Analysis of Algorithms

1. Analysis of Algorithms Rate of Growth of functions Prof. Muhammad Saeed

2. Rate of Growth of functions Analysis of Algorithms

3. Analysis of Algorithms

4. Analysis of Algorithms

5. Analysis of Algorithms

6. Running Times • Assume N = 100,000 and processor speed is 1,000,000 operations per second Analysis of Algorithms

7. Series and Asymptotics Analysis of Algorithms

8. Series I Analysis of Algorithms

9. Series II Analysis of Algorithms

10. Infinite Series Analysis of Algorithms

11. Fundamental DefinitionsAsymptotics • T(n) = O(f(n)) if there are constants c and n0 such that T(n) ≤ cf(n) when n  n0 • T(n) = (g(n)) if there are constants c and n0 such that T(n)  cg(n) when n  n0 • T(n) = (h(n)) if and only if T(n) = O(h(n)) and T(n) = (h(n)) • T(n) = o(p(n)) if T(n) = O(p(n)) and T(n)  (p(n)) Analysis of Algorithms

12. T(n) = (h(n)) T(n) = O(f(n)) T(n) = (g(n)) Asymptotics Analysis of Algorithms

13. Relative Rates of Growth Analysis of Algorithms

14. Relative Growth Rate ofTwo Functions Compute using L’Hopital’s Rule • Limit=0: f(n)=o(g(n)) • Limit=c0: f(n)=(g(n)) • Limit=: g(n)=o(f(n)) Analysis of Algorithms

15. Important Rules 3n3 = O(n3) 3n3 + 8 = O(n3) 8n2 + 10n * log(n) + 100n + 1020 = O(n2) 3log(n) + 2n1/2 = O(n1/2) 2100 = O(1) TlinearSearch(n) = O(n) TbinarySearch(n) = O(log(n)) Analysis of Algorithms

16. Important Rules Rule 1: If T1(n) = O(f(n)) and T2(n) = O(g(n)), then a) T1(n) + T2(n) = max(O(f(n)), O(g(n))) b) T1(n) * T2(n) = O(f(n)*g(n)) Rule 2: If T(x) is a polynomial of degree n, then T(x)=(xn) Rule 3: logk n = O(n) for any constant k. Analysis of Algorithms

17. General Rulesfor • Loops • Nested Loops • Consecutive statements • if-then-else • Recursion Analysis of Algorithms

18. Euclid’s GCD intgcd(int a, int b) { int t; while (b != 0) { t = b; b = a % b; a = t; } return a; } Analysis of Algorithms

19. Binary Search function BinarySearch(a, value, left, right) while left ≤ right mid := floor((right+left)/2) if a[mid] = value return mid if value < a[mid] right := mid-1 else left := mid+1 endwhile return not found Analysis of Algorithms

20. Insertion Sort A case Analysis of Algorithms

21. Insertion Sort Best Case: T(n)=O(n) Worst Case: T(n)=O(n2) Analysis of Algorithms

22. Maximum Subsequence Sum A case Analysis of Algorithms

23. Maximum Subsequence SumAlgorithm 1 intMaxSubsequenceSum( constint A[], const unsigned int N) { int Sum=0, MaxSum=0; for( i = 0; i<N; i++) for( j = i; j<N; j++) { Sum = 0; for( k = i; k<=j; k++) Sum += A[ k ]; if (Sum > MaxSum) MaxSum = Sum; } return MaxSum; } Analysis of Algorithms

24. Maximum Subsequence SumAlgorithm 2 int MaxSubsequenceSum( const int A[], const unsigned int N) { int Sum=0, MaxSum=0; for( i = 0; i<N; i++) Sum = 0; for( j = i; j<N; j++) { Sum += A[ k ]; if (Sum > MaxSum) MaxSum = Sum; } return MaxSum; } Analysis of Algorithms

25. Maximum Subsequence SumAlgorithm 3 intMaxSubsequenceSum( constint A[], const unsigned int N) { int Sum = 0, MaxSum = 0, Start = 0, End = 0; for( End = 0; End<N; End++) { Sum += A[ End ]; if (Sum > MaxSum) MaxSum = Sum; else if (Sum < 0) { Start= end+1; Sum=0; } } return MaxSum; } Analysis of Algorithms

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