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Conditional Asymptotic Notations V. Balasubramanian
Asymptotic Notations • Asym Asympt ptot otic ic nota notati tion on deal deals s wit with h the the behavi behaviour our of a funct function ion in in the limi limit, t, that that is, for sufficiently large values of its parame parameter ter.. Often, Often, when when analy analysin sing g the run run time of an algorithm, it is easier to obtain an approximate formula for the run-time which gives a good indication of the algorithm performance for large problem instances.
Asymptotic Notations • Asym Asympt ptot otic ic nota notati tion on deal deals s wit with h the the behavi behaviour our of a funct function ion in in the limi limit, t, that that is, for sufficiently large values of its parame parameter ter.. Often, Often, when when analy analysin sing g the run run time of an algorithm, it is easier to obtain an approximate formula for the run-time which gives a good indication of the algorithm performance for large problem instances.
Contd… • For For exa examp mple le,, sup suppo pose se the the exa exact ct runrun-ti time me T(n) of an algorithm on an input of size n is • T (n) = 5n 5n2 + 6n 6n + 25 seconds. Then, since n is ≥0 , we have • 5n2 ≤ T (n) ≤ 6n2 for all n≥9. • Thus we can say that T(n) is roughly proportional to • n2 for sufficiently large values of n of n. • We write this as • T (n)ϵ Θ(n2 ), or say that • “T(n) is in the exact order of n of n2”.
Contd… • The main feature of this approach is that we can ignore constant factors and concentrate on the terms in the expression for T(n) that will dominate the function’s behaviour as n becomes large.
Contd… • Generally, an algorithm with a run-time of Θ(n log n) will perform better than an algorithm with a run-time of order Θ(n2 ), provided that n is sufficiently large. However, for small values of n the Θ(n2 ) algorithm may run faster due to having smaller constant factors. The value of n at which the Θ(n log n) algorithm first outperforms the Θ(n2 ) algorithm is called the break-even point for the Θ(n log n) algorithm.
Contd…Big Oh notation
Big-oh
Contd…
Omega lower Bound Law of duality
Theta Notation
Maximum Rule
Contd…
Contd…
Contd…
L’ Hospital Rule
P, NP
Conditional asymptotic notation • Many algorithms easier to analyse if initially we restrict our attention to instances whose size satisfies a certain condition, such as a power of 2.
