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Analysis of Algorithms COMP171 Fall 2006

Introduction ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Introduction ,[object Object],[object Object],[object Object],[object Object]

Example: Selection Problem ,[object Object],[object Object],[object Object],[object Object],[object Object]

Example: Selection Problem… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example: Selection Problem… ,[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm Analysis… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],[object Object],[object Object],[object Object],[object Object],1 2 3 4 1 2N+2 4N 1

Worst- / average- / best-case ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Running-time of algorithms ,[object Object],[object Object],[object Object],[object Object]

Growth Rate ,[object Object],[object Object],[object Object]

Asymptotic notation: Big-Oh ,[object Object],[object Object],[object Object],[object Object],[object Object]

Big-Oh: example ,[object Object],[object Object],[object Object],[object Object],[object Object]

Big Oh: more examples ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Math Review: logarithmic functions

Some rules ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Big-Omega ,[object Object],[object Object]

Big-Omega ,[object Object],[object Object],[object Object],[object Object]

Big-Omega: examples ,[object Object],[object Object],[object Object]

f(N) =  (g(N)) ,[object Object]

Big-Theta ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Some rules ,[object Object],[object Object],[object Object],[object Object]

Growth rates … ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Using L' Hopital's rule ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

General Rules ,[object Object],[object Object],[object Object],[object Object],[object Object]

General rules (cont’d) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another Example ,[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm 1: Simple ,[object Object],[object Object]

Algorithm 2: Divide-and-conquer ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm 2 (cont’d) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm 2 … O(1) T(m/2) O(m) O(1) T(m/2)

Algorithm 2 (cont’d) ,[object Object],[object Object],[object Object]

Algorithm 2 (cont’d) ,[object Object],[object Object],[object Object],[object Object]

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