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This resource, presented by Prof. Muhammad Saeed, delves into the fundamental concepts of algorithm analysis, especially focusing on the rate of growth of functions. It covers key asymptotic notations such as O, Ω, Θ, and o. The analysis includes practical examples of algorithm efficiencies such as binary search, linear search, and insertion sort. Important rules for combining running times and evaluating limits using L'Hopital's Rule are provided, making it an essential guide for students and professionals aiming to deepen their understanding of computational complexity and algorithm performance.
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Analysis of Algorithms Rate of Growth of functions Prof. Muhammad Saeed
Rate of Growth of functions Analysis of Algorithms
Running Times • Assume N = 100,000 and processor speed is 1,000,000 operations per second Analysis of Algorithms
Series and Asymptotics Analysis of Algorithms
Series I Analysis of Algorithms
Series II Analysis of Algorithms
Infinite Series Analysis of Algorithms
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
T(n) = (h(n)) T(n) = O(f(n)) T(n) = (g(n)) Asymptotics Analysis of Algorithms
Relative Rates of Growth Analysis of Algorithms
Relative Growth Rate ofTwo Functions Compute using L’Hopital’s Rule • Limit=0: f(n)=o(g(n)) • Limit=c0: f(n)=(g(n)) • Limit=: g(n)=o(f(n)) Analysis of Algorithms
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
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
General Rulesfor • Loops • Nested Loops • Consecutive statements • if-then-else • Recursion Analysis of Algorithms
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
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
Insertion Sort A case Analysis of Algorithms
Insertion Sort Best Case: T(n)=O(n) Worst Case: T(n)=O(n2) Analysis of Algorithms
Maximum Subsequence Sum A case Analysis of Algorithms
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
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
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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