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Fundamentals of the Analysis of Algorithm Efficiency

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

Order of Growth ,[object Object],[object Object],[object Object],[object Object]

Worst-Case, Best-Case & Average-Case Efficiencies ,[object Object],[object Object],[object Object],[object Object]

Sequential Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sequential Search (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sequential Search (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Asymptotic Notations ,[object Object],[object Object],[object Object]

O-notation (Big Oh) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Ω -notation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Θ -notation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Mappings for n 2 Ω (n 2 ) O(n 2 ) Θ ( n 2 )

Examples of algorithms for sorting techniques and their complexities ,[object Object],[object Object],[object Object],[object Object]

Time Efficiency Analysis ,[object Object],[object Object],[object Object],[object Object],=(n-1) 2 - (n-2)(n-1)/2 = (n-1)n/2 ≈ ½ n 2 Є Θ (n 2 )

Recurrences ,[object Object],[object Object],[object Object]

Mathematical Analysis of Recursive Algorithms ,[object Object],[object Object],[object Object],[object Object],[object Object]

Example 1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example 1 (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example 2 (contd.) ,[object Object]

Example 3 ,[object Object],[object Object],[object Object],[object Object],[object Object]

Example 3 (contd.) ,[object Object],[object Object],[object Object],Whose solution is claimed to be T(n)= Θ(n lg n). The equation 1.0 can be rewritten as follows: T(n)= Where the constant c represents the time required to solve problems of size 1 as well as the time per array element of the divide and combine steps.

Example 3 (contd.) cn T(n) cn T(n/2) T(n/2 ) cn/2 cn/2 T(n/4) T(n/4) T(n/4) T(n/4) (a) (b) (c)

Example 3 (contd.) ,[object Object],[object Object],[object Object]

Example 3 (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Randomized Algorithms ,[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm Visualization ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Assignment ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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