Important Problem Types

Important Problem Types Sorting Searching String processing Graph problems Combinatorial problems Geometric problems Numerical problems

Sorting  The sorting problem is to rearrange the items of a given list in non decreasing order. Advantages of sorting Sorting makes many questions about the list easier to answer. The most important of them is searching eg dictionaries! telephone books!  The sorting algorithms algorithms are are indeed better better than others" others" there there is no algorithm algorithm that would be the best solution in all situations. Some of the algorithms are simple but relatively slow! while others are faster but more comple#" some work better on randomly ordered inputs class lists! and so on are sorted etc. Two properties of sorting algorithms deserve special mention Stable$A Stable$A sorting algorithm is called stable if it preserves the relative order of any two equal elements in %ts input. &#ample we have a list of students sorted alphabetically and we want to sort it according according to student G'A a stable algorithm will yield a list in which students with the same G'A G'A will still be sorted alphabetically. in-place : The second notable feature of a sorting algorithm is the amount of e#tra memory the algorit algorithm hm requi require res. s. An algor algorith ithm m is said said to be in-pla in-place ce if it does not require e#tra memory.

Searching  The searching problem deals with (nding a given value! called a search key ! in a given set )or searching! too! there is no single algorithm that (ts all situations best. Some algorithms work faster than others but require require more memory" some are very fast but applicable only to sorted arrays" and so on.

String Processing A string is a sequence of characters from an alphabet. Strings of particular interest are te#t strings! which comprise letters! numbers! and special characters" bit strings! which comprise *eros and ones" and gene sequences! which can be modeled by strings of characters from the four$character alphabet + A! C! G!  T,. &gsearching for a given word in a te#t-.S tring matching. matching .

Graph Problems grap graph h can be though thoughtt of as a collec collectio tion n of points points called called vertic vertices! es! some of which which are are connected by line segments called edges. Graphs can be used for modeling a wide variety of  applicat applications ions!! includin including g transpor transportatio tation! n! communica communication! tion! social social and economic economic networks networks!! proect scheduling! and games. /asic /asic graph graph algori algorithm thms s includ include e graph graph$tr $trav avers ersal al algor algorith ithms! ms! shorte shortest$ st$pa path th algori algorithm thms! s! topological sorting for graphs with directed edges etc.

Some graph problems are computationally very hard" the most well$known e#amples are the traveling salesman problem and the graph$coloring problem. The traveling salesman  problem (TSP) is the problem of (nding the shortest tour through n cities that visits every city e#actly once.  The graph-coloring problem seeks to assign the smallest number of colors to the vertices of a graph so that no two adacent vertices are the same color.

Geometric Problems Geometric algorithms deal with geometric obects such as points! lines! and polygons.  The main algorithms for classic problems of computational geometry the closest$pair problem and the conve#$hull problem. The closest-pair  problem is self$e#planatory given n points in the plane! (nd the closest pair among them. The convex-hll problem asks to (nd the smallest conve# polygon that would include all the points of a given set.

Numerical Problems !merical problems! another large special area of applications! are problems that involve mathematical obects of continuous nature solving equations and systems of equations! computing de(nite integrals! evaluating functions! and so on.