that can be parsed using LR methods is a proper superset of the class of grammars that can be parsed with predictive parsers. An LR parser can detect a syntactic error as soon as it is possible to do so on a left-to-right scan of the input. The principal drawback of the method is that it is too much work to construct an LR parser by hand for a typical programming-language grammar.

The LR parsing algorithm:     

Input contain the input string Stack contains a string of the form S S … S where each each S is a state Table contain action and go to parts. Action table is indexed by state and terminal symbol. Goto table is indexed by state and non terminal symbol.

is a grammar symbol and

Actions in an LR (shift reduce) parser Assume S is top of stack and a is current input symbol Action [S a can have four values 1. Shift a to the stack and goto state S 2. Reduce by a rule 3. Accept 4. Error Configurations in LR parse Stack :S S . . . S input :a a . . . a If action [S . a ] = shift S then the configuration becomes Stack: S S … S a S a .a $ if action [S a ] = reduce Aβ then the configuration beco es stack :S S . . S AS input : a a … a $ If action [S a ] =accept Then parsing is completed, HALT If action [S a ] = error Then in voke error recovery routine.



Initial state

Complier Design

LR parsing algorithm stack : input: w$

Loop{ if action [S a shift S’ Then ush (a) ush (S’); i else if action [S,a] = reduce Aβ then o ( *|β|) symbol: push (A): push (goto[S” A ) (S” is the state after o ing sy bols) else if action [S,a] = accept then exit else error state id 0

r2

9 10 11

Stack 0 0 id 5 0F3 0T2 0E1

s7

s5

r2

r2

r4

r4

s4 r6

5

8

$

E 1

T F 2 3

8

2

3

9

3

ccc

r4

3

7

( ) s4

s6

2

6

*

s5

1

4

+

r6

s5

s4

s5

s4 s6

r6

10 s11

r1

s7

r1

r1

r3

r3

r3

r3

r5

r5

r5

r5

Input id+id*id$ +id*id$ +id*id$ +id*id$ +id*id$

Action Shift 5 Reduce by Fid Reduce by TF Reduce by ET Shift 6



0 E 1+6 0 E 1+6 id 5 0 E 1+6 F 3 0 E 1+6T9 0E1+6T9*7 0E1+6T9*7id5 0E1+6T9*7F10 0E1+6T9 0E1

id*id$ *id$ *id$ *id$ id$ $ $ $ $

Shift 5 Reduce by Fid Reduce by TF Shift 7 Shift 5 Reduce by F id Reduce by TT*F Reduce by EE+T Accept

$

INPUT

STACK

Complier Design

LR Parsing Program

action

OUTPUT

goto

Fig. 9.2.7 Model of an LR parser.

LR Grammars : A grammar for which we can construct a parsing table is said to be an LR grammar. Intuitively, in order for a grammar to be LR it is sufficient that a left-to-right shift-reduce parser be able to recognize handles when they appear on top of the stack. An LR parser does not have to scan the entire stack to know when the handle appears on top. Rather, the state symbol on top of the stack contains all the information it needs Another source of information that an LR parser can use to help make its shift-reduce decisions is the next K input symbols. The cases k =0 or k=1 are of practical interest, and we shall only consider LR parsers with k ≤1 here. A grammar that can be parsed by an LR parser examining up to k input symbols on each move is called an LR(k) grammar. There is a significant difference between LL and LR grammars. For a grammar to be LR(k), we must be able to recognize the occurrence of the right side of a production, having seen all of what is derived from that right side with it input symbols of lookahead. This requirement is far less than that for LL(k) grammars where we must be able to recognize the use of a production seeing only the first k symbols of what its right side derives. Thus, LR grammars can describe more languages than LL grammars. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 314


Complier Design

Constructing SLR Parsing Tables We now show how to construct from an LR parsing table grammar. We shall give three methods, varying in their power and ease of implementation. The first, called "simple LR" or SLR for short, is the weakest of the three in terms of the number of grammars for which it succeeds, but is the easiest to implement. We shall refer to the parsing table constructed by this method as an SLR table, and to an LR parser using an SLR parsing table as an SLR parser. A grammar for which an SLR parser can be constructed is said to be an SLR grammar. The other two methods augment the SLR method with look ahead information, so the SLR method is a good starting point for studying LR parsing. An LR(0) item (item for short) of a grammar G is a production of G with a dot at some position of the right side. Thus, production A →XYZ yields the four items A →.XYZ A→ X .YZ A → XY. Z A→ XYZ. The production A→ ϵ generates only one item, A → . An item can be represented by a pair of integers, the first giving the number of the production and the second the position of the dot. a is a viable prefix of the grammar if there is a w such that a w is a right sentential form –a.w is a configuration of the shift reduce parse the set of prefixes of right sequential forms that can appear on the stack of a shift-reduce parse are called viable prefixes. It is always possible to add terminal symbol to the end of a viable prefixe to obtain a right-sentential form. Thus as long as we are able to reduce the portion on the stack to a viable prefix, we are safe and no error occurs. The central idea in the SLR method is first to construct the grammar from a deterministic finite automaton to recognize viable prefixes. One collection of sets of LR(0) items, which we call the canonical LR(O) collection, provides the basis for constructing SLR parsers. To construct the canonical LR(0) collection for a grammar, we define an augmented grammar and two functions, closure and goto. If G is a grammar with start symbol S, then G', the augmented grammar for G, is G with a new start symbol S' and production S' →S. The purpose of this new starting production is to indicate to the parser when it should stop parsing and announce acceptance of the input. That is, acce tance occurs when and only when the arser is about to reduce by S’ →S.



Complier Design

The Closure Operation Closure operation requires us to find all such alternate ways to expect further input. If is a set of items for a grammar G then closure(I) is the set of items constructed from I by the two rules:

1. Initially, every item in I is added to closure (I). 2. If Aa. β is in closure (I) and  is a roduction then add the ite . to I if it is not already there. We apply this rule until no more new items can be added to closure (I). Consider the grammar E’E EE+T|T TT*F|F F(E)|id

If I is {E’.E} then closure (I) is E’.E E.E+T E.T T.T*F T.F F.(E) F.id Fig. 9.2.8Computation of closure.

The goto Operation The second useful function is goto(I.X) where I is asset of items and X is a grammar symbol. goto(I.X) is defined to be the closure of the set of all items [Aa .β such that [Aa. β is in I. intuitively, if I is set of items that are valid for some viable prefix a, then goto(I,X) is set of items that are valid for the viable prefix a. consider the following example: if I is the set of two items {E’E.,EE.+T}, then goto(I,+) consists of EE+.T T.T*F T.F F.(E) F.id We computed goto(I,+) by examining I for items with + immediately to the right of the dot. E’E. is not such an item, but E. +T is. we moved the dot over the + to get {EE+.T} and the took the closure of this set. We are now ready to give an algorithm to construct C, the canonical collection of sets of L ( ) ite s for an aug ent gra ar G’; the algorith is as shown below The Sets-of-items Construction C

{closure ({S’.S})}

repeat for each set of items I in C and each grammar symbol X such that goto (I.X) is not empty and not in C do aDD goto (I.X) to C until no more sets of items can be added to C THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 316


Example Grammar ( ) E’ EET.EE+.T EE+T|T TT.*F TT*F|F ( F(E) |id TF. closure (E’.E) ( E’E F(.E) E.E+T E.E+T E.T E.T T.T*F T.T*F T.F T.F F.(E) F.(E) F.id F.id ( ) ( E’E. Fid. EE.+T

) ( ) EE+T. T.T*F ) T.F F(E) ( )F.id ( ) TT*F F.(E) F.id ( ) F.(E) EE.+T ) goto(I . T) is I goto(I F) is I goto(I ( )isI

Complier Design

(

TT.*F goto(I F) is I goto (I ( ) is I goto (I id ) is I ( ) TT*F. goto(I ( )isI goto(I id) is I ( )) F(E). goto (I ) is I goto (I ) is I

goto(I id) is I Fig. 9.2.10The set of items construction  Checking whether the given grammar is LR(0) Grammar or not? 1. No Multiple Entries in the table. It is not LR(0) 2. If RR conflict (or) SR conflict is present, it is not LR(0) Shift-reduce Parsing is a type of bottom up Parsing that Constraint a parse tree for an input beginning at the leaves and working towards the root conflicts. 

Perform shift action when there is no handle on the stack.



Perform reduce action when there is a handle on the top of the stack.

There are two problems that this Parser faces during parsing the string: Shift-Reduce(SR) conflict: are valid?

What actions to take in case both shift and reduce actions

Reduce-Reduce (RR) conflict: Which rule to use for reduction if reduction is possible by more one rule? These conflicts come either because of ambiguous grammars or parsing method is not powerful enough. 3. At Augmented Grammar don’t check SR & RR, because Augmented grammar is dummy. (Augmented Grammar is not is original Grammar.) 4. A state with conflict is referred as inadequate state or error state.



LR (0): SR Conflict

Complier Design

RR Conflict

A→ .𝛃

Shift (S)

𝛃→r .

Reduce (R)

A→ .

Reduce (R)

𝛃→r .

Reduce (R)

SR

RR

SLR Parsing Tables Now we shall show how to construct the SLR parsing action and goto functions from the deterministic finite automaton that recognizes viable prefixes. Given a grammar, G, we augment G to roduce G’ and from G' we construct C, the canonical collection of sets of items for G'. We construct action the arsing action function and goto the goto function fro С using the following algorithm. It requires us to know FOLLOW(A) for each nonterminal A of a grammar. Algorithm Constructing an SLR parsing table. Input: An aug ented gra ar G’. Output: The SLR parsing table functions action and goto for G'. Method: The SLR parse table is constructed for parse states (I to I ) against terminal and non terminal sy bols. For ter inals entries are referred as ‘action’ for that state and ter inal while for non ter inal entries are ‘goto’ for state and non ter inal. The way entries are filled is : If Aa.aB is in I and goto (I a)

I where a is a terminal then action [I,a] = shift j.

If Aa. is in I where a is a string of terminal and non terminals then action [I,b] = reduce Aa for all b in follow (A). If S’  S. is in I where S’ is sy bol introduces for aug enting the gra [I,$] = accept. If goto (I A)

ar then action

I where A is a non terminal then goto[I,A] = j.

The entries which are not filled are errors. The parsing table consisting of the parsing action and goto functions determined by Algorithm is called the SLR(1) table for G. An LR parser using the SLR(1) table for G is called the SLR(1) parser for G, and a grammar having an SLR(l) parsing table is said to be SLR(1) We usually omit the “( )” after the "SL " since we shall not deal here with arsers having ore than one sy bol of lookahead.  Checking whether the given grammar is SLR(1) Grammar or not? 1. No Multiple Entries in the table. It is not SLR(1) 2. If RR conflict (or) SR conflict is present, it is not SLR(1)



Complier Design

NOTE: If two reduced productions are in any state, check for Follow of both productions. Follow symbols for two productions should not contain any symbol common. Constructing Canonical LR Parsing Tables Canonical LR parsing Carry extra information in the state so that wrong reductions by A a will be ruled out Redefine LR items to include a terminal symbol as a second component (looked ahead symbol) The general form of the item becomes [Aa.B,a] which is called LR(1) item. Item [Aa.,a] calls for reduction only if next input is a. the set of symbols Canonical LR parse solve this problem by storing extra information in the state itself. The problem we have with SLR parses is because it does reduction even for those symbols of follow (A) for which it is invalid. So LR items are redefined to store 1 terminal (look ahead symbol) along with state and thus, the items now are LR(!) items. An Lr(1) item has the form : [Aa.B,a] and reduction is done using this rule only if input is ‘a’. clearly the sy bols a’s for a subset of follow(A). To find closure for canonical LR parse: Repeat for each item [Aa. β a in I for each production B in G’ and for each ter inal b in first (βa) add item [B. b to I until no more items can be added to I For the given grammar: S’S SCC CcC | d I closure ([S’S,$]) S’.S $ S.CC $ C.cC c C.cCd C.d c C.dd As first (e$) = {$} As first (C$) = first (C) = {c,d} As first (Cc) = first (C) = {c,d} As first (Cd) = first (C) = {c,d} THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 319


Complier Design

As first (e c) = {c} As first (ed) = {d} Algorithm Construction of the canonical LR parsing table. Input: An aug ented gra ar G’. Construction of canonical LR parse table Construct C ={

.

} the sets of LR(1) items.

If [Aa.aβ b is in and goto ( ,a) = then action [i,a] = shift j If [Aa.,a] is in then action [i,a] reduce Aa If [S’S.,$] is in then action [i,$] = accept If goto (I I;,A) = then goto [i,A] = j for all nonterminals state 0 1 2 3 4 5 6 7 8 9

c s3

d s4

s6 s3 r3

s7 s4 r3

s6

s7

r2

r2

$

S 1

C 2

acc 5 8 r1 9 r3 r2

An LR parse will not make any wrong shift/reduce unless there is an error. But the number of states in LR parse table is too large. To reduce number of states we will combine all states which have same core and different look ahead symbol. If a conflict results from the above rules, the grammar is said not to be LR(1), and the algorithm is said to fail. 1. The goto transitions for state i are determined as follows: If goto(IRRiRR, A) = IRRjRR, then goto[I, A] = j. 2. All entries not defined by rules (2) and (3) are made "error." 3. The initial state of the arser is the one constructed fro the set containing ite [S’ → S, $] The table formed from the parsing action and goto functions produced by Algorithm is called the canonical LR(1) parsing table. An LR parser using this table is called a canonical LR(1) parser. If the parsing action function has no multiply-defined entries, then the given grammar is called as LR(1) grammar. As before, we omit the "(1)" if it is understood. Constructing LALR Parsing Tables The tables obtained by it are considerably smaller than the canonical LR tables, yet most common syntactic constructs of programming languages can be expressed conveniently by an LALR grammar. For a comparison of parser size, the SLR and LALR tables for a grammar always have the same number of states. Thus, it is much easier and more economical to construct SLR and LALR tables than the canonical LR tables. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 320


Complier Design

Consider a pair of similar looking states (same kernel and different look ahead) in the set of LR(1) items → . / → . $ Replace and by a new state consisting of (Cd.,c/d/$) similarly & and & from pairs merge LR(1) items having the same core Algorithm An easy, but space-consuming LALR table construction. Input: An aug ented gra ar G’. Output: The LAL arsing table functions action and goto for G’. Method: Construct LALR parse table Construct C ={ . . }set of LR(1) items For each core present in LR(1) items find all sets having the same core and replace these sets by their union Let C’’ { … } be the resulting set of items Construct action table as was done earlier Let J= U … Since … have same core, goto(J,X) will have he same core Let K=goto ( ) U goto( ) ..goto( ) the goto(J,X)=K The construct rules for LALR parse table are similar to construction of LR(1) parse table. State c d $ S C s36 s47 1 2 0 acc 1 s36 s47 5 2 s36 s47 89 36 r3 r3 r3 47 r1 5 r2 r2 r2 89

Merging items never produces shift/reduce conflicts but may produce reduce/reduce conflicts. Merging states will never give rise to shift-reduce conflicts but may give reduce-reduce conflicts and have some grammars which were in canonical LR parse may becomes ambiguous in LALR parse. To realize this, suppose in the union there is a conflict on lookahead a because there is an item [Aa.,a] calling for a reducing by Aa, and there is another item [Bβ.a b calling for a shift. Then so e set of ite s fro which the union was THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 321


Complier Design

formed has item [Aa.,a], and since the cores of all these states are the same, it must have an item [Bβ.a c for so e c. but then this state has the sa e shift/reduce conflict on a, and the grammar was not LR(1) as we assumed. Thus the merging of states with common core can ever produce a shift/reduce conflict that was not present can have reduce-reduce conflicts. Assume states {[Xa.,a], [Yβ. b[} and {[ a.,b], [Yβ. a }. Now, merging the two states produces {[Xa., a/b], [Yβ. a/b } which generates a reduce-reduce conflict, since reductions by both Xa and Yβ are called for on in uts a and b. Summary Unambiguos s LR(1) LALR(1) SLR(1) Operator Ambiguou precedence s

LR(0) LL(0)

LL(k) LL(2)

LL(k)

LL(1)

Grammar 1) Sa 2) EE+T/T Ti 3) EE+T/T TTF/F FF*/ab 4) SA/a Aa

LL(1)

LR(0)

SCR(1)

LR(k)

LALR(1) LR(1)

√

√

√

√

√

×

√

√

√

√

×

×

√

√

√

×

×

×

×

×

Number of states in LR(0), SLR(1) & LALR(1) are equal. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 322


Complier Design

Operator-Precedence Parsing :      

The operator- precedence parsing is easy to implement. In this kind of parsing, It is hard to handle tokens like the minus sign, which has two different recedence’s One cannot always be sure the parser accepts exactly the desired language. Finally, only a small class of grammars can be parsed using operator-precedence techniques. Often these parsers use recursive descent, for statements and higher-level constructs. Operator-precedence parsers have even been built for entire languages. In operator-precedence parsing, we define three disjoint precedence relations, <, =, and >, between certain pairs of terminals. These precedence relations guide the selection of handles and have the following meanings:

RELATION

MEANING

<

a “yields recedence to” b a “has the sa e recedence as” b a “ takes recedence over” b Fig. 9.2.2

There are two common ways of determining what precedence relations should hold between a pair of terminals. The first method we discuss is intuitive and is based on the traditional notions of associativity and precedence of operators. The second method of selecting operator-precedence relations is first to construct an unambiguous grammar for the language, a grammar that reflects the correct associativity and precedence in its parse trees.

Using Operator-Precedence Relations: The intention of the precedence relations is to delimit the handle of a right- sentential form, with < marking the left end, = appearing in the interior of the handle, and > marking the right end. To be more precise, suppose we have a right-sentential form of an operator grammar. The fact that no adjacent non-terminals appear on the right sides of productions implies that no rightsentential form will have two adjacent non-terminals either. Thus, we may write the rightsentential for as β 0RR aRR1 β 1 . . . .an β nRR, where each β iRR, is either (the empty string) or a single nonterminal, and each a is a single terminal. Suppose that between aRRjRR and aRRi+jRRexactly one of the relations < , =, and > holds. Further, let us use $ to mark each end of the string, and define $ < b and b > $ for all terminals b. Now suppose we remove the non-terminals from the string and place the correct relation < =, or >, between each pair of terminals and between the endmost terminals and the $'s marking the ends of the string. For example, suppose we initially have the right-sentential form id + id * id and the precedence relations are those given in Fig.9.2.3.These relations are some of those that we would choose to parse according to grammar. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 323


id

*

Complier Design

$

id < *

<

$

<

<

<

<

Fig. 9.2.3 Operator-precedence relations. Then the string with the precedence relations inserted is: $ <. id .> + < . id .> * < . id .> $ For example, < is inserted between the leftmost $ and id since < .is the entry in row $ and column id. The handle can be found by the following process. 1. Scan the string fro the left end until the first • is encountered. In(9.2.3) above, this occurs between the first id and + . 2. Then scan backwards (to the left) over any ='s until a < is encountered. In (9.2.3), we scan backwards to $. 3. The handle contains everything to the left of the first > and to the right of the < encountered in step (2), including any intervening or surrounding nonterminals. (The inclusion of surrounding nonterminals is necessary so that two adjacent nonterminals do not appear in a right-sentential form ) In (9.2.3), the handle is the first id. If we are dealing with grammar for arithmetic expression we then reduce id to E. At this point we have the right-sentential form E+ id*id. After reducing the two remaining id's to E by the same steps, we obtain the right-sentential form E + E * E Consider now the string $+*$ obtained by deleting the nonterminals. Inserting the precedence relations, we get $< .+ < . * . >$ indicating that the left end of the handle lies between + and * and the right end between * and $. These precedence relations indicate that, in the right- sentential form E+E *E, the handle is E *E Note how the E's surrounding the * become part of the handle. Since the nonterminals do not influence the parse, we need not to worry about distinguishing among them. A single marker "nonterminal" can be kept on the stack of a shift-reduce parser to indicate placeholders for attribute values. If no precedence relation holds between a pair of terminals (indicated by a blank entry in Fig.) then a syntactic error has been detected and an error recovery routine must be invoked, as discussed later in this section. The above ideas can be formalized by the following algorithm. Operator-precedence parsing algorithm: Input: An input string w and a table of precedence relations. Output: If w is well formed, a skeletal parse tree, with a placeholder non-terminal E labeling all interior nodes; otherwise an error indication. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 324


Complier Design

Method: Initially, the stack contains $ and the input buffer the string w$. To parse, we execute the following program. 1) set ip(input pointer) to point to the first symbol of w$; 2) repeat forever 3) if $ is on top of the stack and ip points to $ then return 4) else begin 5) let a be the topmost terminal symbol on the slack and let b be the: symbol pointed to by ip; 6) if a < b or a = b then begin push b onto the stack; advance ip to the next input symbol; end; 7) else if a > b then /* reduce*/ 8) repeat 9) pop the stack 10) until the top stack terminal is related by < to the terminal most recently popped 11) else error 12) end Operator-Precedence Relations from Associativity and Precedence : 1. If operator has higher precedence than operator , make and < . For example, if * has higher precedence than +, make * .> + and + < *. 2. If and are operators of equal precedence (they may in fact be the same operator), then make > and if the operators are left-associative, or make < and < if they are right- associative. 3. Make <. id, id . > < . (, ( < , ) > , > ), . > $, and $ < for all operators .



9.3

Complier Design

Syntax Directed Translation

There are two notations for associating semantic rules with productions, syntax-directed definitions and translation schemes. Syntax-directed definitions are high-level specifications for translations. Translation schemes indicate the order in which semantic rules are to be evaluated, so they allow some implementation details to be shown. Conceptually, with both syntax-directed definitions and translation schemes, we parse the input token stream, build the parse tree, and then traverse the tree as needed to evaluate the semantic rules at the parse-tree nodes. Evaluation of the semantic rules may generate code, save information in a symbol table, issue error messages, or perform any other activities. The translation of the token stream is the result obtained by evaluating the semantic rules.

Syntax-Directed Definitions (Sd-Definitions): A syntax-directed definition is a generalization of a context-free grammar in which each grammar symbol has an associated set of attributes, partitioned into two subsets called the synthesized and inherited attributes of that grammar symbol. An attribute can represent anything we choose: a string, a number, a type, a memory location, or whatever. The value of an attribute at a parse-tree node is defined by a semantic rule associated with the production used at that node. The value of a synthesized attribute at a node is computed from the values of attributes at the children of that node in the parse tree; the value of an inherited attribute is computed from the values of attributes at the siblings and parent of that node. Semantic rules set up dependencies between attributes will be represented by a graph. From the dependency graph, we derive an evaluation order for the semantic rules. Evaluation of the semantic rules defines the values of the attributes at the nodes in the parse tree for the input string. A parse tree showing the values of attributes at each node is called an annotated parse tree. The process of computing the attribute values at each node is called annotating or decorating the parse tree. Form of a Syntax-Directed Definition: In a syntax-directed definition each gra ar roduction A →α has associated with it a set of semantic rules of the form b := f(c1 , c2……. ck) where f is a function, and either 1. b is a synthesized attribute of A and c1, c2, . . , ck are attributes belonging to the grammar symbols of the production, or 2. b is an inherited attribute of one of the grammar symbols on the right side of the production and c1, c2, . . , ck are attributes belonging to the grammar symbols of the production. In either case, we say that attribute b depends on attributes c1, c2, . . , ck. An attribute grammar is a syntax-directed definition in which the functions in semantic rules cannot have side effects. In a syntax-directed definition, terminals are assumed to have synthesized attributes only, as the definition does not provide any semantic rules for terminals. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 326


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Synthesized Attributes: A syntax-directed definition that uses synthesized attributes exclusively is said to be an Sattributed definition. A parse tree for an S-attributed definition can always be annotated by evaluating the semantic rules for the attributes at each node bottom up, from the leaves to the root. Inherited Attributes: An inherited attribute is one whose value at a node in a parse tree is defined in terms of attributes at the parent and/or siblings of that node. Inherited attributes are convenient for expressing the dependence of a programming language construct on the context in which it appears. Dependency Graphs: If an attribute b at a node in a parse tree depends on an attribute c, then the semantic rule for b at that node ust be evaluated after the se antic rule that defines с. The interde endences among the inherited and synthesized attributes at the nodes in a parse tree can be depicted by a directed graph called a dependency graph, Evaluation Order : A topological sort of a directed acyclic graph in any ordering m1, m2, . . . .mk of the nodes of the graph such that edges go from nodes earlier in the ordering to later nodes; that is, if mi→ mj is an edge from mi to mj, then mi appears before mj in the ordering. Any topological sort of a dependency graph gives a valid order in which the semantic rules associated with the nodes m a parse tree can be evaluated. That is, in the topological sort, the dependent attributes c1, c2. . . .ck in a semantic rule b : = f(c1, c2. . . . ck) are a available at a node before f is evaluated. A syntax-directed definition is said to be circular if the dependency graph for some parse tree generated by its grammar has a cycle.

Construction of Syntax Trees The use of syntax trees as an intermediate representation allows translation to be decoupled from parsing. Translation routines that are invoked during parsing must live with two kinds of restrictions. First, a grammar that is suitable for parsing may not reflect the natural hierarchical structure of the constructs in the language. Second, the parsing method constrains the order in which nodes in a parse tree are considered.



Complier Design

Syntax Trees An (abstract) syntax tree is a condensed form of parse tree useful for representing language constructs. The production S → if В then S1 else S2 might appear in a syntax tree as If-then-else B Fig. 9.3.1 In a syntax tree, operators and keywords do not appear as leaves, but rather are associated with the interior node that would be the parent of those leaves in the parse tree. Another simplification found in syntax trees is that chains of single productions may be collapsed. Syntax-directed translation can be based on syntax trees as well as parse trees. Constructing Syntax Trees for Expressions The construction of a syntax tree for an expression is similar to the translation of the expression into postfix form. We construct sub trees for the sub expressions by creating a node for each operator and operand. The children of an operator node are the roots of the nodes representing the sub expressions constituting the operands of that operator. Each node in a syntax tree can be implemented as a record with several fields. In the node for an operator, one field identifies the operator and the remaining fields contain pointers to the nodes for the operands. The operator is often called the label of the node. When it is used for translation, the nodes in a syntax tree may have additional fields to hold the values (or pointers to values) of attributes attached to the node. In this section, we use the following functions to create the nodes of syntax trees for expressions with binary operators. Each function returns a pointer to a newly created node. 1. mknode(op, left, right) creates an operator node with label op and two fields containing pointers to left and right, 2. mkleaf (id, entry) creates an identifier node with label id and a field containing entry, a pointer to the symbol-table entry for the identifier. 3. mkleaf (num, vat) creates a number node with label num and a field containing val, the value of the number. A Syntax-Directed Definition for Constructing Syntax Trees Figure 9.3.2 contains an S-attributed definition for constructing a syntax tree for an expression containing the operators + and -. It uses the underlying productions of the grammar to schedule the calls of the functions mknode and mkleaf to construct the tree. The synthesized attribute nptr for E and T keeps track of the pointers returned by the function calls.



PRODUCTION

SEMANTIC RULES

E →E

T

E.nptr:

knode(‘ ’ E . n tr. T. n tr)

E→E

T

E.n tr

knode(‘ ’ E . n tr. T. n tr)

E →T

E.nptr := T.nptr

T → (E)

T.nptr := E.nptr

T → id

T.nptr := mkleaf (id, id.entry)

T → nu

T.nptr := mkleaf (num, num.val)

Complier Design

Fig.9.3.2 Syntax-directed definition for constructing a syntax tree for an expression. Directed Acyclic Graphs for Expressions A Directed Acyclic Graph (DAG) for an expression identifies the common sub expressions in the expression. Like a syntax tree, a DAG has a node for every sub expression of the expression; an interior node represents an operator and its children represent its operands. The difference is that a node in a DAG representing a common sub-expression has more than one "parent;" in a syntax tree, the common sub-expression would be represented as a duplicated sub tree. Algorithm: Value-number method for constructing a node in a DAG. Suppose that nodes are stored in an array, as in Fig, and that each node is referred to by its value number. Let the signature of an operator node be a triple consisting of its label op, left child l, and right child r. Input: Label op node l and node r. Output: A node with signature . Method: Search the array for a node m with label op, left child l, and right child r. If there is such a node, return m; otherwise, create a new node m with label op, left child l, right child r, and return m. An obvious way to determine if node m is already in the array is to keep all previously created nodes on a list and to check each node on the list to see if it has the desired signature. The search for m can be made more efficient by using k lists, called buckets, and using a hashing function A to determine which bucket to search. The hash function h computes the number of a bucket from the value of op, l and r. It will always return the same bucket number, given the same arguments. If m is not in the bucket h(op, l, r) then a new node m is created and added to this bucket, so subsequent searches will find it there. Each bucket can be implemented as a linked list as shown in Fig.



Complier Design

Each cell in a linked list represents a node. The bucket headers, consisting of pointers to the first cell in a list, are stored in an array. The bucket number returned by h(op, l, r) is an index into this array of bucket headers. 0

List elements representing nodes

9 ... Array of bucket heads, indexed by hash value

25

3

...

20

2 ...

Fig. 9.3.3 Data structure for searching buckets. This algorithm can be adapted to apply to nodes that are not allocated sequentially from an array. In many compilers, nodes are allocated as they are needed, to avoid pre-allocating an array that may hold too many nodes most of the time and not enough nodes some of the time. In this case, we cannot assume that nodes are in sequential storage, so we have to use pointers to refer to nodes. If the hash function can be made to compute the bucket number from a label and pointers to children, then we can use pointers to nodes instead of value numbers, Otherwise, we can number the nodes in any way and use this number as the value number of the node. DAG can also be used to represent sets of expressions, since a DAG can have more than one root.

Bottom-Up Evaluation of S-Attributed Definitions Synthesized attributes can be evaluated by a bottom-up parser as the input is being parsed. The parser can keep the values of the synthesized attributes associated with the grammar symbols on its stack. Whenever a reduction is made, the values of the new synthesized attributes are computed from the attributes appearing on the stack for the grammar symbols on the right side of the reducing production. Only synthesized attributes appear in the syntax-directed definition for constructing the syntax tree for an expression. The approach of this section can therefore be applied to construct syntax trees during bottom-up parsing. Synthesized Attributes on the Parser Stack: A bottom-up parser uses a stack to hold information about subtrees that have been parsed. We can use extra fields in the parser stack to hold the values of synthesized attributes. Figure shows an example of a parser stack with space for one attribute value. Let us suppose, as in the figure, that the stack is implemented by a pair of arrays state and vat. Each state entry is a pointer (or index) to an LR (1) parsing table. (Note that the grammar symbol is implicit in the state and need not be stored in the stack.) It is convenient, however, to refer to the state by the unique grammar symbol that it covers when placed on the parsing stack. If the statei symbol is A, then val [i] will hold the value of the attribute associated with the parse tree node corresponding to this A. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 330


top

State

Val

...

...

X

X.x

Y

Y.y Z.z ...

Z ...

Complier Design

Fig. 9.3.4 Parser stack with a field for synthesized attributes. The current top of the stack is indicated by the pointer top. We assume that synthesized attributes are evaluated just before each reduction.

L-Attributed Definitions When translation takes place during parsing, the order of evaluation of attributes is linked to the order in which nodes of a parse tree are "created" by the parsing method. A natural order that characterizes many top-down and bottom-up translation methods is the one obtained by applying the procedure dfvisit in Fig. to the root of a parse tree. We call this evaluation order the depth-first order. Even if the parse tree is not actually constructed, it is useful to study translation during parsing by considering depth-first evaluation of attributes at the nodes of a parse tree. proceduredfvisit {n: node); begin for each child m of n, Train left to right do begin evaluate inherited attributes of m; dfvisit(m) end; evaluate synthesized attributes of n end Fig. 9.3.5 Depth-first evaluation order for attributes m a pane tree. We now introduce a class of syntax-directed definitions, called L-attributed definitions; whose attributes can always be evaluated in depth-first order. (The L is for “left ” because attribute information appears to flow from left to right.) L-attributed definitions include all syntaxdirected definitions based on LL (1) grammars. L-Attributed Definitions A syntax-directed definition is L-attributed if each inherited attribute of Xj, 1≤ j ≤ n, on the right side of A→X1 ,X2 …….. Хn, depends only on 1. The attributes of the symbols Xl ,X2 …….. Х1-1 to the left of Xj in the production and 2. The inherited attributes of A. Note that every S-attributed definition is L-attributed, because the restrictions (1) and (2) apply only to inherit attributes. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 331


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Translation Schemes A Translation scheme is a context-free grammar in which attributes are associated with the grammar symbols and semantic actions enclosed between braces {} are inserted within the right sides of productions.

PRODUCTION

SEMANTIC RULES

A→LM

L.i : = l(A.i) M.i := m(L.s) A.s := f(M.s)

A→Q

R.i := r(A.i) Q.i := q(R.s) A.s := f(Q.s)

Fig.9.3.6 A Non-L-attributed syntax-directed definition. The easiest case occurs when only synthesized attributes are needed. For this case, we can construct the translation scheme by creating an actionconsisting of an assignment for each semantic rule, and placing this action at the end of the right side of the associated production. For example, the production and semantic rule Production Rule

Semantic Rule

T → T1 * F

T.val := T1.val

F.val

yield the following production and semantic action T → T1 * F {T.val := T1.val

F.val }

If we have both inherited and synthesized attributes, we must be more careful: 1. An inherited attribute for a symbol on the right side of a production must be computed in an action before that symbol. 2. An action must not refer to a synthesized attribute of a symbol to the right of the action. 3. A synthesized attribute for the non-terminal on the left can only be computed after all attributes of it reference have been computed. The action computing such attributes can usually be placed at the end of the right side of the production.



Complier Design

Differences between S-attributed and L-attributed definition: S-attributed L-attributed 1. It uses synthesized attributes 1. It uses both synthesized and only inherited attributes only 2. Semantic actions can be 2. Semantic actions can be placed at the end of rules places anywhere of R.H.S 3. Attribute can be evaluated 3. Each inherited attribute is during “ UP” restricted to inherit either from parent/left sibling 4. Both BUP, TDP plays role 4. Attributes are evaluated by traversing the parse tree depth first left to right Fig. 9.3.7

Top-Down Translation In this section, L-attributed definitions will be implemented during predictive parsing. We work with translation schemes rather than syntax-directed definitions so we can be explicit about the order in which actions and attribute evaluations take place. We also extend the algorithm for left-recursion elimination to translation schemes with synthesized attributes. Eliminating Left Recursion from a Translation Scheme Since most arithmetic operators associate to the left, it is natural to use left- recursive grammars for expressions. We now extend the algorithm for eliminating left recursion in to allow for attributes when the underlying grammar of a translation scheme is transformed;the transformation applies to translation schemes with synthesized attributes. The next example motivates the transformation. Suppose we have the following translation scheme A → A1 Y { A.a : = g(A1.a, Y.y) } A → x { A.a := f(X.x)} Each grammar symbol has a synthesized attribute written using the corresponding lower case letter and f and g are arbitrary functions. The generalization to additional A-productions and to productions with strings in place of symbols X and Y can be done as below. The algorithm for eliminating left recursion constructs the following grammar A→XR R→YR|ϵ Taking the semantic actions into account, the transformed scheme becomes A → X {R.i. := f(X.x)} R { A.a := R.s} R →Y {R1.i : = g(R.i, Y.y)} R1 { R.s := R1.s} R → ϵ {R.s := R.i} THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 333


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BOTTOM-UP EVALUATION OF INHERITED ATTRIBUTES In this section, we present a method to implement L-attributed definitions in the framework of bottom-up parsing. It can implement any L-attributed definition based on an LL (1) grammar. It can also implement many (but not all) L-attributed definitions based on LR (1) grammars. The method is a generalization of the bottom-up translation technique. Removing Embedding Actions from Translation Schemes In the bottom-up translation method, we relied upon all translation actions being at the right end of the production, while in the predictive parsing method we needed to embed actions at various places within the right side. To begin our discussion of how inherited attributes can be handled bottom up, we introduce a transformation that makes all embedded actions in a translation scheme occur at the right ends of their productions. Inheriting Attributes on the Parser Stack A bottom-up parser reduces the right side of production A →XY by removing X and Y from the top of the parser stack and replacing them by Y. Since the value of X.s is already on the parser stack before any reductions take place in the sub tree be to Y, this value can be inherited by Y. That is, if inherited attribute Y.i is defined by the copy rule Y.i := X.s, then the value X.s can be used where Y.i is called for. As we shall see, copy rules play an important role in the evaluation of inherited attributes during bottom-up parsing. Replacing Inherited by Synthesized Attributes It is sometimes possible to avoid the use of inherited attributes by changing the underlying grammar. For example, a declaration in Pascal can consist of a list of identifiers followed by a type, e.g., m, n; integer. A grammar for such declarations may include productions of the form D →L : T T → integer | char L →L , id | id Since identifiers are generated by L but the type is not in the sub tree for L, we cannot associate the type with an identifier using synthesized attributes alone. In fact, if non-terminal L inherits a type from T to its right in the first production, we get a syntax-directed definition that is not Lattributed, so translations based on it cannot be done during parsing. A solution to this problem is to restructure the grammar to include the type as the last element of the list of identifiers: D → ML L → , id L | : T T → integer | char Now, the type can be carried along as a synthesized attribute L.type. As each identifier is generated by L, its type can be entered into the symbol table.



9.4

Complier Design

Run Time Environment

SOURCE LANGUAGE ISSUES For specificity, suppose that a program is made up of procedures, as in Pascal. This section distinguishes between the source text of a procedure and its activations at run time. Activation Trees We make the following assumptions about the flow of control among procedures during the execution of a program: 1. Control flows sequentially; that is, the execution of a program consists of a sequence of steps, with control being at some specific point in the program at each step. 2. Each execution of a procedure starts at the beginning of the procedure body and eventually returns control to the point immediately following the place where the procedure was called. This means the flow of control between procedures can be depicted using trees, as we shall soon see.  Each execution of a procedure body is referred to as an activation of the procedure.  The lifetime of an activation of a procedure p is the sequence of steps between the first and last steps in the execution of the procedure body, including time spent executing procedures called by p, the procedures called by them, and so on.  More precisely, each time control flows from an activation of a procedure p to an activation of a procedure q, it returns to the same activation of p.  If a and b are procedure activations, then their lifetimes are either non- overlapping or nested. That is, if b is entered before a is left, then control must leave b before it leaves a.  A procedure is recursive if a new activation can begin before an earlier activation of the same procedure has ended.  A recursive procedure p need not call itself directly; p may call another procedure q which may then call p through some sequence of procedure calls. We can use a tree, called an activation tree, to depict the way that control enters and leaves activations. In an activation tree, 1. Each node represents an activation of a procedure, 2. The root represents the activation of the main program, 3. The node for a is the parent of the node for b if and only if control flows from activation a to b, and 4. The node for a is to the left of the node for b if and only if the lifetime of a occurs before the lifetime of b. The Scope of a Declaration   

The scope rules of a language determine which declaration of a name applies when the name appears in the text of a program. The portion of the program to which a declaration applies is called the scope of that declaration. An occurrence of a name in a procedure is said to be local to the procedure if it is in the scope of a declaration within the procedure; otherwise, the occurrence is said to be nonlocal.





Complier Design

At compile time, the symbol table can be used to find the declaration that applies to an occurrence of a name. When a declaration is seen, a symbol- table entry is created for it. As long as we are in the scope of the declaration, its entry is returned when the name in it is looked up.

Activation Records Information needed by a single execution of a procedure is managed using a contiguous block of storage called an activation record consisting of the collection of fields shown in Fig. Not all languages or all compilers use all of these fields; often registers can take the place of one or more of the . For languages like Pascal and С it is custo ary to ush the activation record of a procedure on the run-time stack when the procedure is called and to pop the activation record of the stack when control returns to the caller. The purpose of the fields of an activation record is as follows, starting from the field for temporaries. 1. Temporary values, such as those arising in the evaluation of expressions, are stored in the field for temporaries. 2. The field for local data holds data that is local to an execution of a procedure. 3. The field for saved machine status holds information about the state of the machine just before the procedure is called. This information includes the values of the program counter and machine registers that have to be restored when control returns from the procedure. 4. The optional access link is used in to refer nonlocal data held in other activation records. 5. The optional control link points to the activation record of the caller. returned value actual parameters optional control link optional access link saved machine status local data temporaries

Fig. 9.4.1 A general activation record. 6. The field for actual parameters is used by the calling procedure to supply parameters to the called procedure. We show space for parameters in the activation record, but in practice parameters are often passed in machine registers for greater efficiency. 7. The field for the returned value is used by the called procedure to return a value to the calling procedure. Again, in practice this value is often returned in a register for greater efficiency. The sizes of each of these fields can be determined at the time a procedure is called. In fact, the sizes of almost all fields can be determined at compile time. An exception occurs if a procedure may have a local array whose size is determined by the value of an actual parameter, available only when the procedure is called at run time. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 336


9.5

Complier Design

Intermediate Code Generation

Although a source program can be translated directly into the target language, some benefits of using a machine-independent intermediate form are: 1. Retargeting is facilitated; a compiler for a different machine can be created by attaching a back end for the new machine to an existing front end, 2. A machine-independent code optimizer can be applied to the intermediate representation. For simplicity, we assume that the source program has already been parsed and statically checked, as in the organization of Fig.9.5.1 intermediate intermediate code code generator

Static checker

parser

code generator

Fig 9.5.1 Position of intermediate code generator

Intermediate languages Syntax trees and postfix notation, respectively, are two kinds of intermediate representations. A third, called three-address code, will be discussed here. The semantic rules for generating threeaddress code from common programming language constructs are similar to those for constructing syntax trees or for generating postfix notation. Graphical Representations A syntax tree depicts the natural hierarchical structure of a source program. A DAG gives the same information but in a more compact way because common sub expressions are identified. A syntax tree and DAG for the assignment statement a := b*-c + b* - с a ear in Fig. 9.5.2 assign a

assign + *

* b

+

a

uminus b

* uminus

c (a) Syntax tree

c

b

uminus c (b) DAG

Fig. 9.5.2 Graphical representations of a := b*-c + b*-с. Postfix notation is a linear end representation of a syntax tree; it is a list of the nodes of the tree in which a node appears immediately after its children. The postfix notation for the syntax tree in Fig. 9.5.2(a) is a b с u inus * b с u inus * assign THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 337


Complier Design

Three-Address Statements Three-address statements are similar to assembly code. Statements can have symbolic labels and there are statements for flow of control. A symbolic label represents the index of a threeaddress statement in the array holding inter-mediate code. Actual indices can be substituted for the labels either by making a separate pass, or by using "back patching". Here are the common three-address statements used in the remainder of this book: 1. Assign ent state ents of the for x у o z where o is a binary arith etic or logical operation. 2. Assignment instructions of the form x := op y, where op is a unary operation. Essential unary operations include unary minus, logical negation, shift operators, and conversion operators that, for example, convert a fixed-point number to a floating-point number. 3. Co y state ents of the for x y where the value of у is assigned to x. 4. The unconditional jump goto L The three-address statement with label L is the next to be executed. 5. Conditional jumps such as if x reloo у goto L. This instruction a lies a relational operator (<, =, > =, etc) to x and y, and executes the statement with label L next if x stands in relation reloop to y. If not, the three-address statement following if x reloo у goto L is executed next, as in the usual sequence. 6. param x and call р n for rocedure calls and return y where у re resenting a returned value is optional- Their typical use is as the sequence of three-address statements param x1 param x2 ara хn callp,n generated as pan of a call of the procedure p(x1,x ……….., xn). The integer n indicating the number of actual parameters in "call p, n" is not redundant because calls can be nested. Indexed assignments of the form x :=y[i] and x[i] := y. The first of these sets x to the value in the location i memory units beyond location y. The statement x[i] := y sets the contents of the location i units beyond x to the value of y. In both these instructions, x, y, and i refer to data objects. 7. Address and pointer assignments of the form x : = &y, x := *y, and *x := y. The first of these sets the value of x to be the location of y. presumably у is a na e erha s a temporary, that denotes an expression with an l-value such as A[i, j], and x is a pointer name or temporary. That is, the r-value of x is the l-value (location) of some object. In the state ent x *y resu ably у is a ointer or a te orary whose r- value is a location. The r-value of x is ade equal to the contents of that location. Finally *x у sets the rvalue of the object pointed to by x to the r-value of y.



Complier Design

Syntax-Directed Translation into Three-Address Code: When three-address code is generated, temporary names are made up for the interior nodes of a syntax tree. The value of non-ter inal E on the left side of E → E1+E2 will be computed into a new temporary t, In general, the three-address code for id := E consists of code to evaluate E into some temporary t, followed by the assignment id.place : = t. If an expression is a single identifier say y then у itself holds the value of the expression. For the moment, we create a new name every time a temporary is needed. The S-attributed definition in Fig.Generates three-address code for assignment statements. Given in ut а b*-с b*-с it roduces the code in Fig. (a). The synthesized attribute S.code represents the three- address code for the assignment S, The non terminal has two attributes: 1. E.place, the name that will hold the value of E, and 2. E.code, the sequence of three-address statements evaluating E. The function new temp returns a sequence of distinct names t1, t2,,..in response to successive calls. For convenience we use the notation gen(x ' ' у ' ' z) in Fig 9.5.3 to represent the threeaddress statement x := y + z. Expressions appearing instead of variables like x, y, and z are evaluated when passed to gen, and quoted operators or operands, like +, are taken literally. In practice, three- address statements might be sent to an output file, rather than built up into the code attributes. Flow-of-control statements can be added to the language of assignments in Fig.9.5.3 by roductions and se antic rules like the ones for while state ents in the figure the code for S→ while E do S1, is generated using new attributes S.begin and S .after to mark the first statement in the code for E PRODUCTION S → id E E→E E

E→E

E→

E

E

E → (E ) E → id

SEMANTIC RULES S.code E.code || gen(id. lace ‘ ’ E. lace) E.place := newtemp; E.code:= E . code||E . code|| gen (E. lace ‘ ’ E . lace E . lace) E.place := newtemp; E.code:= E . code||E . code|| gen (E. lace ‘ ’ E . lace E . lace) E.place:= newtemp; E.code: = E . code||gen(E. lace u inus E / lace) E.place:= E . lace; E.code:= E . code E.place := id.place; E.code “

Fig. 9.5.3 Syntax-directed definition to produce three-address code for assignments.



Complier Design

and the statement following the code for S, respectively. These attributes represent labels created by a function new label that returns a new label every time it is called. Note that S.after becomes the label of the statement that comes after the code for the while statement. We assume that a non-zero expression represents true; that is, when the value of E becomes zero, control leaves the while statement. Expressions that govern the flow of control may in general be boolean expressions containing relational and logical operators. Postfix notation can be obtained by adapting the semantic rules in Fig. 9.5.3. The postfix notation for an identifier is the identifier itself. The rules for the other productions concatenate only the operator after the code for the operands. For example, associated with the production E → -E1 is the semantic rule E.code := E1.code || 'uminus' In general, the intermediate form produced by the syntax-directed translations in this chapter can be changed by making similar modifications to the semantic rules.

Implementations of Three-Address Statements A three-address statement is an abstract form of intermediate code. In a compiler, these statements can be implemented as records with fields for the operator and the operands. Three such representations are quadruples, triples, and indirect triples.

1) Quadruples A quadruple is a record structure with four fields, which we call op, arg1 arg2, and result. The op field contains an internal code for the operator. The three-address state ent x у o z is re resented by lacing у in arg z in arg2, and x in result. Statements with unary operators like x ; = -y or x у do not use arg . O erators like ara use neither arg nor result. Conditional and unconditional jumps put the target label in result. The quadruples in Fig.9.5.4(a) are for the assignment a := b*-c + b*-c. The contents of field’sarg 1, arg 2, and result are normally pointers to the symbol-table entries for the names represented by these fields;if so, temporary names must be entered into the symbol table as they are created.

2) Triples To avoid entering temporary names into the symbol table, we might refer to a temporary value by the position of the statement that computes it. If we do



op

arg 1

arg 2

(0)

uminus

c

(1)

*

b

(2)

uminus

c

(3)

*

b

t

t

(4)

+

t

t

t

(5)

:=

t

Complier Design

result t

t

t t

a

(a) Quadruples op

arg 1

arg 2

(0)

uminus

c

(1)

*

b

(2)

uminus

c

(3)

*

b

(2)

(4)

+

(1)

(3)

(5)

assign

a

(4)

(0)

(b) Triples Fig. 9.5.4 Quadruple and triple representations of Three-address statements.

So, three-address statements can be represented by records with only three fields: op, arg1 and arg2. The field’s arg1 and arg2, for the arguments of op, are either pointers to the symbol table (for programmer- defined names or constants) or pointers into the triple structure (for temporary values). Since three fields are used, this intermediate code format is known as triples. Except for the treatment of programmer-defined names, triples correspond to the representation of a syntax tree or dag by an array of nodes. Parenthesized numbers represent pointers into the triple structure, while symbol-table pointers are represented by the names themselves. In practice, the information needed to interpret the different kinds of entries in the arg1 and arg2 fields can be encoded into the op field or some additional fields.



Complier Design

The triples in Fig. 9.5.4(b) correspond to the quadruples in Fig. 9.5.5(a), Note that the copy statement a := t5 is encoded in the triple representation by placing a in the arg1 field and using the operator assign. A ternary o eration like x[i ; у requires two entries in the tri le structure as shown in Fig 9.5.5(a), while x := y[ i] is naturally represented as two operations in Fig.9.5.5(b).

Op

arg 1

arg 2

(0)

[ ]=

x

i

(1)

Assign

(0)

y

(a) x[i] := y

op

arg 1

arg 2

(0)

[ ]=

x

i

(1)

assign

(0)

y

(b) x := y[i] Fig. 9.5.5More triple representation

3) Indirect Triples Another implementation of three-address code that has been considered is that of listing pointers to triples, rather than listing the triples themselves. This implementation is naturally called indirect triples.



9.6

Complier Design

Code Optimization

The performance improvement of a program is achieved by program transformations that are traditionally called optimizations.

The Principal Sources Of Optimization In this section, we introduce some of the most useful code-improving transformations. Techniques for implementing these transformations are presented in subsequent sections. A transformation of a program is called local if it can be performed by looking only at the statements in a basic block; otherwise, it is called global. Many transformations can be performed at both the local and global levels. Local transformations are usually performed first. Function-Preserving Transformations There are a number of ways in which a compiler can improve a program without changing the function it computes. Common sub expression elimination, copy propagation, dead-code elimination, and constant folding are common examples of such function-preserving transformations. The dag representation of basic blocks showed how local common sub expressions could be removed as the dag for the basic block was constructed. The other transformations come up primarily when global optimizations are performed, and we shall discuss each in turn. Frequently, a program will include several calculations of the same value, such as an offset in an array. Some of these duplicate calculations cannot be avoided by the programmer because they lie below the level of detail accessible within the source language. Common Sub expressions An occurrence of an expression E is called a common sub expression if E was previously computed, and the values of variables in E have not changed since the previous computation. We can avoid re computing the expression if we can use the previously computed value. Directed acyclic graph is used to eliminate common sub expression elimination. Copy Propagation The idea behind the copy-propagation transformation is to use g for f, wherever possible after the copy statement f :=g. In the following program g can be used in place of f. g=19; f=g; x=x+f; THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 343


Complier Design

Copy Constant Propagation Constant can be propagated to the program if possible as in the following example. g can be replaced by a constant(19) in the following program. g=19; x=x+g; Dead-Code Elimination A variable is live at a point in a program if its value can be used subsequently; otherwise, it is dead at that point. A related idea is dead or useless code, statements that compute values that never get used. While the programmer is unlikely to introduce any dead code intentionally, it may appear as the result of previous transformations. debug := false if (debug) print print statement never been reached in the program because always if going to be fail, called as dead code.

Loop Optimizations Three techniques are important for loop optimization;  Code motion - which moves code outside a loop  Induction-variable elimination -which we apply to eliminate i and j  Reduction in strength, which replaces an expensive operation by a cheaper one, such as a multiplication by an addition.



9.7

Complier Design

Code Generation

It takes as input an intermediate representation of the source program and produces as output an equivalent target program, The output code must be correct and of high quality, meaning that it should make effective use of the resources of the target machine. Moreover, the code generator itself should run efficiently. Issues in code generation: Memory Management Mapping names in the source program to addresses of data objects in run-time memory is done cooperatively by the front end and the code generator. Instruction Selection The nature of the instruction set of the target machine determines the difficulty of instruction selection. The uniformity and completeness of the instruction set are important factors. If the target machine does not support each data type in a uniform manner, then each exception to the general rule requires special handling. Instruction speeds and machine idioms are other important factors. If we do not care about the efficiency of the target program, instruction selection is straightforward. For each type of threeaddress statement, we can design a code skeleton that outlines the target code to be generated for that construct. Register Allocation Instructions involving register operands are usually shorter and faster than those involving operands in memory. Therefore, efficient utilization of registers is particularly important in generating good code. The use of registers is often subdivided into two sub problems: 1. During register allocation, we select the set of variables that will reside in registers at a point in the program. 2. During a subsequent register assignment phase, we pick the specific register that a variable will reside in. Choice of Evaluation Order The order in which computations are performed can affect the efficiency of the target code. Some computation orders require fewer registers to hold intermediate results than others  The Target Machine Familiarity with the target machine and its instruction set is a prerequisite for designing a good code generator. Unfortunately, in a general discussion of code generation it is not possible to describe the nuances of any target machine in sufficient detail to be able to generate good code for a complete language on that machine. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 345


Complier Design

Flow Graphs We can add the flow-of-control information to the set of basic blocks making up a program by constructing a directed graph called a flow graph. The nodes of the flow graph are the basic blocks. One node is distinguished as initial; it is the block whose leader is the first statement. There is a directed edge from block B1 to blockB2 if B2 can immediately follow B1 in some execution sequence; that is, if 1. there is a conditional or unconditional ju fro the last state ent of В 1 to the first statement of B2, or 2. B2 immediately follows B1in the order of the program, and B1 does not end in an unconditional jump. We say that В1 is a predecessor of B2 and В2 is a successor of В 1. Representation of Basic Blocks Basic blocks can be represented by a variety of data structures. For example, after partitioning the three-address statements by Algorithm, each basic block can be represented by a record consisting of a count of the number of quadruples in the block, followed by a pointer to the leader (first quadruple) of the block, and by the lists of predecessors and successors of the block. An alternative is to make a linked list of the quadruples in each block. Explicit references to quadruple numbers in jump statements at the end of basic blocks can cause problems if quadruples are moved during code optimization. Loops In a flow graph, what is a loop, and how does one find all loops? Most of the time, it is easy to answer these questions. For exa le in Fig. there is one loo consisting of block В2 the general answers to these questions, however, are a bit subtle, and we shall examine them in detail in the next chapter. For the present, it is sufficient to say that a loop is a collection of nodes in a flow graph such that 1. All nodes in the collection are strongly connected; that is, from any node in the loop to any other, there is a path of length one or more, wholly within the loop, and 2. The collection of nodes has a unique entry, that is, a node in the loop such that the only way to reach a node of the loop from a node outside the loop is to first go through the entry. A loop that contains no other loops is called an inner loop. Overview of Phases of a Compiler: A Compiler takes as input a source program and produces as output an equivalent Sequence of machine instructions. This process is so complex that it is divided into a series of sub process called Phases. The following gives brief explanation of phases of compiler with an example.



Complier Design

1. Lexical Analysis: It is the first phase of a Compiler. Lexical analyzer or Scanner reads the characters in the source program and groups them into a stream of tokens. The usual tokens are identifiers, keywords, Constants, Operators and Punctuation Symbols such as Comma and Parenthesis. Each token is a Sub-String of the source program that is to be treated as a single unit. Tokens are of two types: 1. Specific Strings Eg: If, Semicolon 2. Classes of Strings Eg: identifier, Constants, Label. A token is treated as a pair consisting of two parts. 1. Token type 2. Token Value. The character sequence forming a token is called the lexeme for the token. Certain tokens will be increased by a lexical value. The lexical analyzer not only generates a token, but also it enters the lexeme into the symbol table. Symbol table 1. a 2. b 3. c Token values are represented by pairs in square brackets. The second component of the pair is an index to the symbol table where the infor ation’s are ke t. For eg. Consider the ex ression a = b + c * 20 After lexical Analysis it will be. id1 = id2 + id3 *20 The lexical phase can detect errors where the characters remaining in the input do not form any token of the language. Eg: Unrecognized Keyword. 2. Syntax Analysis: It groups tokens together into Syntactic Structures called an Expression. Expressions might further be combined to form statements. Often the syntactic structure can be regarded as a tree where leaves are tokens, called as parse trees. The parser has two functions. It checks if the tokens, occur in pattern that are permitted by the specification of the source language. i.e., Syntax checking. For eg., Consider the expire the each position A+/B. After lexical Analysis this will be, as the token sequence id+/id in the Syntax analyzer. On seeing / the syntax analyzer should detect an error. Because the presence of two adjacent binary operators violates the formulation rules. The second aspect is to make explicit the hierarchical Structure of incoming token stream by identifying which parts of the token stream should be grouped. The Syntax analysis can detect syntax errors. Eg., Syntax error. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 347


Complier Design

3. Semantic Analysis: An important role of semantic analysis is type checking. Here the computer checks that the each operator has operands that are permitted by the source language specification. Consider the eg: x= a+b The language specification may permit some operand coercions. For eg: When a binary arithmetic operator is applied to an integer and real. In this case, the compiler array needs to convert the integer to a real.In this phase, the compiler detects type mismatch error. 4. Intermediate Code generation: It uses the structure produced by the syntax analyzer to create a stream of simple instructions.Many styles are possible. One common style uses instruction with one operator and a small number of operands.The output of the previous phase is some representation of a parse tree. This phase transforms this parse tree into an intermediate language. One popular type of intermediate language is called three address code. A typical three- address code statement is A = B op C.Where A, B, C are operands. OP is a binary Operator. Eg: A = B + C * 20 Here, T1, T2, T3 are temporary variables. Id1, id2, id3 are the identifiers corresponding to A, B, C. 5. Code Optimization: It is designed to improve the intermediate code. So that the Object program less space. Optimization may involve: 1. Detection & removal of dead code. 2. Calculation of constant expressions & terms. 3. Collapsing of repeated expressions into temporary storage. 4. Loop unrolling. 5. Moving code outside the loops. 6. Removal of unnecessary temporary-variables. For e.g.: A: = B+ C * 20 will be T1 = id3 * 20.0 Id1 = id2 + T1



Complier Design

6. Code generation: Once optimizations are completed, the intermediate code is mapped into the target languages. This involves; Allocation of registers & memory, Generation of connect references, Generation of correct types, Generation of machine code. Eg: MOVF id3, R2 MULF # 20.0, R2 MOVF id2, R1 ADDF R2, R1 MOVF R1, id1. Few tips while you are checking for parsing grammars:  To check LL (1) grammar is explained with the example in the top down parsing section. Remember that predictive parser or top down parser are by default called as LL (1) parser.  To check bottom up parsing grammars: Either you can check as following order or reverse of the following order: CLR(1),LALR(1),SLR(1),LR (0) grammar To check CLR (1), construct transition diagram with lookaheads and check whether all the states are without conflict. To check LALR (1), Take the CLR (1) transition diagram and combine those states which are having common core part with different lookaheads in production. Then check all the states should not contain any conflict (apply the rules given in the class). To check SLR(1), Take the same LALR(1) transition diagram and then check all the states should not contain any conflict (apply rules given in the class). To check LR (0), Take the same SLR (1) transition diagram without any lookahead for each production and then check for conflict (apply rules given in the class).



Computer Networks

Part. 10 Computer Networks 10.1 Introduction to Computer Networks A computer network is created when several computers and terminal device are connected together by data communication system. A network is basically a communication system for computers. Goals / Advantages of Networking 1. 2. 3. 4. 5. 6. 7. 8.

Program and file sharing Network Resource sharing Database sharing Ability to use network software Ability to use electronic mail Centralized Management Security Access to more than one operating system

Components of Network Computer networks consist of the following fundamental components: 1. 2. 3. 4. 5.

Server Workstations Network Interface Cards Cabling system Shared Resources and Peripherals.

Classification of Networks Networks are classified upon the geographical area they span and can fall into the following categories: 1. Local Area Network (LAN) 2. Metropolitan Area Network (MAN) 3. Wide Area Network (WAN) LAN The number of computers in LAN varies from small LAN’s that connect 2 to 25 computers, to large LAN’s that may connect more than 10,000 computers. Normally LANs are owned by single organization. The speed of data transfer ranges from several thousands bit per second to several Mbps (Mega bits per second). LAN Topologies:  

Each computer or device in a network is called a node. The geometrical arrangement of computer resources, remote device, and communication facilities is known as network topology. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 350




Computer Networks

All network topologies derived from two basic types: The bus and the point-to-point and a network can be made in one of the two different topologies: o Bus topology o Ring topology

Bus Topology A bus network consists of a single medium that all the stations share. Advantages of Bus Network (a) Short cable length and simple wiring layout (b) Resilient Architecture (c) Easy to extent Disadvantages of the Bus Network (a) Fault diagnosis is difficult (b) Fault isolation is difficult (c) Repeater configuration (d) Nodes must be intelligent Ring Network    

In a ring network, several devices or computers are connected to each other in a closed loop by a single communication cable. A ring network is also called loop network. A ring can be unidirectional or bi-directional. In the unidirectional ring, special software is needed if one computer should break down, whereas, in the bi-directional ring, message can be sent in the opposite direction to meet the requirement.

Advantages of the Ring Network: (a) Short cable length (b) Suitable for optical fibers: Optical fibers offer the possibility of very high speed transmissions. Because traffic on a ring travel in one direction, it is easy to use optical fibers as medium of transmission. Disadvantages of the Ring Network: (a) Node failure causes network failure: (b) Difficult to diagnose faults: (c) Network reconfiguration is difficult: For a very high big ring network it is not possible to shut down a small section of the ring while keeping the majority of it working normally while rectifying the problem at some node. Metropolitan Area Network (MAN) 

A MAN is basically a bigger version of a LAN and normally uses similar technology.



  

Computer Networks

A special standard has been adopted for MAN is called DQDB (Distributed Queue Dual Bus). DQDB consists of two unidirectional buses (cables) to which all computers are connected. A key aspect of a MAN is that there is broadcast medium to which all the computer are attached. A network can be made in one of the two different topologies: 1. Star Network 2. Tree Network 3. Mesh Network

1. Star Network  In a star network, devices or computers are connected to one centralized computer. Advantages of the Star Network: (a) Ease of service (b) One device per connection (c) Centralized control / problem diagnosis Disadvantages of the Star Network: (a) Long cable Length (b) Difficult to expand (c) Central node dependency: If the central node in a star network fails, the entire network will go down. 2. Tree Network  In a tree network, several devices or computers are linkedhierarchical fashion. Tree network is also known as hierarchical network.  This type of distribution system is commonly used in the organization where headquarters communicate with regional offices and regional offices communicate with distinct offices and so on. Advantages of Tree Network: (a) Easy to extend (b) Fault isolation Disadvantages of Tree Network: (a) Dependent on the root 3. Mesh Network  A mesh network has point-point connections between every device in the network. Each device requires an interface for every other device on the network, mesh topologies are not usually considered practical.  esh Topology re uire cables 

Numbers of ports per device = n

1



Computer Networks

Advantages and Disadvantages of Mesh Network: (a) Units Affected by Media Failure: Mesh topology resist media failure better than other topologies. Implementations that include more than two devices will always have multiple paths to send signals from one device to another. If one path fails, the transmission signals can be routed around the failed link. (b) Ease of Installation: Mesh networks are relatively difficult to install because each device must be linked directly to all other devices. As the number of devices increases, the difficulty of installation increases geometrically. (c) Ease of Troubleshooting: Mesh topologies are easy to troubleshooting because each medium link is independent of all others. You can easily identify faults and can rectify it. (d) Difficulties of Reconfiguration: Mesh topologies are difficult to reconfigure for the same reasons that they are difficult to install. Wide Area Network (WAN)        

A WAN spans a large geographical area, often a country or continent. It contains a collection of machines intended for running user programs, these machines are called hosts. The hosts are connected by a communication subnet or just subnet. The job of subnet is to carry messages from host to host just as the telephone system carries words from speaker to listener. In most wide area networks, the subnet consists of two distinct components: transmission lines and switching elements. Transmission lines moves bits between machines. The switching elements are specialized computers used to connect two or more transmission lines. When the data arrive on an incoming line, the switching element must choose an outgoing line to forward them on. These switching elements are popularly called routers. The collection of communication lines and routers form the subnet

Layered Architecture   

The OSI model is built of seven ordered layers: physical (layer 1), data link (layer 2), network (layer 3), transport (layer 4), session (layer 5), and presentation (layer 6), and application (layer 7). Each layer defines a family of functions distinct from those of the other layers. By defining and localizing functionality in this fashion, the designers created an architecture that is both comprehensive and flexible. The interface between the layers facilitates passing of data downwards and upwards between the adjacent layers



Computer Networks

Functions of the Layers Physical Layer 1. Physical characteristics of Interfaces and media 2. Representation of bits. 3. Data rate 4. Synchronization of bits 5. Line configuration 6. Physical topology 7. Transmission mode Data Link Layer 

Framing: The data link layer divides the stream of bits received from the network layer into manageable data units called frames.



Physical addressing: If frames are to be distributed to different systems on the network, the data link layer adds a header to the frame to define the physical address of the sender (source address) and/or receiver (destination address) of the frame. If the frame is intended for a system outside the sender’s network, the receiver address is the address of the device that connects one network to the next.



Flow control: If the rate at which the data are absorbed by the receiver is less than the rate produced in the sender, the data link layer imposes a flow control mechanism to prevent overwhelming the receiver.



Error control: The data link layer adds reliability to the physical layer by adding mechanisms to detect and retransmit damaged or lost frames. It also uses a mechanism to prevent duplication of frames. Error control is normally achieved through a trailer added to the end of the frame.



Access control: When two or more devices are connected to the same link, data link layer protocols are necessary to determine which device has control over the link at any given time.

Network Layer 

Logical addressing: The physical addressing implement by the data link layer handles the addressing problem locally. If a packet passes the network boundary. We need another addressing system to help distinguish the source and destination systems. The network layer adds a header to the packet coming from the upper layer that, among other things, indicators the logical addresses of the sender and receiver.





Computer Networks

Routing: When independent networks or links are connected together to create an internetwork (a network of networks) or a large network, the connecting devices (called routers or gateways) route the packets to their final destination. One of the functions of the network layer is to provide this mechanism.

Transport Layer 

Service-point addressing: Computers often run several programs at the same time. For this reason, source-to-destination delivery means delivery not only from one computer to the next but also from a specific process (running program) on one computer to a specific process (running program) on the other. The transport layer header therefore must include a type of address called a service-point address (or port address). The network layer gets each packet to the correctly to the computer, the transport layer gets the entire message to the correct process on that computer.



Segmentation and reassembly: A message is divided into transmittable segments, each segment containing a sequence number. These numbers enable the transport layer to reassemble the message correctly upon arriving at the destination and to identify and replace packets that were lost in the transmission.



Connection control: The transport layer can be either connectionless or connection-oriented. A connectionless transport layer treats each segment as an independent packet and delivers it to the transport layer at the destination machine. A connection-oriented transport layer makes connection with the transport layer at the destination machine first before delivering the packets. After all the data are transferred, the connection is terminated.



Flow control: Like the data link layer, the transport layer is responsible for flow control. However, flow control at this layer is performed end to end rather than across a single link.



Error control: Like the data link layer, the transport layer is responsible for error control. However, error control at this layer is performed end-to-end rather than across a single link. The sending transport layer makes sure that the entire message arrives at the receiving transport layer with out error. (damage, loss, or duplication). Error correction is usually achieved through retransmission.

Session Layer 

Dialog control



Synchronization

Presentation Layer 

Translation:Different host computer uses different format to present the data. The presentation layer at the source host converts the source specific formatting of data to a well known common format and the receiving host converts common data to its own format.



Encryption: To carry sensitive information, a system must be able to assure privacy. Decryption reverses the original process to transform the message back to its original form.





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Compression: Data compression reduces the number of bits to be transmitted. Data compression becomes particularly important in the transmission of multimedia such as text, audio, and video.

Application Layer 

Network Virtual Terminal



File Transfer Access Management (FTAM)



Mail services



Directory services



And lot more

Networks Connecting Devices Internetwork commonly uses special devices called repeaters, bridges, routers and gateways to connect independent networks. Repeaters: Devices that amplify signals in this way are called repeaters. Repeaters fall into two categories : amplifiers and signal-regenerating repeaters. 

Amplifiers simply amplify the entire incoming signal i.e. they amplify both the signal and noise.



Signal-regenerating repeaters differentiate between data and noise, and retransmitting only the desired information. This reduces the noise. The original signal is duplicated, boosted to its original strength, and sent.

Bridges: Bridges connect two networks that use the same technology. The use of a bridge increases the maximum possible size of your network. A bridge selectively determines the appropriate segment to which it should pass a signal. It does this by reading the address of all the signals it receives. The bridges read the physical location of the source and destination computer from this address. There are two basic types of bridges: 

Transparent bridges: That keeps a table of addresses in memory to determine where to send data.



Source-routing bridges: That requires the entire route to be included in the transmission and do not route packets intelligently. IBM Token Ring networks use this type of bridge.

Routers: A router routes data between networks. Routers use logical and physical addressing to connect two or more logically separate networks. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 356


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B-Routers: Many routers may be appropriately called brouters. A brouter is a router that can also bridge. Gateways: A gateway is a device that can interpret and translate the different protocols that are used on two distinct networks. The gateway can actually convert data so that it works with an application on a computer on the other side of the gateway. Hubs: All networks require a central location to bring media segment (i.e. computers) together. These central locations are called hubs. There are three categories of hubs: 

Passive hubs: A passive hub simply combines the signals of network segments. There is no signal processing or regeneration. With a passive hub, each computer receives the signals sent from all other computer connected to hub.



Active hubs: Active hubs are like passive hubs except that they have electronic components that regenerate or amplify signals. Because of this, the distance between devices can be increased.



Intelligent hubs: In addition to signal regeneration, intelligent hubs perform some network management and intelligent path selection. Many switching hubs can choose which alternative path will be the quickest and send the signal that way.

Connection-oriented and Connectionless Services There are two ways that communication between computers can be arranged: connectionless and connection-oriented. 

The connection oriented network service, the service user first establishes a connection, uses the connection, and then releases the connection.



The connection less service each message carries the full destination address, and each one is routed through the system independently.



epeater elay



ridge elay

bit delay transmission delay T.



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10.2 Multiple Access Protocols Many algorithms for allocation for multiple access channels are known. ALOHA Protocol Pure ALOHA   

A node transmits wheneverithas data to be sent .There will be collision and the colliding frames will be destroyed. If the channel supports feedback property sender can find out whether or not its frame was destroyed by listening to the channel. If the frame was destroyed, the sender just waits the random interval of time and sends again. The throughput of ALOHA system is maximized by having the uniform frame size rather than allowing variables length frames.

Efficiency of ALOHA channel:   

  

Efficiency means, the number of successful transmission without collision. Let the infinite population of users generates the new frames according to Poisson distribution with mean N frames per frame-time. Let that the probability of k transmission attempts per frame-time, old and new frames combined, is also Poisson, with mean G per frame-time. e [ ] robability of ero frames e 0 G is always greater than N. At low load there will be few collision, so G and N would b approximately equal on the other hand at high load will be many collisions so G>N. Throughput = G times the probability of transmission being successful. That is GP, Since P = e–2G, the throughput is The maximum throughout occurs at G=0.5, with S = 1/2e, which is about 0.184. i.e., 18% approx

Slotted ALOHA  

In case of slotted ALOHA a computer is not permitted to send whenever it is ready to send, but it is required to wait for a beginning of next slot., the probability of no other traffic during the same slot as out test frame is e-G which leads efficiency toS = Ge-G Slotted ALOHA peaks at G=1, with throughput of S = 1/e or about 0.368 or 37 percent approx.

Carrier Sense Multiple Access (CSMA) Protocols Protocol in which stations listen for a carrier and act accordingly are called carrier sense protocols. A number of them have been proposed.



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Persistent and Non Persistent CSMA Persistent CSMA (Carrier Since Multiple Access).      

When a station has data to send, if first listen to the channel to see if anyone else is transmitting at the moment. If the channel is busy, the station waits until it becomes idle If the channel is free, it transmits the frame. If a collision occurs the station waits a random interval of time and starts all over again. The protocol is called 1-persistent if station transmits with the probability of 1 whenever it finds the channel free. The protocol is call p-persistent if the station transmits with the probability p, whenever it finds the channel free.

Non-persistent CSMA.    

In this protocol, continues attempt is made to be less greedy than in the 1-persisent If no one else is sending, the station begins doing so itself. If the channel is already in use it waits for random period of time and then repeats the algorithm. This algorithm should lead the better channel utilization and longer delays than 1persisent CSMA.

CSMA with Collision Detection (CSMA-CD)     

If two station sense the channels to be idle and begin transmitting simultaneously, they will both detects the collision almost immediately. Rather than finish transmitting the frames, they should abruptly stop as soon as collision is detected, quickly terminating damaged frames saves time and bandwidth. This protocol, known as CSMA/CD (Carrier sense Multiple Access Collision Detection), is widely used in Ethernet. The collision detection puts a restriction on minimum size of the frame a source can transmit. In order to correctly detect the collision, the transmission duration of a frame should be greater than round trip delay With CSMA/CD the collisions are still possible

IEEE 802.4 Token Bus  This standard, 802.4 describes a LAN called a Token buses a linear or tree-shaped cable into which a station are attached.  Logically, the stations are organized into a ring with each station knowing the address of the station to its “left” or “right”.  The token propagates around the logical ring, with only the token holder being permitted to transmit frames. Since only one station at a time holds a token, collision do not occur.  An important point to realize is that the physical order in which the stations are connected to the cable is not important. Since the cable is inherently a broadcast medium, each station receives each frame, discarding those not addressed to it.



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IEEE 802.5 Token ring    

Unlike token bus case, a ring really consists of collection of ring interfaces, connected through point to point lines in a physical manner. Each bit arriving at interfaces is copied into a 1 bit buffer and then copied out onto the ring again. This coping step introduces a 1 bit delay. At each interface. A special bit pattern, called a token, circulates around the ring whenever all station is idle. When station wants to transmit the frame, it is required to seize the token and remove it from the ring before transmitting. Ring latency can be defined as the time required by the one bit traveling complete ring.



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10.3 The Data Link Layer (Flow and error control techniques) Functions of Data Link Layer   

Framing: determining how the bits of the physical layer are grouped into frames, Error control: detection and correction of transmission errors. Flow control: Regulating the flow of frame so that slow receivers are not swapped by fast sender, and general link management.

Framing Commonly used methods:  Character count : The first framing method uses the field is the header to specify the number of character in the frame, when the data link layer at the destination sees the character count, it knows how many character follow, and hence the end of the frame. The trouble with this algorithm is that the count can be garbled by a transmission error. 



Character based Start and Stop: Start each frame start with the ASCII character sequence DLESTX and end with sequence DLEETX (DLE is data link escape, STX is Start of Text, ETX is End of Text). A serious problem occurs with this method when binary data, such as object programs or floating point numbers, are being transmitted, It may easily happen that the character for DLESTX or DLRETX occur in the data, which will interface with the farming. One possible solution to this problem is character stuffing. The Sender data lin layer insert one ASCII LE character just before each “accidental” LE character in the data, the data link layer in the receiving end removes the DLE before the data are given to the network layer. Bit pattern based start and stop: This technique allows the frames to contain any arbitrary bits and allows the character codes with an arbitrary numbers of bits per character. Each frame begins and ends with specific bit pattern, 01111110, called the flag byte. Whenever the sender’s data link layers encounter five consecutive ones in the data, it automatically stuffs the 0 bit into the outgoing bit stream. This bit stuffing is analogous to character stuffing, in which the DLE is stuffed into the outgoing character stream before DLE in the data.Whenever the receiver sees five consecutive 1 bits, followed by zero bit, it automatically de-stuffs (i.e. deletes) the 0 bit, if the user data contains the flag pattern, 01111110, this flag is transmitted as 011111010 but stored in the receiver memory as 0111111.

Error Detection  Parity Bit:Most of errors result in change of a bit from 0 to 1 or 1 to 0. One of the simplest error detection codes which is used in common is called as parity bit.This can be done by counting number of 1’s in message string.There are two types of parity bit:  Even parity bit: If in a message string number of 1’s are even than it is called even parity bit e.g. 101000 has even parity bit since, there are two 1’s  Odd parity bit: If in a message string number of 1’s are odd than it is called odd parity bit e.g. 101100 has odd parity bit since, there are three number 1’s .  The single parity bits can detect all even (odd) errors  Multiple parity bit schemes can also be used instead of one bit, to improve the error detection efficiency THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 361


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Cyclic Redundancy Check (CRC) CRC is based on binary division   

A sequence of redundant bits, called the CRC or the CRC remainder, is appended to the end of a data unit so that the resulting data unit becomes exactly divisible by a second, predetermined binary number. At its destination, the incoming data unit is divided by the same number. If at this step there is no remainder, the data unit is assumed to be intact and is therefore accepted. A remainder indicates that the data unit has been damaged in transit and therefore must be rejected.

CRC Mechanism     

First, a string of n 0’s is appended to the data unit. The number n is one less than the number of bits in the predetermined divisor, which are n + 1 bits. Second, the newly appended data unit is divided by the divisor using a process called binary division. The remainder resulting from this division is the CRC. Third, the CRC of n bits derived in step 2 replaces the appended 0’s at the end of the data unit. Note that the CRC may consist of all 0s. The receiver divides the received data whole string as a unit and divides it by the same divisor that was used to find the CRC remainder. If the string arrives without error, the CRC checker yields a remainder of zero and the data unit passes. If the string has been changed in transit, the division yields a non-zero remainder and the data unit does not pass.

CRC generator or divisor   

  

The CRC generator (the divisor) is most often represented not as a string of 1’s and 0’s, but as an algebraic polynomial. The polynomial format is useful for two reasons: It is short, and it can be used to prove the conceptmathematically The selected polynomial should be at least divisible by x and x+1. The first condition guarantees that all burst errors of length equal to the degree of the polynomial are detected. The second condition guarantees that all burst errors affecting an odd number of bits are detected CRC can detect all burst errors that affect an odd number of bits. CRC can detect all burst errors of length less than or equal to the degree of the polynomial. CRC can detect with a very high probability burst errors of length greater than the degree of the polynomial.

Flow Control  

Flow control refers to a set of procedures used to restrict the amount of data, the sender can send before waiting for acknowledgement. Two methods have been developed to control the flow of data across communications links: stop- and- wait and sliding window



  



Computer Networks

In the stop–and–wait method of flow control, the sender sends one frame and waits for an acknowledgement before sending the next frame. The advantage of stop–and–wait is simplicity each frame is checked and acknowledged before the next frame is sent. The disadvantage is inefficiency: stop- and-wait is slow. In the sliding window method of flow control, the sender can transmit several frames before needing an acknowledgement. Frames can be sent one right after another. The link can carry several frames at once and its capacity can be used efficiently. The receiver acknowledges only some of the frames, using a single ACK to confirm the receipt of multiple data frames. The advantage of sliding window is its higher efficiency. The disadvantage is its complexity. For stop and wait T

Channel utli ation

T

S S 1 T here TT

T TT robability of frame error oun trip time

Throught put

C

.

Where CU = Channel utilization BW = Band width Transmission elay T

ac et length and width

Sliding Window Protocol      

The sliding window refers to imaginary windows at both the sender and receiver. At sender side, this window can hold frames which are either transmitted but not received the acknowledgment and frames that can be transmitted before requiring an acknowledgement. Frames may be transmitted as long as the window is not yet full. To keep track of which frames have been transmitted and which received, sliding window introduces an identification scheme based, on the size of the window. The frames are numbered modulo-n which means they are numbered from 0 to n – 1. When receiver sends an ACK, it includes the number of next frame it expects to receive. When the sender sees an ACK with the number 5, it knows that all frames up through number 4 have been received. A maximum of n – 1 frame may be sent before an acknowledgement is required. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 363


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 

At the beginning of a transmission, the senders window contains n – 1 frame. The sliding window of the sender shrinks from the left, when frames of data are sent. The sliding window of the sender expands to the right when acknowledgements are received.  The receiver window therefore represents not the number of frames received but the number offrames that may still be received between an ACK must be sent.  The sliding window of the receiver shrinks from the left when frames of data are received. The sliding window of the receiver expands to the right when acknowledgements are sent.  In the sliding window method of flow control, the size of the window is one less than modulo range, so that there is no ambiguity in the acknowledgement of the received frames. TT ptimal window ⌈ ⌉ T TT Sender windows si e 1 ⌈ ⌉ T 1 ⌈ ⌉ T Sliding window protocal channel utili ation T TT N T T TT

Error Control         



In the data link layer, the term error control refers primarily to methods of detection and retransmission. Anytime an error is detected in an exchange, a negative acknowledgement (NAK) is returned and the specified frames are retransmitted. This process is called automatic repeat request (ARQ). Error control in the data link layer is based on automatic repeat request (ARQ), which means retransmission of data in three cases: damaged frame, lost frame and lost acknowledgement. ARQ error control is implemented in the data link layer as an adjacent to flow control Usually stop-and-wait flow control is usually implemented as stop-and-wait ARQ and sliding window is usually implemented as one of two variants of sliding window ARQ, called Go-back-N or selective-reject Stop-and-wait ARQ is a form of stop-and-wait flow control extended to include retransmission of data in case of lost or damaged fames. For retransmission to work, four features are added to the basic flow control mechanism: The sending device keeps a copy of the last frame transmitted until it receives an acknowledgement for that frame. For identification purposes, both data frames and ACK frame are numbered after 0 and 1. A data 0 frame is acknowledged by an ACK 1 frame, indicating that the receiver has gotten data 0 and is now expecting data 1. This numbering allows for identification of data frames in case of duplicating transmission. If an error is discovered in a data frame, indicating that it has been corrupted in transmission, a NAK frame is returned. NAK frames, which are not numbered, tell the sender to retransmit the last frame sent. Stop-and-wait ARQ requires that the sender wait until it receives an acknowledgement for the last frame transmitted before it



            

Computer Networks

transmits the next one. When the sending device receives a NAK, it resends the frame transmitted after the last acknowledgement, regardless of number. The sending device is equipped with a timer. If an expected acknowledgement is not received within an allotted time period the sender assumes that the last data frame was lost in transmit and sends it again. Go-back-N ARQ and selective reject ARQ both are based on sliding window flow control. To extend sliding window to cover retransmission of lost or damaged frames, three features are added to the basic flow control mechanism: The sending device keeps copies of all transmitted frames, until they have been acknowledged. In addition to ACK frames, the receiver has the option of returning a NAK frame the data have been received damaged. NAK frame tells the sender to retransmit a damaged frame. In this sliding window Go-Back-NARQ method, if one frame is lost or damaged, all frames sent since the last frame acknowledged are retransmitted. In selective-Reject ARQ, only the specific damaged or lost frame is retransmitted. The receiving device must be able to sort the frames it has and insert the retransmitted frame into its proper place in the sequence. To make such selectivity possible, a selective reject ARQ system differs from a Go-back-N ARQ system in the following ways: The receiving device must contain sorting logic to enable it to recorder frame received out of sequence. It must also be able to store frame received after a NAK has been sent until the damaged frame has been repaired. The sending device must contain a searching mechanism that allows it to find and select only the requested frame for retransmitted A buffer in the receiver must keep all previously receives frame on hold until all retransmissions have been sorted and any duplicate frames have been identified and discarded. To aid selectivity, ACK numbers, like NAK numbers, must refer to the frame receive (or lost) instead of the next frame expected. This complexity requires a smaller window size than is needed by the Go-back-N method if it is to work efficiently. It is recommended that the window size be less than or equal ton /2, where n – 1 is the Go-back-N window size.



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10.4 Routing & Congestion Control Routing Algorithms 

The main function of the network layer is routing packets from the source machine to the destination machine.



The routing algorithm is that part of the network layer software responsible for deciding which output line an incoming packet should be transmitted on.



In case of data datagrams (packet switched network), the routing decision must be made new for every arriving data packet since the best route may have changed since last time.



In case of virtual circuits (circuit switched network), routing decisions are made only when a new virtual circuit is being set up. Thereafter, data packets just follow the previously established route.

Routing algorithms can be grouped into two major classes: non-adaptive and adaptive. 

Nonadaptive algorithms do not base their routing decisions on measurements or estimates of the current traffic and topology. Instead, the choice of the route to use to get from I to J (for all I and J) is computed in advance, off-line, and downloaded to the routers when the network is booted. This procedure is sometimes called static routing.



Adaptive algorithms, in contrast, change their routing decisions to reflect changes in the topology, and usually the traffic as well.

Flooding 

Another static algorithm is flooding, in which every incoming packet is sent out on every outgoing line except the one it arrived on.



Flooding generates vast numbers of duplicate packets, in fact, an infinite number unless some measures are taken to damp the process.



A variation of flooding that is slightly more practical is selective flooding. In this algorithm the routers do not send every incoming packet out on every line, only on those lines that are going approximately in the right direction.

Dynamic Routing Algorithms Two dynamic algorithms in particular, distance vector routing and link state routing, are the most popular. Distance Vector Routing 

Distance vector routing algorithms operate by having each router maintain a table (i.e, a vector) giving the best known distance to reach destination and which line to use to get there. These tables are updated by exchanging information with the neighbours.



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In distance vector routing, each router maintains a routing table indexed by, and containing one entry for, each router in the subnet. This entry contains two parts: the preferred outgoing line to use for that destination and an estimate of the time or distance to that destination. 

The metric used might be number of hops, time delay in milliseconds, total number of packets queued along the path, or something similar.



The router is assumed to know the ''distance'' to each of its neighbours. If the metric is hops, the distance is just one hop. If the metric is queue length, the router simply examines each queue. If the metric is delay, the router can measure it directly with special ECHO packets that the receiver just timestamps and sends back as fast as it can.

Link State Routing The idea behind link state routing is simple and can be stated as five parts. Each router must do the following:     

Discover its neighbours and learn their network addresses. Measure the delay or cost to each of its neighbours. Construct a packet telling all it has just learned. Send this packet to all other routers. Compute the shortest path to every other router.

In effect, the complete topology and all delays are experimentally measured and distributed to every router. Then Dijkstra's algorithm can be run to find the shortest path to every other router. Below we will consider each of these five steps in more detail. Learning about the Neighbours 

Send HELLO packet on each point-to-point line.



The router on the other end is expected to send back a reply telling who it is.

Measuring Line Cost 

The most direct way to determine this delay is to send over the line a special ECHO packet that the other side is required to send back immediately. By measuring the round-trip time and dividing it by two, the sending router can get a reasonable estimate of the delay.

Building Link State Packets The packet starts with the identity of the sender, followed by a sequence number and age, and a list of neighbours. For each neighbour, the delay to that is given. Distributing the Link State Packets  The fundamental idea is to use flooding to distribute the link state packets.  When a new link state packet comes in, it is checked against the list of packets already seen. If it is new, it is forwarded on all lines except the one it arrived on. If it is a duplicate, it is discarded. If a packet with a sequence number lower than the highest one seen so far ever arrives, it is rejected as being obsolete since the router has more recent data.



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This algorithm has a few problems, but they are manageable. 

If the sequence numbers wrap around, confusion will arise. The solution here is to use a 32bitsequence number. With one link state packet per second, it would take 137 years to wrap around, so this possibility can be ignored.



If a router ever crashes, it will lose track of its sequence number. If it starts again at 0, the next packet will be rejected as a duplicate.



Third, if a sequence number is ever corrupted and 65,540 is received instead of 4 (a 1-bit error), packets 5 through 65,540 will be rejected as obsolete, since the current sequence number is thought to be 65,540.



The solution to all these problems is to include the age of each packet after the sequence number and decrement it once per second. When the age hits zero, the information from that router is discarded. Normally, a new packet comes in, say, every 10 sec, so router information times out when a router is down (or six consecutive packets have been lost, an unlikely event). The Age field is also decremented by each router during the initial flooding process, to make sure no packet can get lost and live for an indefinite period of time (a packet whose age is zero is discarded).



Some refinements to this algorithm make it more robust. When a link state packet comes in to a router for flooding, it is not queued for transmission immediately. Instead it is first put in a holding area to wait a short while. If another link state packet from the same source comes in before the first packet is transmitted, their sequence numbers are compared. If they are equal, the duplicate is discarded. If they are different, the older one is thrown out.



To guard against errors on the router-router lines, all link state packets are acknowledged. When a line goes idle, the holding area is scanned in round-robin order to select a packet or acknowledgement to send.

Computing the New Routes 

Construct the entire subnet graph after collecting link state for every link.



Use Dijkstra's algorithm to compute shortest path to all possible destinations. The results of this algorithm can be installed in the routing tables, and normal operation resumed.

Hierarchical Routing 

As networks grow in size, the router routing tables grow proportionally.



In above case, hierarchical routing is used, the routers are divided into what we will call regions, with each router knowing all the details about how to route packets to destinations within its own region, but knowing nothing about the internal structure of other regions.



When different networks are interconnected, it is natural to regard each one as a separate region in order to free the routers in one network from having to know the topological structure of the other ones.



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Broadcast Routing 

Sending a packet to all destinations simultaneously is called broadcasting; various methods have been proposed for doing it.



One broadcasting method that requires no special features from the subnet is for the source to simply send a distinct packet to each destination. Not only is the method wasteful of bandwidth, but it also requires the source to have a complete list of all destinations. In practice this may be the only possibility, but it is the least desirable of the methods.



A third algorithm is multi-destination routing. If this method is used, each packet contains either a list of destinations or a bit map indicating the desired destinations. When a packet arrives at a router, the router checks all the destinations to determine the set of output lines that will be needed.



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10.5 TCP/IP, UDP And Sockets, IP(V4) Overview of TCP/IP TCP/IP and the Internet An internet under TCP/IP operates like a single network connecting many computers of any size and type. Internally, an internet (or, more specifically, the internet) is an inter-connection of independent physical networks (such as LANs) linked together by internet working devices. TCP/IP and OSI  The TCP/IP protocol is made of 4 layers: physical& data link, network, transport and application.  The application layer in TCP/IP can be equated with the combination of session, presentation and application layers of the OSI model.  At the physical and data link layers, TCP/IP does not define any specific protocol. It supports all of the standard and proprietary protocols discussed earlier in this book.  A network in a TCP/IP internetwork can be a local area network (LAN), a metropolitan area network (MAN) or a wide area network (WAN).

Network Layer At the network layer TCP/IP supports the internetwork protocol (IP). IP, in turn contains four supporting protocols: ARP, RARP, ICMP and IGMP. Internetwork Protocol (IP)  IP is the transmission mechanism used by the TCP/IP protocols. It is an unreliable and connectionless datagram protocol – a best- effort delivery service.  IP transport data in packets called datagram (describe below) each of which is transported separately.  Datagrams may travel along different routes and may arrive out of sequence or duplicated. Datagram Packets in the IP layer are called datagrams. A datagram is a variable-length packet (up to 65,536 bytes) consisting of two parts: header and data. The header can be from 20 to 60 bytes and contains information byte sections. A brief description of each field is in order.  Version: The first field defines the version number of the IP. The current version is 4 (IPv4), with a binary value of 0100.  Header length (HLEN): The HLEN field defines the length of the header in multiples of four bytes. The four bits can represent a number between 0 and 15, which when multiplied by 4, gives a maximum of 60 bytes.  Service type: The service type field defines how the datagram should be handled. It includes bits that define the priority of the datagram. It also contains bits that specify the type of service the sender desires, such as the level of throughput, reliability and delay.  Total length: The total length defines the total length of the IP datagram. It is a two-byte field (16 bits) and can define up to 65,535 bytes.



   

    

Computer Networks

Identification: The identification field is used in fragmentation. A datagram, when passing through different network, may be divided into fragments to match the network frame size. When this happens, each fragment is identified with a sequence number in this field. Flags: the bits in the flags field deal with fragmentation (the datagram can or cannot be fragmented; can be the first, middle or last fragment, etc). Fragmentation offset: The fragmentation offset is a pointer that shows the offset of the data in the original datagram (if it is fragmented). Time to live: The time – to –live field defines the number of hops a datagram can travel before it is discarded. The source host, when it creates the datagram, sets this field to an initial value. Then, as the datagram travels through the Internet, router by router, each router decrements this value by 1. If this value becomes 0 before the datagram reaches its final destination, the datagram is discarded. This prevents a datagram from going back and forth forever between routers. Protocol: The protocol field defines which upper-layer protocol data are encapsulated in the datagram (TCP, UDP, ICMP, etc). Header Checksum: This is a 16-bit field used to check the integrity of the header, not the rest of the packet. Source address: The source address field is a four-byte (32-bit) Internet address. It identifies the original source of the datagram Destination address: The destination address field is a four-byte (32-bit) Internet address. It identifies the final destination of the datagram. Options: The options field gives more functionality to the IP datagram. It can carry fields that control routing, timing, management and alignment.

Addressing In addition to the physical addresses (contained on NISs) that identify individual services, the Internet requires an additional addressing convention; an address that identifies the connection of a host to its network.Each Internet address consists of four bytes (32-bits) defining three fields class type, netid, and hostid. These parts are of varying lengths, depending on the class of the addresses Classes There are currently five different field-length patterns in use, each defining a class of address. The different classes are designed to cover the needs of different types of organizations.    



Class ‘A’ addresses are numerically the lowest. They use only one byte to identify class type and netid, and leave three bytes available for hostid numbers. Class B uses two bytes to identify the netid and leave two bytes available for hostid numbers Class C uses one three bytes to identify the netid and leave one byte available for hostid numbers Class D is reserved for multicast addresses. Multicasting allows copies of a datagram to be passed to a select group of hosts rather than to an individual host. It is similar to broadcasting, but where broadcasting requires that a packet be passed to all possible destinations, multicasting allows transmission to a selected subset. Class E addresses is reserved for future use.



Computer Networks

Range of Different Classes Class A : 0.0.0.0 – 127.255.255.255 Class B : 128.0.0.0 – 191.255.255.255 Class C: 192.0.0.0- 223.255.255.255

Subnetting As we previously discussed, an IP address is 32 bits long. One portion of the address indicates a network (netid) and the other portion indicates the host (or router) on the network (hostid). This means that there is a sense of hierarchy in IP addressing. To reach a host on the Internet, we must reach the network using the portion of the address (netid). Then we must reach the host itself using the second portion (hosted). In other words, classes A,B and C in IP addressing are designed with two levels of hierarchy.

However, in many cases, these two levels of hierarchy are not enough. For example imagine an organization with a class B address. The organization has two-level hierarchical addressing, but it cannot have more than one physical network With this scheme, the organization is limited of two levels of hierarchy. The hosts cannot be organized into groups, and all of the hosts are at the same level. The organization has one network with many hosts. One solution to this problem is subnetting, the further division of a network into smaller networks called subnetworks. Three Levels of Hierarchy Adding subnetworks creating an intermediate level of hierarchy in the IP addressing system. Now we have three levels: netid, subnetid and hostid. The netid is the first level; it defines the site. The second level is the subnetid; it defines the physical subnetwork. The routing of an IP datagram now involves three steps: delivery to the site, delivery to the subnetwok and delivery to the host. Masking Masking is a process that extracts the address of the physical network from an IP address. Masking can be done whether we have subnetting or not. If we have not subnetted the network, masking extracts the subnetwork address from an IP address



Computer Networks

Finding the Subnetwork Address To find the subnetwork address, apply the mask to the IP address. Boundary – Level masking If the masking is at the boundary level (the mask numbers are either 255 or 0), finding the subnetwork address is very easy. Follow these two rules: 1. The bytes in the IP address that correspond to 255 in the mask will be repeated in the subnetwork address. 2. The bytes in the IP address that correspond to 0 in the mask will change to 0 in the subnetwork address. Nonboundary –Level Masking If the masking is not at the boundary level (the mask numbers are not just 255or 0). Finding the subnetwork address involves using the bit-wise AND operator. Follow these three rules: 1. The bytes in the IP address that correspond to 255 in the mask will be repeated in the subnetwork address. 2. The bytes in the IP address that correspond to 0 in the mask will change 0 in the subnetwork address. 3. For other bytes, use bit-wise AND operator.

Other Protocols in the Network Layer TCP/IP supports four other protocols in the network layer, ARP, RARP, ICMP and IGMP. Address Resolution Protocol (ARP)     

The address resolution protocol (ARP)associates an IP address with the physical address. ARP is used to find the physical address of the node when its Internet address is known. When a host or a router needs to find the physical address of another host on its network, it formats as ARP query packet that includes the IP address and broadcasts it over the network. Every host on the network receives and processes the ARP packet, but only the intended recipient recognizes its internet address and sends back its physical address. The host holding the datagram adds the address of the target host both to its cache memory and to the datagram header, then sends the datagram on its way.

Reverse Address Resolution Protocol (RARP)   

The reverse address resolution protocol (RARP)allows a host to discover its internet address when it knows only its physical address. Required for a diskless computer or the computer is being connected to the network for the first time RARP works much like ARP. The host wishing to retrieve its internet address broadcasts an RARP query packet that contains its physical address to every host on its physical network. A server on the network recognizes the RARP packet and returns the host internet address. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 373


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Internet Control Message Protocol (ICMP)   

 

The internet control message protocol (ICMP)is a mechanism used by hosts and routers to send notifications of network problems back to the sender. IP is essentially an unreliable and connectionless protocol. ICMP. However, allows IP to inform a sender. If a datagram is undeliverable. A datagram travels from router to router until it reaches one that can deliver it to its final destination. If a router is unable to route or deliver the datagram because of unusual conditions or because of network congestion, ICMP allows it to inform the original source. ICMP uses echo test/reply to test whether a destination is reachable and responding. It also handles both control and error messages, but its sole function is to report problems, not correct them, responsibility for correction lies with the sender. Note that a datagram carries only the addresses of the original sender and the final destination. It does not know the addresses of the previous routers that passed it along. For this reason, ICMP can send messages only to the source, not to an intermediate router.

Internet Group Message Protocol (IGMP)  

 

The IP protocol can be involved in two type of communication; unicasting and multi-casting. Unicasting is the communication between one sender and one receiver. It is a one-to-one communication. However, some processes some time need to send the same message to a large number of receivers simultaneously. This is called multicasting, which is a one-tomany communication. IP addressing supports multicasting. All 32-bit IP addresses that start with 1110 (class D) are multicast addresses. With 28 bits remaining for the group address, more than 250 million addresses are available for permanently assigned. Internet group message protocol (IGMP) has been designed to help a multi-cast router identify the hosts in a LAN that are members of a multi-cast group. It is a companion to the IP protocol.

Transport Layer The transport layer is represented in TCP/IP by two protocols: TCP and UDP. Of these UDPis the simpler; it provides non-sequenced transport functionality when reliability and security are less important than size and speed. Most applications however require reliable end-to-end delivery and so make use of TCP. The IP delivers a datagram from a source host to a destination host, making it a host-to-host protocol. A host receiving a datagram may be running several different concurrent processes, any one of which is a possible destination for the transmission. In fact, although we have been talking about hosts sending messages to other hosts over a network, it is actually a source process that is sending a message to a destination process. TCP/IP transport level protocols are port-to-port protocols that work on top of the IP protocols to deliver the packet from the originating port to the IP services at the start of a transmission and from the IP services to the destination port at the end. Each port is defined by a positive integer address carried in the header of a transport layer packet. An IP datagram uses the host 32-bit internet address. A frame at the transport level uses the process port address of 16-bits, enough to allow the support up to 65,536 to 65535) ports. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 374


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User Datagram Protocol (UDP) It is an end-to-end transport level protocol that adds only port addresses, checksum error control, and length information to the data from the upper layer. The packet produced by the UDP is called a user datagram. A brief description of its fields is in order.    

Source port address: The source port address is the address of the application program that has created the message. Destination port address: The destination port address is the address of the application program that will receive the message. Total length: The total length field defines the total length of the user datagram in bytes. Checksum: The checksum is a 16-bit field used in error detection.

UDP provides only the basic functions needed for end-to-end delivery of a Transmission. It does not provide any sequencing or reordering functions and cannot specify the damaged packet when reporting an error (for which it must be paired with ICMP).UDP can discover that an error has occurred; ICMP can then inform the sender that a user datagram has been damaged and discarded. Neither, however, has the ability to specify which packet has been lost. UDP contains only a checksum; it does not contain an IP or sequencing number for a particular data segment. Transmission Control Protocol (TCP) The Transmission Protocol (TCP) provides full transport layer service to applications.TCP is a reliable stream transport port-to-port protocol. The term stream in this context, means connection-oriented : a connection must be established between both ends of a transmission before either may transmit data. By creating this connection .TCP generates a virtual circuit between sender and receiver that is active for the duration of a transmission, (Connections for the duration of an entire exchange are different and are handled by session functions in individual application.) TCP begins each transmitted by altering the receiver that datagrams are on their way (connection establishment) and ends each transmission with a connection termination. In this way the receiver knows to expect the entire transmission rather than a single packet. IP and UDP treat multiple diagrams belonging to a single transmission as entirely separate units, unrelated to each other. The arrival of each datagram at the destination is therefore a separate event, unexpected by the receiver. TCP, on the other hand, as a connection – oriented service, is responsible for the reliable delivery of the entire stream of bits contained in the message originally generated by the sending application. Reliability is ensured by provision for error detection and retransmission of damaged frames; all segments must be received and acknowledged before the transmission is considered complete and the virtual circuit is discarded.At the sending ends of each transmission. TCP divides long transmission s into smaller data units and packages each into a frame called a segment. Each segment includes a sequencing number for recording after receipt, together with as acknowledgement ID numbers and a window-size field for sliding window ARQ. Segments are carried across network link inside of IP datagrams. At the receiving end, TCP collects each datagram as it comes in and reorders the transmission based on sequence numbers. The TCP Segment The scope of the services provided by TCP requires that the segment header be extensive. A comparison of the TCP segment format with that of a UDP datagram shows the differences between the two protocols. TCP provides a comprehensive range of reliability functions but THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 375


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sacrifices speed (connections must be established, acknowledgements waited for, etc). Because of its smaller frame size. UDP is much faster than TCP, but at the expense of reliability. A brief description of each field is in order.    



 

  



Source port address: The source port address defines the application program in the source computer. Destination port address: The destination port address defines the application program in the destination computer. Sequence number: A stream of data from the application program may be divided into two or more TCP segments. The sequence number field shows the position of the data in the original data stream. Acknowledgement number: The 32-bit acknowledgement number is used to acknowledge the receipt of data from the other communicating device. This number is valid only if the ACK but in the control field (explained later) is set. In this case it defines the byte sequence number that is next expected. Header length (HLEN): The four-bit HLEN field indicates the number of 32-bit (four-byte) words in the TCP header. The four bits can define the number up to 15. This is multiplied by 4 to give the total number of bytes in the header. Therefore, the size is of the header is 20 bytes; 40 bytes are thus available for the option sections. Reserved: A six-bit field is reserved for future use. Control: Each bit of the six-bit control field functions individually and independently. A bit can either define the use of a segment or serve as a validity check for other fields. The urgent bit, when set, validates the urgent pointer field. Both this bit and the pointer indicate that the data in the segment are urgent. The ACK bit, when set, validates the acknowledgement number field. Both are used together and have different functions, depending on the segment type. The PSH bit is used to inform the sender that a higher throughput is needed. If possible data must be pushed through paths with higher throughput is needed throughput. The reset bit is used to reset the connection when there is confusion in the sequence numbers. The SYN bit is used for sequence number synchronization in three types of segments; connection request, connection confirmation (with the ACK bit set), and confirmation acknowledgement (with the ACK bit set). The FIN bit is used in connection termination in three types of segments; termination request, termination confirmation (with the ACK bit set). And acknowledgement of termination confirm. Window size: The window is a 16-bit field that defines the size of the sliding window. Checksum: The checksum is a 16-bit field is used in error detection. Urgent pointers: This is the last required field in the header. Its value is valid only if the URG bit in the control field is set. In this case, the sender is informing the receiver that there are urgent data in the data portion of the segment. This pointer defines the end of urgent data and start of the normal data. Options and padding: The remainder of the TCP header defines the optional fields. They are used to convey additional information to the receiver of for alignment purposes.



Computer Networks

10.6 Application Layer The Domain Name Service (DNS)   



Instead of using the full 32-bit IP address, many systems adopt meaningful names for their devices and networks. To solve the problem of network names; the Network Information Centre (NIC) maintains a list of network names and the corresponding network gateway addresses. DNS uses a hierarchical architecture, much like the UNIX file system. The first level of hierarchy divides networks into the category of sub-networks, such as com for commercial, mil for military, edu for education, and so on. Below each of these is another division that identifies the individual sub network, usually one for each organization. This is called the domain name. The organi ation’s system manager can further divide the company’s sub-networks as desired, with each network called a sub-domain.

File Transfer Protocol (FTP) 

FTP is an internet services that transfer a data file from the disk on one computer to the disk on another. File transfer services, which can copy a large volume of data efficiently, only require a single person to manage the transfer.

Internet Remote login is called TELNET 

The TELNET protocol specifies exactly how a remote login client and a remote login server interact.

Uniform Resource Locator (URL) 



A URL identifies a particular Internet resource; for example a web page, a Gopher server, a library catalog, an image, or a text file. URLs represent a standardized addressing scheme for Internet resources, and help the users to locate these resources by indicating exactly where they are. Every resource available via the World Wide Web has a unique URL. URLs consist of letters, numbers, and punctuation. The basic structure of a URL is hierarchical and hierarchy moves from left to right :Protocol:// server- name.domainname.top-level-domain; port/directory/filename.

Simple Mail Transfer Protocol (SMTP)      

The SMTP is the defined Internet method for transferring electronic mail. SMTP is similar to FTP in many ways, including the some simplicity of operation. SMTP uses TCP port numbers. When a message is sent to SMTP, it places it in a queue. SMTP attempts to forward the message from the queue whenever it connects to remote machines. If it cannot forward the message within a specified time limit, the message is returned to the sender or removed. When a connection is established, the two SMTP systems exchange authentication codes. Following this, one system sends a MAIL command to the other to identify the sender and provide information about the message. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 377


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Computer Networks

The receiving SMTP returns an acknowledgment, after which a RCTP is sent to identify the recipient. If more than one recipient, at the receiver location is identified, several RCTP messages are sent, but the message itself is transmitted only once. After each RCTP there is an acknowledgment. A DATA command is followed by the message line, until a single period on a line by itself indicates the end of the message. The connections to closed with a QUIT command.

Hypertext Transfer Protocol (HTTP)    



HTTP is short for Hyper Text Transfer Protocol. It is the set of rules, or protocol that governs the transfer of hypertext between two or more computers. Hyper is the text that is specially coded using a standard system called Hypertext Markup Language (HTML). The HTML codes are used to create links. These links can be textual or graphic and when clic ed on can “lin ” the user to another resource such as other HT L documents text files, graphics, animation and sound. HTT is used on the client/serves principle. HTT allows “computer A” the client) to establish a connection with “computer ” the server and ma e a re uest. The server accepts the connection initiated by the client is interested in and tells the server what “action” to ta e on the se uence. When a user select a hypertext link, the client programs on their computer uses HTTP to contact the server, identify a resource , and ask the server to respond with an action. The server accepts the request, and then uses HTTP to respond to or perform the action.



Computer Networks

10.7 Network Security Cipher Modes 

Electronic Code Book Method



Cipher feedback mode



Output feed back mode

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A stream cipher. (a) Encryption. (b) Decryption



Counter mode

Asymmetric Key Algorithm (or ) Public key Algorithm: Requirements: 1. D(E(P)) = P. 2. It is exceedingly difficult to deduce D from E. 3. E cannot be broken by a chosen plaintext attack. The RSA Algorithm: 1. 2. 3. 4.

Choose two large primes, p and q. (typically greater than 10100). Compute n = p x q and z = (p-1) x (q-1). Choose a number relatively prime to x and call it d. Find e such that e x d = 1 mod z.

Cipher text (Sender side) C

m mod n

Receiver side

Digital Signatures Requirements: 1. The receiver can verify the claimed identity of the sender. 2. The sender cannot later repudiate the contents of the message. 3. The receiver cannot possibly have concocted the message himself. 1. Digital signatures using public-key cryptography.. 2. Message Digests (MD5) THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 379


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3. SHA-1 4. Mutual authentication using public-key cryptography. Firewalls Aims:   

Establish a controlled link Protect the premises network from Internet-based attacks Provide a single choke point

Firewall Characteristics 

Design goals: –

All traffic from inside to outside must pass through the firewall (physically blocking all access to the local network except via the firewall)

–

Only authorized traffic (defined by the local security police) will be allowed to pass

–

The firewall itself is immune to penetration (use of trusted system with a secure operating system)

Types of Firewalls Three common types of Firewalls: 1. Packet-filtering routers 2. Application-level gateways 3. Circuit-level gateways 4. Bastion host 1. Packet-filtering Router Packet-filtering Router 

Possible attacks and appropriate countermeasures –

IP address spoofing

–

Source routing attacks

–

Tiny fragment attacks



Computer Networks

1. Application-level Gateway 2. Circuit-level Gateway 3. Bastion Host –

A system identified by the firewall administrator as a critical strong point in the network´s security.

–

The bastion host serves as a platform for an application-level or circuit-level gateway.

Firewall Configurations   

Screened host firewall, single-homed based configuration Screened host firewall system (dual-homed bastion host) Screened-subnet firewall system



SE & WT

Part – 11: Software Engineering 11.1 Introduction Software engineering is defined as an engineering discipline that applies sound scientific, mathematical, management, and engineering principles to the successful building of large computer programs (software). Software engineering includes software requirements analysis; software design; modern programming methods; testing procedures; verification and validation; software configuration management; software quality assurance; tools for analysis and design; corporate software policies; strategies and goals; project management planning; organizing; staffing; directing; and controlling; as well as the foundations of computer science.

Software Development The Waterfall Model The Waterfall Model (WM) is an early lifecycle model. WM is based on engineering practice; it works well if the requirements are well-understood and do not change — this rarely happens in practice. The Waterfall Model is important in the same sense as Newton’s Theory of Gravity: it’s wrong, but you can’t understand relativistic gravitation if you do not understand Newtonian gravitation. A software project is divided into phases. There is feedback from each phase to the previous phase, but no further. 1. Requirements Analysis. 2. Design and Specification. 3. Coding and Module Testing . 4. Integration and System Testing. 5. Delivery and maintenance. WM is document driven. Requirements analysis yields a document that is given to the designers; design yields a document that is given to the implementers; implementation yields documented code. Requirements Analysis (SRS) Write a System Requirements Document (SRD) that describes in precise detail what the customer wants. Find out what the client wants. This should include what the software should do and also:  likely and possible enhancements;  platforms (machines, OS, programming language, etc);  cost;  delivery schedule;  terms of warranty and maintenance;  user training;



SE & WT

Note: The SRD does not say how the software works. Major deliverable: SRD. Design Design a software system that satisfies the requirements. Design documentation has three parts: Architecture Document (AD): An overall plan for the components of the system. The AD is sometimes called High-level Design Document (HDD). Module Interface Specifications (MIS): Description of the services provided by each software module. Internal Module Design (IMD): Description of how the module implements the services that it provides. In the AD, each module is a “black box”. The MIS describes each module as a black box. The IMD describes each module as a “clear box”. Major deliverable: AD, MIS, and IMD. Implementation Implement and test the software, using the design documents. Testing requires the development of test plans, based on SRD, which must be followed precisely. Roughly: for each requirement, there should be a test. Major deliverable: Source code and Test results. Delivery and Maintenance The product consists of all the documentation generated and well-commented source code. Maintenance includes: Correcting: removing errors; Adapting: For a new processor or OS or to new client requirements; Perfecting: Improving performance in speed or space. Maintenance is 60% to 80% of total budget for typical industrial software. This implies the need for high quality work in the early stages. Good documentation and good coding practice make maintenance easier, cheaper, and faster. Reverse engineering is a rapidly growing field. Many companies have MLOCs of legacy code developed 20 or 30 years ago in old languages (e.g. COBOL, FORTRAN, PL/I) with little supporting documentation. Tools are used to determine how it works. Delivery also includes customer assistance in the form of manuals, tutorials, training sessions, response to complaints.



SE & WT

The requirements of high quality are the same as the requirements for maintainability. Maintenance is the solution, not the problem. Software Tools Software tools are an important part of software development. The larger the project, the more important it is to use tools in its development.  Editor.  Compiler and Linker.  Version control system (RCS).  Software measurement (DATRIX).  Specification checkers (OBJ3, Larch Prover).  Test generators.  Graph editors for DFDs and other diagrams.  CASE tools for integrated development.  Browsers, library managers, etc.

The Software Requirements Document The SRD has a number of important functions. It provides the basis for:  Agreement between customer and supplier. There may be other components of the agreement, such as legal documents.  Costing and scheduling.  Validation and verification. You cannot test software unless you know what it is supposed to do.  All forms of maintenance. A well-written SRD will reduce development effort by avoiding (expensive) changes later, in design and implementation phases. Notation: ◊ = good, ♣ = bad. Characteristics of a good SRD:  The SRD should define all of the software requirements but no more. In particular, the SRD should not describe any design, verification, or project management details. ♣ “The table is ordered for binary search.” ♣ “The table is organized for efficient search.” ◊ “The search must be completed in time O(log N).”     

The SRD must be unambiguous. There should be exactly one interpretation of each sentence. Special words should be defined. Some SRDs use a special notation for words used in a specific way: !cursor!. Avoid “variety” — good English style, but not good SRD style. Careful usage.

♣ “The checksum is read from the last record.” Does “last” mean (a) at end of file, (b) most recently read, or (c) previous? ◊ “. . . from the final record of the input file.” THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 384


◊

SE & WT

“. . . from the record most recently processed.”

 The SRD must be complete. It must contain all of the significant requirements related to functionality (what the software does), performance (space/time requirements), design constraints (“must run in 640Kb”), and external interfaces.  The SRD must define the response of the program to all inputs.  The SRD must be verifiable. A requirement is verifiable if there is an effective procedure that allows the product to be checked against the SRD. ♣ “The program must not loop” . ♣ “The program must have a nice user interface.” ◊ “The response time must be less than 5 seconds for at least 90% of queries.”  The SRD must be consistent. A requirement must not conflict with another requirement. ♣“When the cursor is in the text area, it appears as an I-beam. During a search, the cursor appears as an hour-glass.”        

The SRD must be modifiable. It should be easy to revise requirements safely — without the danger of introducing inconsistency. This requires: Good organization; Table of contents, index, extensive cross-referencing; Minimal redundancy. The SRD must be traceable. The origin of each requirement must be clear. (Implicitly, a requirement comes from the client; other sources should be noted.) The SRD may refer to previous documents, perhaps generated during negotiations between client and supplier. The SRD must have detailed numbering scheme.



The SRD must be usable during the later phases of the project. It is not written to be thrown away! A good SRD should be of use to maintenance programmers. The SRD is prepared by both the supplier with help and feedback from the client.  The client (probably) does not understand software development.  The supplier (probably) does not understand the application. Writing Requirements   

Include input/output specifications. Give representative, but possibly incomplete, examples. Use models: mathematical (e.g. regular expressions); functional (e.g. finite state machines); timing (e.g. augmented FSM).

 Distinguish mandatory, desirable, and optional requirements.  “The user interface must use X Windows exclusively.”  “The software must provide the specified performance when executed with 16Mb of RAM. Preferably, it should be possible to execute the software with 8Mb of RAM.”  “Sound effects are desirable but not required.”



SE & WT

  

Anticipate change. Distinguish what should not change, what may change, and what will probably change. “The FSM diagram will contain at least nodes and arcs.” “The software may be eventually required to run on machines with the EBCDIC character set.”

Capability Maturity Model (CMM): CMM that defines key activities required at different level of process maturity Level 1: Initial: The software process is characterized as adhoe and occasionally even chaotic. Few process are defined, and success depends on individual effort. Level 2: Repeatable: Basic project management processes are established to track cost, schedule and functionality Level 3: Defined: The software process for both management and engineering activities is documented, standardized, and integrated into an organization. Level 4: Managed: Detailed measures of the software process and product quality are collected Level 5: Optimizing: Continuous process improvement is enabled by quantities feedback from the process and from testing innovative ideas and technologies. This level include all characteristics defined for level 4.

Definitions, Qualities and Principles Definitions for Software Engineering Product: What we are trying to build. Process: The methods we use to build the product. Method: A guideline that describes an activity. Methods are general, abstract, widely applicable. Example: Top-down design. Technique: A precise rule that defines an activity. Techniques are precise, particular, and limited. Example: Loop termination proof. Tool: A mechanical/automated aid to assist in the application of a methodology. Examples: editor, compiler, . . . Methodology: A collection of techniques and tools. Rigor: Careful and precise reasoning. Example: An SRD should be rigorous.



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Formal: Reasoning based on a mechanical set of rules (“formal system”). Example: programming language, predicate calculus. Software Qualities Good software is: Correct: The software performs according to the SRD. The SRD may be too vague (although it should not be) — in this case, conformance to a specification is needed. Reliable: This is a weaker requirement than “correct”. E-mail is reliable — messages usually arrive — but probably incorrect. Robust: The software behaves well when exercised outside the requirements. For example, software designed for 10 users should not fall apart with 11 users. Performance: The software should have good space/time utilization, fast response times, and the worst response time should not be too different from the average response time. Friendly: The software should be easy to use, should not irritate the user, and should be consistent.  “The screen always mirrors the state.”  “One key — one effect. E.g. F1 for help.” Verifiable: A common term that is not easily defined; it is easier to verify a compiler than a wordprocessor. Maintainable:  Easy to correct or upgrade.  Code traceable to design; design traceable to requirements.  Clear simple code; no hacker’s tricks.  Good documentation.  Simple interfaces between modules.  More later. Reusable: Programmers tend to re-invent wheels. We need abstract modules that can be used in many situations. Sometimes, we can produce a sequence of products, each using code from the previous one. Example: accounting systems. OO techniques aid reuse. Portable: The software should be easy to move to different platforms. Recent developments in platform standards (PCs, UNIX, X, . . .) have aided portability. Interoperable: The software should be able to cooperate with other software (word-processors, spread-sheets, graphics packages, . . .). Visibility: All steps must be documented.



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Software Engineering Principles There are a number of general principles that apply to many areas, including aspects of software engineering. Separation of Concern Frequently, we have a large, complex problem with many inter-related aspects. To deal with such problems, separate concerns and look at each concern separately. Modularity Every large system must be divided into modules so we can understand it. Each module performs a set of tasks. Modules may be nested. Nesting suggests a tree-structure, but this is misleading. Usually, modules are constructed on layers, with each layer using the modules below it, but not above it. The implied topology is a Directed Acyclic Graph or DAG. The important attributes of modules are cohesion and coupling. Cohesive is a property of modules. A cohesive module provides a small number of closely related services.  “Create, add entry, find entry, delete entry.” ♣ “store a variable in the array, update the screen, and sound an alarm after 5 p.m.” Coupling is a property of systems. Modules are loosely coupled if the communication between them is simple.  “Modules form clusters with few interconnections.” ♣ “One modules needs all of the others.” ♣ “Every module needs every other module. Cf. spaghetti code.” The goal is: high cohesion and low coupling. Cohesion: A cohesive module perform a single task within a software procedure, requiring little interaction with procedures being performed in other parts of a program Types of cohesion: (i) Functional cohesion: Each element in a module is a necessary and essential part of one and only one function (ii) Coincidentally cohesion: Module that performs a set of tasks that relate to each other loosely (iii) Logically cohesive: A module that performs tasks that are related logically (iv) Temporal cohesion: When a module contains tasks that are related by the fact that all must be executed with the same span of time. The module exhibits temporal cohesion (v) Procedural cohesion: When processing element of a module are related and must be executed in specific order, procedural cohesion exists (vi) Communicational cohesion: When all processing element concentrate on one area of a data structure, communicational cohesion is present (vii) Sequential cohesion: The elements of a module are related by performing different parts of a sequence of operations where the output of one operation is the input to next Coupling: Coupling is a measure of interconnection among modules in a software structure Types of coupling Content coupling: It occurs when one module make use of data or control information maintained within the boundary of another method. This is highest degree of coupling THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 388


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Control coupling: At moderate level, coupling is characterized by passage of control between of coupling Common coupling: Two modules are common coupled if they share the same global data areas. Data coupling: Two modules are data coupled if they communicate via a variable or array that is passed directly as a parameter between the two modules. Content Common Control Stamp Data High low Stamp coupling: Two modules are stamp coupled if they communicate via a composite data item External coupling, import coupling, Routine call coupling are other form. Language Support for Modularity: Standard Pascal: Almost none. Nested procedures provide hierarchical structure. Turbo Pascal: Units provide quite good modularity. C: Separate compilation helps, but all responsibility is left to programmer. C++: Classes provide modularity, but C++ is still file based. Modular Languages: modularity.

These emerged during the 70s: Ada, Modula-n and provide “true”

Object oriented languages: The “pure” OOLs provide classes, which are a good basis for modularity. Abstraction: It is sometimes best to concentrate on general aspects of the problem while carefully removing detailed aspects.



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11.2 Process Models and Software Estimation Process Model A process model is a description of a way of developing software. A process model may also be a methodology for software development.

Waterfall Model The waterfall model is document-driven; Good features:  Simple to understand;  Phases are important even if their sequence is not;  Works for well-understood problems;  Keeps managers happy. Bad features:  Does not allow for change;  Does not work for novel or poorly understood problems;  Produces inaccurate estimates;  Does not allow for changing requirements;  Plethora of documents lead to “bureaucratic” project management with more concern for the existence/size of documents than their meaning.

Evolutionary Model The evolutionary model is increment driven and cyclical: 1. Deliver something (this is the “increment”); 2. Measure “added value” to customer (may be positive and negative); 3. Adjust design and objectives as required. Evolution often requires prototypes. Prototypes A prototype is a preliminary version that serves as a model of the final product. Examples:  Software Prototypes:  Emulate the user interface (UI) and see if people like it.  Develop application code without UI to assess feasibility There are several kinds of prototypes. Throwaway Prototype A throwaway prototype is not part of the final product. Throwaway prototypes should:  Be fast to build;  Help to clarify requirements and prevent misunderstanding;  Warn implementers of possible difficulties. Some languages are suited to prototyping: APL, LISP, SML, Smalltalk. Others are not: FORTRAN, COBOL, C. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 390


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A prototype meets a clearly identified subset of requirements. Examples: It may provide a realistic UI but not provide full functionality; or it may provide a functional subset without meeting performance criteria. Evolutionary Prototype: Evolutionary prototypes become part of the final product. They are usually written in the final language of the application. They fit well into the evolutionary model: 1. Develop a system that meets a well-understood (and possibly small) subset of the requirements. 2. Deliver the system and obtain feedback from the client. 3. Choose next-best understood requirement and work on that. Incremental Prototype: Even if all requirements are understood, the product may be developed as a sequence of working components. The idea is to avoid a sudden shock at the end of development when the client sees the product for the first time. Spiral Model The spiral model is Barry Boehm’s (1986) formalization of the evolutionary model. The spiral model is based on risks. In risk analysis, we identify risks and respond to them before they endanger the whole project. The spiral model envisaged by Boehm has four phases: 1. Identify objectives, alternatives, and constraints. 2. Evaluate alternatives and assess risks. 3. Develop according to established objectives and verify that these objectives are met. 4. Review results obtained during the current cycle. Plan another iteration if required. WM is roughly “once around the spiral”. A typical industrial-scale project requires from three to six iterations. Rapid Application Development Model (RAD): The RAD model is proposed when requirements and solution can be modularized as independent system or software components, each of which can be developed by different teams. After these smaller system components are developed, they are integrate to produce the larger software system solution Drawbacks:  RAD is not appropriate when technical risk is high  Not all types of applications are appropriate for RAD  RAD requires developers and customers who are committed to get a system in a much abbreviated time frame. If commitment is lacking from either constituency, RAD projects will fail Assessment of Models It is hard to do large-scale comparative studies in software engineering, but there have been a few attempts.  Waterfall development provides: good management of the process; poor response to clients; a large final product; a short test phase.



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Spiral development provides: short development time; good response to changes in requirements; a small final product. Consensus: the spiral is better than the waterfall, especially for products that are not well understood.

Design Design is conveniently split into three parts: the architecture of the system, the module interfaces, and the module implementations. We give an overview of theses and then discuss each part in more detail. Design Documentation The design documentation consist of: AD — Architectural Design MIS — Module Interface Specifications IMD — Internal Module Design The document names also provide a useful framework for describing the design process. Architectural Design The AD provides a module diagram and a brief description of the role of each module. Module Interface Specifications Each module provides a set of services. A module interface describes each service provided by the module. “Services” are usually functions (used generically: includes “procedure”). A module may also provide constants, types, and variables. To specify a function, give:  name;  argument types;  a requires clause — a condition that must be true on entry to the function;  an ensures clause — a condition that will be true on exit from the function;  further comments as necessary. This requires clause is a constraint on the caller. If the caller passes arguments that do not satisfy the requires clause, the effect of the function is unpredictable. This ensures clause is a constraint on the implementer. The caller can safely assume that, when the function returns, this ensures clause is true. This requires and ensures clause constitute a contract between the user and implementor of the function. The caller guarantees to satisfy this requires clause; in return, the implementer guarantees to satisfy this ensures clause. Internal Module Design The IMD has the same structure as the MIS, but adds:  data descriptions (e.g. binary search tree for NameTable); THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 392


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data declarations (types and names); a description of how each function will work (pseudocode, algorithm, narrative, . . .).

Remarks on Design What designers actually do:  Construct a mental model of a proposed solution.  Mentally execute the model to see if it actually solves the problem.  Examine failures of the model and enhance the parts that fail.  Repeat these steps until the model solves the problem. Design involves:  Understanding the problem;  Decomposing the problem into goals and objects;  Selecting and composing plans to solve the problem;  Implementing the plans;  Reflecting on the product and the process. But when teams work on design:  The teams create a shared mental model;  Team members, individually or in groups, run simulations of the shared model;  Teams evaluate the simulations and prepare the next version of the model.  Conflict is an inevitable component of team design: it must be managed, not avoided.  Communication is vital.  Issues may “fall through the cracks” because no one person takes responsibility for them.

Varieties of Architecture The AD is a “ground plan” of the implementation, showing the major modules and their interconnections. An arrow from module A to module B means “A needs B” or, more precisely, “a function of A calls one or more of the functions of B”. The AD diagram is sometimes called a “Structure Chart”. The AD is constructed in parallel with the MIS. A good approach is to draft an AD, work on the MIS, and then revise the AD to improve the interfaces and interconnections. Hierarchical Architecture The Structure Diagram is a tree.  Top-down design tends to produce hierarchical architectures.  Hierarchical architectures are easy to do.  May be suitable for simple applications.  Do not scale well to large applications.  Leaves of the tree tend to be over-specialized and not reusable.



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Layered Architecture The structure diagram has layers. A module may use only modules in its own layer and the layer immediately below (“closed” architecture) or its own layer and all lower layers (“open” architecture).  Layers introduced by THE system (Dijkstra 1968) and Multics (MIT, Bell Labs, General Electric) (Corbato et al. 1965). (UNIX was designed in opposition to Multics).  Programs with “utility functions” are (informal) layered systems.  Requires a combination of top-down and bottom-up design. Top-down ensures that overall goals are met. Bottom-up ensures that lower layers perform useful and general functions.  High layers perform high-level general tasks. Lower layers perform specialized (but not too specialized!) tasks.  Modules in lower layers should be reusable. General Architecture Arbitrary connections are allowed between modules.  Not recommended: “cf. spaghetti code”.  May be an indication of poor design.  Avoid cycles. Parnas: “nothing works until everything works”. Event-Driven Architecture In older systems, the program controlled the user by offering a limited choice of options at any time In a modern, event-driven system, the user controls the program. User actions are abstracted as events, where an event may be a keystroke, a mouse movement, or a mouse button change. The architecture consists of a module that responds to events and knows which application module to invoke for each event. For example, there may be modules related to different windows. This is sometimes called the Hollywood approach: “Don’t call us, we’ll call you”. Calling sequences are determined by external events rather than internal control flow. Modules in an event-driven system must be somewhat independent of one another, because the sequence of calls is unknown. The architecture may be almost inverted with respect to a hierarchical or layered architecture. Subsumption Architecture A subsumption architecture, sometimes used in robotics, is an extension of a layered architecture. The lower layers are autonomous, and can perform simple tasks by themselves. Higher layers provide more advanced functionality that “subsumes” the lower layers. Designing for Change What might change? 1. Users typically want more:  commands;  reports;  options;  fields in a record. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 394


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Solutions include: Abstraction Example: abstract all common features of commands so that it is easy to add a new command. The idea would be to add the name of a new command to a table somewhere and add a function to implement the command to the appropriate module. Constant Definitions There should be no “magic numbers” in the code. Parameterization If the programming language allows parameterization of modules, use it. C++ provides templates. Ada packages can be parameterized. 2. Unanticipated errors may occur and must be processed. Incorporate general-purpose error detection and reporting mechanisms. It may be a good idea to put all error message text in one place, because this makes it easier to change the language of the application. 3. Algorithm changes might be required. Usually a faster algorithm is needed. As far as possible, an algorithm should be confined to a single module, so that installing a better algorithm requires changes to only one module. 4. Data may change. Usually a faster or smaller representation is needed. It is easy to change data representations if they are “secrets” known to only a small number of modules. 5. Change of platform (processor, operating system, peripherals, etc) Keep system dependencies localized as much as possible. 6. Large systems exist in multiple versions:  different releases  different platforms  different devices Module Design Ideas about module design are important for both AD and MIS. Language Support The programming language has a strong influence on the way that modules can be designed.  Turbo-Pascal provides units which can be used as modules. A unit has an “interface part” and an “implementation part” that provide separation of concern.  C does not provide much for modularization. Conventional practice is to write a .h file for the interface of a module and a .c file for its implementation. Since these files are used by both programmers and compilers, the interfaces contain information about the implementation. For example, we can (and should) use typedefs to define types, but the typedef declarations must be visible to clients.  Ada provides packages that are specifically intended for writing modular programs. Unfortunately, package interfaces and package bodies are separate.  The modern approach is to have one physical file for the module and let the compiler extract the interfaces. This is the approach used in Eiffel and Dee. A Recipe for Module Design 1. Decide on a secret. 2. Review implementation strategies to ensure feasibility. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 395


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Design the interface. Review the interface. Is it too simple or too complex? Is it coherent? Plan the implementation. E.g. choose representations. Review the module.  Is it self-contained?  Does it use many other modules?  Can it accommodate likely changes?  Is it too large (consider splitting) or too small (consider merging)?

Design Notations A design can be described by diagrams, by text, or (preferably) by both. Diagrams are useful to provide an overview of the design, showing how the parts relate to one another. Text (“Textual Design Notation”, TDN) can be more precise, and as detailed as necessary, but it may not be easy to see the overall plan from a textual design. Text and graphics are complementary, with graphics working at a higher level of abstraction. Design Strategies We need a strategy, or plan, to develop the design. Strategies and architectures are related, in that a particular strategy will tend to lead to a particular architecture, but there is not a tight correspondence. For example, functional design tends to give a hierarchical architecture, but does not have to. Functional Design  Base the design on the functions of the system.  Similar to writing a program by considering the procedures needed.  A functional design is usually a hierarchy (perhaps with some layering) with “main” at the root.  Compatible with “top-down design and stepwise refinement”. Good feature: Functional design works well for small problems with clearly-defined functionality. Weaknesses:  Leads to over-specialized leaf modules.  Does not lead to reusable modules.  Emphasis on functionality leads to poor handling of data. For example, data structures may be accessible throughout the system.  Poor “information hiding”.  Control flow decisions are introduced early in design and are hard to change later. Structured Analysis/Structured Design (SA/SD) Structured Design/Structured Analysis (SA/SD) is a methodology for creating functional designs that is popular in the industry. 1. Formalize the design as a Data Flow Diagram (DFD). A DFD has terminators for input and output, data stores for local data, and transformers that operate on data. Usually, terminators are squares, data stores are parallel lines, and transformers are round. These components are linked by labelled arrows showing the data flows. 2. Transform the DFD into a Structure Chart (SC). The SC is a hierarchical diagram that shows the modular structure of the program. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 396


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Jackson Structured Design (JSD) Jackson Structured Design (JSD) has been pioneered by Michael Jackson2(1975, 1983) JSD is data-driven—the software design is based on relationships between data entities — but is not (as some believe) object oriented. Design by Data Abstraction Data abstraction (i.e. abstract data types) historically preceded object oriented design. 1. Choose data structures needed for the application. 2. For each data structure, design a module that hides the representation and provides appropriate functions. 3. Build layers using these modules. Note that this is at least partly a bottom-up approach. Strengths:  Data representations are hidden inside modules.  Control flow decisions are deferred until late in design.  Data is less likely to change than functionality.  Code is likely to be reusable. Weaknesses:  Must have a clear vision of the finished product, otherwise unnecessary or inappropriate data structures may be introduced.  A module can either implement one instance of an ADT (restricted) or export a type (leads to awkward notation). Object Oriented Design Object Oriented Programming The object oriented approach extends ADTs. A module in an OOL is called a class. A class declaration contains declarations of instance variables, so-called because each instance of the class gets a copy of them. An instance of a class is called an object. An object has:  state (i.e. data);  identity (e.g. address);  methods(procedures and functions). Methods are usually characterized as:  Constructors create new objects.  Inspectors returns values.  Mutatorschange objects. A stack object might have:  Constructor: Create.  Inspectors: Empty, Full ,Top.  Mutators: Push, Pop. Classes in object oriented programming play the role of modules in procedural programming. In fact, a class is somewhat more restricted than a module: it is essentially a module that exports one type and some functions and procedures.



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A class:  is a collection of objects that satisfy the same protocol (or provide the same services);  may have many instances (which are the objects). All of this can be done with conventional techniques. OOP adds some new features. Inheritance Suppose we have a class Window and we need a class ScrollingWindow. We could:  rewrite the class Window from scratch;  copy some code from Window and reuse it;  inheritWindow and implement only the new functionality. Class Scrolling Window Inherits Window  new code to handle scroll bars etc.  redefine methods of Window that no longer work Inheritance is important because it is an abstraction mechanism that enables us to develop a specialized class from a general class. Inheritance introduces a two new relationships between classes.  The first relationship is called is-a. For example: “a scrolling window is a window.” If X is-a Y it should be possible to replace Y by X in any sentence without losing the meaning of the sentence. Similarly, in a program, we should be able to replace an instance of Y by an instance of X, or perform an assignment Y :=X. For example, a dog is an animal. We can replace “animal” by “dog” in any sentence, but not the other way round.5 In a program, we could write A :=D where A:Animal and D: Dog.  The second relationship is called inherits-from. This simply means that we borrow some code from a class without specializing it. For example, Stack inherits from Array: it is not the case that a stack is an array, but it is possible to use array operations to implement stacks. Organization of Object Oriented Programs Consider a compiler. The main data structure in a compiler is the abstract syntax tree; it contains all of the relevant information about the source program in a DAG. The DAG has various kinds of node and, for each kind of node, there are several operations to be performed. Using object oriented techniques, we can reverse the relationship between the library and the main program. Instead of the application calling the library, the library calls the application. A library of this kind is referred to as a framework. A framework is a coherent collection of classes that contain deferred methods. The deferred methods are “holes” that we fill in with methods that are specific to the application. The organization of a program constructed from a framework is shown in Fig. 11.2.1. The structure of the system is determined by the framework; the application is a collection of individual components. The best-known framework is the MVC triad of Small talk . It is useful for simulations and other applications. The components of the triad are a model, a view and a controller — see Fig. 11.2.2.



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Library

Applicatio n Figure 11.2.1: A Program Using a Framework from a Library

Controller

Model

View Figure 11.2.2: The Model-View-Controller Framework

The Model is the entity being simulated. It must respond to messages such as step (perform one step of the simulation) and show (reveal various aspects of its internaldata). The View provides one or more ways of presenting the data to the user. View classes might include DigitalView,GraphicalView, BarChartView, etc. The Controller coordinates the actions of the model and the view by sending appropriate messages to them. Roughly, it will update the model and then display the new view. Smalltalk provides many classes in each category and gives default implementations of their methods. To write a simulation, all you have to do is fill in the gaps. A framework is a kind of upside-down library. The framework calls the application classes, rather than the application calling library functions. Frameworks are important because they provide a mechanism for design re-use. Functional or Object Oriented? Functional Design (FD)  FD is essentially top-down. It emphasizes control flow and tends to neglect data.  Information hiding is essentially bottom-up. It emphasizes encapsulation (“secrets”) in low-level modules.  In practice, FD must use both top-down and bottom-up techniques. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 399


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FD uses function libraries. FD is suitable for small and one-off projects.

Object Oriented Design (OOD)  OOD is also top-down and bottom-up, but puts greater emphasis on the bottom-up direction, because we must define useful, self-contained classes.  OOD emphasizes data and says little about control flow. In fact, the control flow emerges implicitly at quite a late stage in the design.  Inheritance, used carefully, provides separation of concern, modularity, abstraction, anticipation of change, generality, and incrementality.  OOD is suitable for large projects and multi-version programs.

Risk A risk is the possibility that an undesirable event could happen. Risk Estimation: Risk estimation invokes two tasks in rating a risk. The first task is estimating the probability of the occurrence of a risk called risk probability and risk impact, cost of risk event happening. Risk Exposure: Risk exposure is the expected value of risk event. Risk exposure = Risk probability Risk impact Risk Decision Tree: A decision tree gives a graphic view of the processing logic involved in decision making and the corresponding actions taken. A technique that can be used to visualize the risks of alternatives is to build a risk decision tree. The top –level branch splits based on alternatives available. Reliability Metrics: 1. Mean Time To Failure(MTTF): MTTF is the average time between two successive failures, observed over a large number of failures. To measure MTTF we can record the failure data for n failures. Let the failure occur at the time instants t , t , t . . . . . t . Then MTTF = ∑ (

)

2. Mean Time To Repair(MTTR): MTTR measures the average time it takes to track the errors causing the failure and then to fix them. 3. Mean Time Between Failure(MTBF): MTBF = MTTF + MTTR MTBF of t hours indicates that once a failure occurs, the next failure is expected to occur only after t hours. 4. Availability: Availability of a system is a measure of how likely will the system be available for use over a given period of time. Availability = 100% = 100%



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LOC – BASED COST ESTIMATION The basic formula has three parameters cost = ∝ * (KLOC) Alpha ∝, is the marginal cost per KLOC (thousand line of code). This is the added cost for an additional thousand lines of code. β - is an exponent that reflects the non-linear of the relationship. Y - reflects fixed cost of doing any project. Function Point Metric The conceptual idea of underlying the function point metric is that the size of a software product is directly dependent on the number of different functions or features it supports. Function point is computed in two steps: The first step is to compute the unadjusted function point (UFP) after that technical complexity factor (TCF) is computed UFP = 4* (Number of inputs) +5 * (Number of outputs) + 4 * (Number of inquiries) + 10* (Number of files) + 10 * (Number of interfaces) Number of inputs – Each data item input by the user is counted. Number of output – The outputs considered refer to reports printed, screen outputs, error messages produced etc. Number of inquiries – Number of inquiries is the number of distinct interactive queries which can be made by users. Number of files – Each logical file is counted. Number of interfaces- Here the interfaces used to exchange information with other systems. To compute the function point the following relationship is used. FP = UFP * TCP = UFP * [ 0.65 + 0.01*∑(Fi) COCOMO Model: In 1981 Barry Boehm introduced a hierarchy of software estimation models bearing the name COCOMO for Constructive Cost Model. The latest version of COCOMO model is COCOMO II. It address the following areas. Application composition model Early design stage model Post – architecture – stage model.



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11.3 Validation and Verification 

Validation = have we built the right product? = check that product matches SRD

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Verification = have we built the product right? = are all internal properties satisfied?

We can also say, approximately:  validation ∼ black box ∼ semantics  verification ∼ clear box ∼ syntax Ideally, all properties of the program should be validated: correctness, performance, reliability, robustness, portability, maintainability, user friendliness. In practice, it is usually not feasible to validate everything. Some results may be precise: tests passed or failed, or a percentage of tests passed. Other results may be subjective: user friendliness.

Varieties of Testing We can perform verification and validation by testing. A single test can provide both validation and verification. A failure (e.g. system crashes) reveals an internal fault in the system (verification). An incorrect result indicates that the software does not meet requirements (validation). The main problem with testing is that testing can never be complete. 1. Goal-driven testing:  Requirements-driven testing: Develop a test-case matrix (requirements vstests) to insure that each requirement undergoes at least one test. Tools areavailable to help build the matrix.  Structure-driven testing: Construct tests to cover as much of the logical structure of the program as possible. A test coverage analyzer is a tool that helps to ensure full coverage.  Statistics-driven testing: These tests are run to convince the client that the software is working by running typical applications. Results are often statistical.  Risk-driven testing: These tests check “worst case” scenarios and boundary conditions. They ensure robustness. 2. 1. 2. 3.

Phase-driven testing: Unit testing. Test individual components before integration. Integration testing. Assemble the units and ensure that they work together. System testing. Test the entire product in a realistic environment.

Designing Tests A test has two parts:  A procedure for executing the test. This may include instructions for getting the system into a particular state, input data, etc.  An expected result, or permitted range of results. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 402


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There are two possibilities Black box testing: We choose tests without knowledge of how the program works, i.e. based on requirements only. It is also called behavioral testing focuses on functional requirement of the software. White box testing: We choose test based on our knowledge of how the program works, it is also called glass box testing. Equivalence portioning Equivalence portioning is a black box testing method that divides the input domain of a program into classes of data from which test cases can be derived. Equivalence portioning strives to define a test cases that uncovers classes of errors, thereby reducing the total number of test must be developed. Guidelines for Black Box Testing  Test for success and failure. Requirement: “The maximum length of a line is 255 characters.” Tests: lines with length l such that l ≤ 255 and l > 255.  Test boundary conditions. Test: l = 255.  Test as many combinations as feasible. Requirement: An editor requires special treatment of tab characters and has special actions at the right side of the window. Test: tabs, right side of window, and tab character at extreme right. Guidelines for White Box Testing The general idea is to ensure that every component of the program is exercised. Possible criteria include: All statements Every statement in the program must be exercised during testing. All Edges All edges of the control graph must be exercised. (This is very similar to “all statements”.) All Branches Each possibility at a branch point (if or case statement) should be exercised. All Paths Exercise all paths: usually intractable.

Stages of Testing Unit Testing: Test an individual unit or basic component of the system. Example: test a function such as sqrt. Module Testing: Test a module that consists of several units, to validate the interaction of the units and the module interface. Subsystem Testing: Test a subsystem that consists of several modules, to validate module interaction and module interfaces. Integration Testing: Test the entire system.



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Acceptance Testing: Test the entire system thoroughly with real data to satisfy the customer that the system meets the requirements.

Testing Strategies The strategies described here apply to development testing, not acceptance testing. Top-down Testing  Goes from system to subsystem to module to unit;  Requires that we write stubs for parts of the system that are not yet completed. A stub:  Replaces a module or unit for the purposes of testing;  Must return with a valid response;  May do nothing useful;  May always do the same thing;  May return random values;  May handle specific cases. Advantages (+) and disadvantages (−) of top-down testing:  Catches design errors (but these should have been caught in design reviews).  Enables testing to start early in the implementation phase. − It is hard to write effective stubs. − Stubs take time to design and write. Bottom-up Testing Test units, then modules, then subsystems, then system. We require drivers to exercise each unit or module because its environment does not exist yet. Drivers must  provide environment and simulated input  check outputs Advantages ( ) and disadvantages (−) of bottom-up testing:  Each component is tested before it is integrated into a larger component.  Debugging is simplified because we are always working with reliable components. − Drivers must be written; drivers are usually more complex than stubs and they may contain errors of their won. − Important errors (e.g. design errors) may be caught late in testing — perhaps not until integration. Mixed strategies are also possible. We can aim at gradual refinement of the entire system, doing mostly top-down testing, but with some bottom-up testing. The order of testing and the order of implementation must be chosen together. Clearly, topdown testing requires top-down coding. Alpha test: It conducted at the developer’s site by a customer in a controlled environment Beta test: It is a line application of the software in an environment that can’t be controlled by developer, and unlike alpha testing developer is generally not present customer records all the problem encounter during beta testing THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 404


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When do we stop? Ideal: Stop when all tests succeed. Practice: stop when the cost of testing exceeds the cost of shipping with errors. Preparing Test Cases A Test Set consists of a list of tests. Each test should include the following three components: Purpose: For an acceptance test, the purpose is an SRD item. For a unit or subsystem test, the purpose is an internal requirement based on the design. Data: The environment (i.e. state of the system when the test is conducted), inputs to functions, etc. Expected Result: The effect of conducting the test, the value returned by a function, the effect of a procedure, etc. A Test Plan has the following components:  A description of the phases of testing. For example: unit, system, module.  The objectives of the testing phase (verify module, validate subsytem, etc).  A schedule that specifies who does what to which and when.  The relationship between implementation and testing schedules (don’t schedule a test before the component is written).  Tracing from tests to requirements.  How test data are generated.  How test results are recorded. Reviews, Walkthroughs, and Inspections The basic idea of reviews, walkthroughs, and inspections is the same: A team examines a software document during a meeting. Studies have shown that errors are found more effectively when a group of people work together than when people work individually. Common features include:  A small group of people;  The person responsible for the document (analyst, designer, programmer, etc) should attend;  One person is responsible for recording the discussion;  Managers must not be present, because they inhibit discussion;  Errors are recorded, but are not corrected. During a review:  The author of the document presents the main themes;  Others criticize, discuss, look for omissions, inconsistencies, redundancies, etc.  Faults and potential faults are recorded. During a walkthrough:  Each statement or sentence is read by the author;  Others ask for explanation or justification if necessary. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 405


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Example: “Since n > 0, we can divide . . . .” “How do you know that n > 0?” During an inspection:  Code is carefully examined, with everyone looking for common errors. As usual, there is variation in usage: an IBM “inspection” is close to what we have called a “walkthrough”.  Some general rules:  Teams prepare in advance, e.g. by reading the documentation.  Meetings are not long — at most 3 hours — so concentration can be maintained.  A moderator is advisable to prevent discussions from rambling.  The author may be required to keep silent except to respond to questions. If the author explains what she/he thought she/he was doing, others may be distracted from what is actually written.  All members must avoid possessiveness and egotism, cooperating on finding errors, not defending their own contributions. McCabe’s CYCLOMATIC NUMBER (Product metric) McCabe’s cyclomatic complexity is based on fact that complexity is related to the control flow of the program. It defines an upper band on the number of independent paths in a program. Method I Given a control flow graph G of a program, the cyclomatic complexity can be computed as = − 2 Where E is the number of edges and N is the number of nodes in the control flow graph. Method II Cyclomatic complexity = Total No. of bounded areas +1 (for planar graphs) Method III If N is the number of decision statements of a program then the McCabe’s metric is equal to N 1 Software Maturity Index Software maturity index (SMI) provides an indication of the stability of a software product based on changes that occur for each release of the product. M = Number of modules in the current release. F = Number of modules in the current release that have been changed. F = Number of modules in the current release that have been added. F = Number of modules from the preceding release that were deleted in the current release. Software Maturity Index is computed in the following manner: ( ) SMI = Coupling Metric: Module coupling provides an indication of the “connectedness” of a module to other module, global data, and outside environment. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 406


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For data and control flow coupling d = number of input date parameters. c = number of input control parameters. d = number of output date parameters. c = number of output control parameters. Reliability: For global coupling g = number of global variables used as date. g = number of global variables used as contrl. For environmental coupling w= no. of modules called (fan-out) r = no. of modules calling under consideration (fan-in) module coupling indicator m m = Where k = 1, a proportionality constant M = d + (a c ) + d + ( b c ) + g +(c g ) + w +r Where a = b = c = 2 Higher the value of m lower is the overall module coupling



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11.4 HTML HTML Structure HTML uses tags that are encased in brackets like the following: <> HTML documents consist of elements which are constructed with tags. For instance, a paragraph is considered to be an html element constructed with the tags
and
. The
tag begins the paragraph element and the
tag ends the element. Not all tags have a tag for ending the element such as the line break,
tag. The HTML document is begun with the tag and ended with the tag. Elements of an HTML document include the HEAD, BODY, paragraphs, lists, tables, and more. Some elements have attributes embedded in the tag that define characteristics of the element such as the placing of text, size of text, source of an image, and other characteristics depending on the element. An HTML document is structured with two main elements: 1. HEAD 2. BODY An Example HTML File <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="GENERATOR" content="Arachnophilia 3.9"> <meta name="description" content="Comprehensive Documentation and information about HTML."> <meta name="keywords" content="HTML, tags, commands"> The CTDP HTML Guide
HTML Document Structure

This is a sample HTML file.
Comments begin with bracket. The tags "HTML", "BODY", and all others may be in capital or small letters, however the new XHTML standard requires small letters so small letters are recommended. In the above file, there is a header and a body. Normally you can copy this file and use it as a template to build your new file while being sure to make some modifications. You can edit HTML using a standard editor, but it is easier to use an HTML editor like Arachnophilia since it displays the tags with different colors than the text is displayed in. Also note the LINK element above which specifies a style sheet to be used. This is a file with a name "style.css". This is a text file that may be created with a text editor but must be saved in plain text format.



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HTML Header The HTML header contains several notable items which include: 1. doctype - This gives a description of the type of HTML document this is. 2. meta name="description" - This gives a description of the page for search engines. 3. meta name="keywords" - This line sets keywords which search engines may use to find your page. 4. title - Defines the name of your document for your browser. The Line The line is used to define the type of HTML document or page. It has no ending tag. The three document types that are recommended by the World Wide Web Consortium (W3C) are: 1. . This implies strict adherence with HTML 4 standards. 2. . This supports frameset tags. 3. . This is used to support depreciated HTML 3.2 features. It does not support frames. Elements in the Header Elements allowed in the HTML 4.0 strict HEAD element are:  BASE - Defines the base location for resources in the current HTML document. Supports the TARGET attribute in frame and transitional document type definitions.  LINK - Used to set relationships of other documents with this document.  META - Used to set specific characteristics of the web page and provide information to readers and search engines.  SCRIPT - Used to embed script in the header of an HTML document.  STYLE - Used to embed a style sheet in the HTML document.  TITLE - Sets the document title.  The additional element allowed by the HTML 4.0 transitional standard is:  ISINDEX (Depreciated) - Allows a single line of text input. Use the INPUT element rather than ISINDEX. The <META> Element The <META> element is used to set specific characteristics of the web page and provide information to readers and search engines. It has no ending tag. Attributes  http-equiv - Possible values are: o refresh - The browser will reload the document after the specified seconds that is specified with the CONTENT value have elapsed. Ex: <META HTTP-EQUIV=refresh CONTENT=45> o expires - Gives the date that content in the document is considered unreliable. o reply-to - A an email address of the responsible party for the web page. This attribute is not commonly used. Ex: <META HTTP-EQUIV=reply-to CONTENT="[email protected]">  Name - Provides non-critical information about the document possibly useful to someone looking at it. Possible values are: o Author - The person who made the page or the HTML editor name . Ex: <META NAME=author CONTENT="Mark Allen"> THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 409


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description - An explanation of the page or its use, used by search engines at times to provide a page summary. Ex: <META NAME=description CONTENT="The CTDP Home Page"> o copyright - A copyright notice for the page. Ex: <META NAME=copyright CONTENT="Copyright 2000, Mark Allen"> o keywords - A list of keywords which are separated by commas. These keywords are used by search engines. EX: <META name="keywords" CONTENT="computer documentation, computers, documentation, computer help"> This section is very important if you want your web page to be found by search engines. Please note that keywords are separated by commas, not spaces and that the words "computer documentation" are treated by search engines as one word. If someone enters the phrase "computer documentation" when doing a search, it gives the web page a much greater chance of being found than just having the separate keywords "computer" and "documentation". o date - <META name="date" CONTENT="2000-05-07T09:10:56+00:00"> CONTENT - Specifies a property's value such as the content of this document is text/HTML. scheme - Names a scheme to be used to interpret the property's value. o

 

The Element This element defines the way relative hyper links are handled in the body of the document. It has no ending tag. It supports the TARGET attribute in frame and transitional document type definitions. Only one BASE element may exist in the header. If you copy the websites page from the CTDP website at http://ctdp.tripod.com/websites.html to your website at http://yoursite.yourwebserver.com/websites.html, then any relative links on the copied page will try to link to the http://ctdp.tripod.com page rather than the http://yoursite.yourwebserver.com page and you may get the "file not found" error from your browser. Inserting the tag into the copied page will fix the problem. Without this tag, if there were a relative link to "linux/index.html" on this page, the user's web browser would look for the file in "http://yoursite.yourwebserver.com/linux/index.html" which would not exist unless it were also copied over and placed at the same relative location on your website. The Element Used to set relationships of other documents with this document. For example a style sheet that is used to control element style (as in the above example) may be defined with this element. Attributes:  CHARSET - The character encoding of the link. An example is "ISO-8859-1".  HREF - The location and name of the resource.  HREFLANG - Describes the expected language. An example value is "en" for English.  MEDIA - Describes the type of display the content is designed for. Possible values include all, aural, braille, handheld, projection, print, screen, tty, and tv.  REL - Relationship to the linked resource.  REV - Relationship from the linked resource to the document. Possible values of REL and REV include alternate, appendix, stylesheet, bookmark, chapter, contents, copyright, glossary, help, index, next, prev, start, section, and subsection. The value "made" supported by HTML 3.2 is not supported by HTML 4.0.  TYPE - Describes the expected content of the resource the link points to. Typical values include application/java, text/html, text/css, image/jpeg, and text/javascript. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 410


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An additional attribute allowed by the HTML 4.0 transitional standard is: TARGET - Specifies the referenced page will be put into. The target may be a named window or one of the special reserved names below. If the target is a named window that exists the web page the link points to will be loaded into that window. If a window by that name does not exist, a new window with that name will be created. o blank - The web linked page loads in a new window and does not name the window. o parent - The new page will load in the parent frame or window. o self - The new page will load in the same window that the link is in. If the link is in a frame, it will load inside the frame. This is the default target. o top - The new page will load into the top window reguardless of how many framesets deep the link is embedded. An example:

The Element The text between the <title> and tags is the title of the document. An example: The CTDP HTML Guide The <SCRIPT> Element Used to embed script in the header of an HTML document. Attributes:     

CHARSET - The character encoding of the script such as "ISO-8859-1". DEFER - The script will not be parsed until the document is loaded. LANGUAGE - Describes the name of the script language. SRC - The external location where the script may be. In this case the script in not includes in between the SCRIPT tags, but an external file is loaded. TYPE - Describes the content type of the script language.

The tags. HTML Body The HTML body element will define the rest of the HTML page which is the bulk of your document. It will include headers, paragraphs, lists, tables, and more. The BODY Element Tags and Attributes The tag is used to start the BODY element and the tag ends it. It is used to divide a web page within one or more sections. Its tags and attributes are:


Quick Refresher Guide 



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- Designates the start of the body. o ONLOAD - Used to specify the name of a script to run when the document is loaded. o ONUNLOAD - Used to specify the name of a script to run when the document exits. o BACKGROUND="clouds.gif" - (Depreciated) Defines the name of a file to use for the background for the page. The background can be specified as in the following line. o BGCOLOR="white" - (Depreciated) Designates the page background color. o TEXT="black" - (Depreciated) Designates the color of the page's text. o LINK="blue" - (Depreciated) Designates the color of links that have not been visited. o ALINK="red" - (Depreciated) Designates the color of the link currently being visited. o VliNK="green" - (Depreciated) Designates the color of visited links. - Designates the end of the body.

HTML Element Categories It is important to be aware, when using HTML and style sheets that HTML elements are categorized into several categories. Some style properties apply to some categories of elements and not to others. The following types of elements exist:  Block - Include a line break before and after the element.  Inline - Included with the text as part of a line.  List Item - Elements that support providing a list of items. List item elements are block level elements. Block HTML Elements A block with centered contents are defined as: Name ADDRESS BLOCKQUOTE CENTER DIV

Description Supplies contact information for the document Used to quote in block form Depreciated A container allowing specific style to be added to a block of text. A container allowing specific style to be added to a block of text. A container allowing multiple frames (HTML documents) to be placed on a web browser.

Comment -

H1, H2, H3, H4, H5, H6

Headings

-

HR

Horizontal rule

-

ISINDEX

An input prompt for a single line of text

Depreciated

Alternate content for browsers that do not support frames.

-

Alternate content for browsers that cannot run script programs

-

DIR FRAMESET

NOFRAMES NOSCRIPT

-



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Name P

Description

Comment

Paragraph - Should not contain other block elements including tables, but may contain list elements

-

PRE

Preformatted text is rendered with spaces and carriage returns as typed. Used to present an ordered set of data. Table subelements work as block elements. Used to present a form to the client. Form subelements work as block elements.

TABLE FORM

Includes table sub elements -

List item elements are also considered to be block elements. List-item Elements Name Description Comment DIR Directory List Depreciated DL Definition List LI List Item OL Ordered (numbered) List UL Unordered List -



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11.5 XML and DTDs Definitions: When talking about XML, here are some terms that would be helpful:  XML: Extensible MarkupLanguage, a standard created by the W3C Group for marking up data.  DTD: Document Type Definition, a set of rules defining relationships within a document. DTDs can be “internal” (within a document) or “external” (links to another document).  XML Parser: Software that reads XML documents and interprets or “parse” the code according to the XML standard. A parser is needed to perform actions on XML, such as comparing an XML document to a DTD. XML Anatomy Like HTML, XML is based on SGML, Standard Generalized Markup Language, and designed for use with the Web. XML documents, at a minimum are made of two parts: the prolog and the content. The prolog or head of the document usually contains the administrative metadata about the rest of document. It will have information such as what version of XML is used, the character set standard used, and the DTD, either through a link to an external file or internally. Content is usually divided into two parts, that of the structural markup and content contained in the markup, which is usually plain text. Let’s take a look at a simple prologue for an XML document: is different from , which is different from . Note: The characters that are excluded from element names in XML are &, <, “, and >, which are used by XML to indicate markup. The character :should be avoided as it has been used for special extensions in XML. If you want to use these restricted characters as part of the content within elements but do not want to create new elements then you would need to use the following entities to have them displayed in XML: 2. Elements containing content must have closing and opening tags. (opening) (closing) Note that the closing tag is the exact same as the opening tag, but with a backslash in front of it. The content within elements can be either elements or character data. If an element has additional elements within it, then it is considered as a parent element; those contained within it are called child elements. For example, This is a sample of simple XMLcoding. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 414


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So in this example, is the parent element. is the child of elementName, because it is nested within elementName. Elements can have attributes attached to them in the following format: When using attributes in XML, the value of the attributes must always be contained in quotes. The quotes can be either single or double quotes. For example, the attribute version= “1.0” in the opening XML declaration could be written version= ‘1.0’ and would be interpreted the same way by the XML parser. However, if the attribute value contains quotes, it is necessary to use the other style of quotation marks to indicate the value. For example, if there was an attribute name with a value of John “Q.” Public then it would need to be marked up in XML as name=‘John “Q” Public’, using the symbols for quotes to enclose the attribute value that is not being used in the value itself. Creating a Simple XML Document Like most, if not all, standards developed by the W3Group, you can create XML documents using a plain text editor like Notepad (PC), TextEdit (Mac), or pico (Unix). You can also use programs like Dreamweaver and Cooktop, but all that is necessary to create the document is a text editor. Let’s say we have two types of documents we would like to wrap in XML: emails and letters. We want to encode the emails and letters because we are creating an online repository of archival messages within an organization or by an individual. By encoding them in XML, we hope to encode their content once and be able to translate it to a variety of outputs, like HTML, PDFs, or types not yet created. To begin, we need to declare an XML version: Now, after declaring the XML version, we need to determine the root element for the documents. Let’s use message as the root element, since both email and letters can be classified as messages. Parent and child relationships A way of describing relationships in XML is the terminology of parent and child. In our examples, the parent or “root” element is , which then has two child elements, , and . An easy way of showing how elements are related in XML is to indent the code to show that an element is a child of another. For example, Now that we have the XML declaration, the root element, and the child element (email). Let’s determine the information we want to break out in an email. Say we want to keep information about the sender, recipients, subject, and the body of the text. Since the information about the sender and recipients are generally in the head of the document, let’s consider them children elements of a parent element that we will call
. In addition to
, the other child elements of will be and . When creating XML documents, it’s always useful to spend a little time thinking about what information you want to store, as well as what relationships the elements will have. Now that we’ve made some XML documents, let’s talk about “well formed” XML and valid XML. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 415


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“Well Formed” vs. Valid When talking about XML documents, two commonly-used terms are “well formed” and “valid.” As in “Is your document marked up in valid and well formed XML?” Well formed in relation to XML means that it has no syntax, spelling, punctuation, grammar errors, etc. in its markup. These kinds of errors can cause your XML document to not parse. Note: An XML Parser is a software that reads XML documents and interprets or “parses” the code according to the XML standard. A parser is needed to perform actions on XML. For example, a parser would be needed to compare an XML document to a DTD. In the next section, we will talk about some common errors that prevent an XML document from being well formed. When you say an XML document is valid, you’re saying that the element structure and markup of the XML document matches a defined standard of relationships, in addition to having well formed markup. In other words, is this XML document a quality document? One standard used to validate XML is a DTD, or Document Type Declaration, although XML Schemas are also used. These standards are useful when dealing with the creation of a number of XML documents for they provide a quality control measure to ensure that all the documents meet a minimum standard. Another benefit is that it allows for errors to be detected in the process of creating the XML document, rather than at the end. Note: An important thing to remember is that when a document is valid it is also “well formed,” but a “well formed” document is not necessarily valid. Additionally, you can create XML documents without a DTD, but the XML document can’t be considered valid without a document type. Creating a DTD (Document Type Definition) Why would you want to create a DTD? The benefits of DTDs are that it allows you to create numerous documents and make sure that the information contained in them will be comparable. For example, all the information about dates are in tags called rather than , , or . By creating XML documents that meet a DTD’s requirements, you can also share information between institutions. Rules for Creating DTDs When creating a DTD, you need to define all the elements and attributes you’ll have in the XML documents. So let’s create a DTD for our message XML documents. Some syntax to remember when creating DTDs are the following: Symbol Meaning Example header (sender, AND , recipient*, date) message (email | OR | letter) () + ? *

Occurs only Once must occur at least once occurs either once or not at all can occur zero or more times

(email | letter) (header, subject?, text+) (header, recipient* , date?) (sender, recipient*, date)



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Elements are declared in the following manner: Attributes are declared like this: An important thing to remember when making DTDs is that unless you use | when defining the element parts, the order of the elements you have within that section is required in your XML. So in a letter, the element must occur before the element . There are some tools that will automatically generate DTDs from XML. HitSoftware, a W3Group member, has created this XML to DTD tool. Linking your XML document to a DTD Now that you’ve created a DTD for an XML document, you need to link the XML to the DTD. This takes place in the prolog of the XML document. Remember that the prolog starts off with the XML declaration: Immediately following the XML declaration, you would then either link to a DTD or write an internal DTD. While DTDs can be both internal or external, if you are using a DTD for multiple documents, it makes more sense to have the DTD in a separate “external” file. Otherwise, you will have to put the full DTD in the prolog of every XML document, rather than just one line of code. To link to an external DTD, the declaration goes like this: Note: SYSTEM in the DTD declaration can be replaced by PUBLIC if the DTD is available via the Internet. You would then need to have a public name for the DTD in the file. Assuming that the message.dtd file we created was in the same folder as our XML files for our email and letters, we would add the following line to the XML code: Validating with a DTD Now that you have the XML linked, you’ll need a full parser to validate the XML files. While most browsers can check for well formed XML, only Internet Explorer 5.0 and higher has a complete XML parser built in to the program that checks against DTDs. You can also use programs such as Dreamweaver, Cooktop, and a variety of other XML authoring software. There are also online resources, such as this online XML Validator, but you need an internal DTD or a DTD available on the Internet in order to compare it against. An internal DTD is located in the same place as a link to an external DTD but follows the following structure. To link to a DTD on the Internet, you first need to have an account that can serve items to the web, such as an iSchool account. Then you would need to link to the external DTD with the full URL address. So if we were hosting message.dtd in your personal iSchool account, its link would appear as the following: .



Reference Book

Reference Books Mathematics:

  

Higher Engineering Mathematics – Dr. BS Grewal Advance Engineering Mathematics – Erwin Kreyszig Advance Engineering Mathematics – Dr. HK Dass

Programming and Algorithms:

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Data Structure & Algorithm in C - Mark Allen Weiss Data Structures Using C and C++ - YedidyahLangsam, Moshe J. Augenstein, and Aaron M. Tenenbaum Programming Languages - Ravi Sethi Introduction to Algorithms - Corman, Rivest Fundamentals of computer Algorithms - Ellis Horowitz and SartajSahni C programming - Kernighan and Ritchie

Operating Systems:

   

Operating System Concept – Silbersehatz Galvin and Gagne. Operating Systems – Stallings. Operating Systems – Gary Nutt. OPERATING SYSTEMS:A Concept-Based Approach - D M Dhamdhere

Discrete Math and Graph Theory:

    

Discrete and Combinatorial Mathematics – Ralph Grimaldi Discrete Mathematical Structures with applications to Computer Science - Tremblay and Manohar. Discrete Mathematics - Rosen Discrete Mathematics for Computer Scientists – Mott Elements of Discrete Mathematics : C. L. Liu

Computer Organization:

     

Digital Electronics – Morris Mano Digital circuits – Herbert Taub& Donald L Schilling Microprocessor Architecture, Programming and Applications with the 8085 – Ramesh Gaonker Computer Organisation – Hamacher Computer Architecture A quantitative approach - PATTERSON and HENNESSY Computer Organization and Architecture : William Stallings



Reference Book

Database:

  

Database Systems - Korth DBMS - Ramakrishnan and Gerke Fundamentals of Database Systems : Shamkant B. NavatheBipin, Desai

Computer Networks:

 

Computer Networks – Tanenbaum TCP/IP Illustrated (volume 1) : W. Richard Stevens

Theory of Computations:

  

Introduction to automata theory, languages and computation – Hopcroft, Motwani and Ullman Theory of Computation – Dexter. C. Kozen. The Theory of Computation – Bernard M Moret

Compiler Design:



Principles of Compiler Design - Aho and Ullman and ravisethi

Software engineering and Web Technology:

   

A Practitioner's Approach - Roger Pressman Web Engineering - Roger Pressman and David Lowe Fundamentals of Software Engineering - Rajib Mall An Integrated Approach to Software Engineering - PankajJalote


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