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Lecture 6. Analysis of Iterative Algorithms Sorting Algorithms. Some useful terminology. Time and Space Complexity : analyze not just the number of steps, but also the space usage. Worst case analysis versus other types ( best case, average ). Time and Space Complexity.
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Lecture 6 Analysis of Iterative Algorithms Sorting Algorithms
Some useful terminology • Time and Space Complexity: analyze not just the number of steps, but also the space usage. • Worst case analysis versus other types (best case, average)
Time and Space Complexity • Time: number of steps • Space: number of memory cells used by the algorithm, besides the input • Examples of space usage: • Find Min, Search, Selection Sort, Insertion Sort: no extra space (constant, in fact: O(1) ) • Merge: extra n space, O(n) • Merge Sort: O(n) extra space
Best case, Average case • Sequential Search: worst case O(n), best case O(1), average case n/2= O(n) • Binary Search: best case O(1), worst case and average O(log n) • QuickSort: worst case O(n 2), average case O(n log n)
Analysis of Iterative Algorithms • Examples: • Selection Sort • Insertion Sort • Correctness proof using loop invariant for Selection Sort (pb. 1.2 textbook) • Summation formulas
Selection Sort • Applet and Pseudocode from Dominique’s page • Proof using loop invariant • Analysis: • Time to find minimum M(n)=n-1 • Time to do Selection Sort: T(n) = M(n)+M(n-1)+…+ M(2)+M(1) = = 1+2+…+(n-1) = n(n-1)/2 = O(n2)
Insertion Sort • Applet and pseudocode from Dominique’s page • Proof of correctness using loop invariant • Analysis: • Time to search and shift at step k: S(k)=k • Time to do Insertion Sort: T(n) = S(1)+S(2)+…+S(n-1)=1+2+…+(n-1)= = n(n-1)/2 = O(n 2)
