Conditional Asymptotic Notations

V. Balasubramanian

Asymptotic Notations

• Asymptotic notation deals with the behaviour of a function in the limit, that is, for sufficiently large values of its parameter. Often, when analysing the 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 example, suppose the exact run-time T(n) of

an algorithm on an input of size n is• T(n) = 5n2 + 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. • We write this as• T(n)ϵ Θ(n2 ), or say that• “T(n) is in the exact order of n2”.

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• 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 theΘ(n log n) algorithm.

Contd…Big Oh notation

Big-oh

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Omega lower Bound Law of duality

Theta Notation

Maximum Rule

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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.

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Example

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Smooth functions

Non-smooth functions

Smoothness rule

Multiplication

Another technique

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Example

Example

Example

Example

Master Theorem