Quick Sort Algorithm - Florida Institute of Technology

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1|void quicksort (int[] a, int lo, int hi)2|{3| int i=lo, j=hi, h;4| int x=a[k];// partition5| do6| {7| while (a[i]x) j--;9| if (i
6 5 1 26|8 3 6 4 9 5 71|2|3|4|5|Sorted11SortedSorted16 5 1 2Sorted1Pivot ItemPivot Item222SortedPivot Item6 5 25556Sorted6Sorted67| 8 3 6 4 9 5 78|9|10|11|Sorted3Sorted3Pivot Item5 3 43Sorted6Pivot Item SortedFigure 4.Best Case of Quicksort9 8 7Sorted5 4 6 7 8 9sorted4Pivot Item455Sorted6Sorted6Sorted7Pivot ItemPivot ItemSorted87 8 99Figure 3.Worst Case of QuicksortThe worst case of quick sort occur when we choose the smallestor the largest element as the pivot. It needs the mostcomparisons in each iteration. See figure 3, we want to sorta=[6,5,1,2] . From line1 to 2, the smallest item 1 is chosen as thepivot. All elements need to compare with the pivot item once.From line 2 to 3, the largest item 6 is chosen as the pivot. Allelements need to compare with the pivot item once. From line 3to 4, 1 comparison is needed.Let T(N) be the number of comparison needed to sort an arrayof N elements.In the worst case, we need N-1 comparisons in the first round.Then we need T(N-1) comparisons to sort N-1 elements.TNN‐1TN‐11We need 0 time to sort 1 element:T102The best case of quicksort occurs when we choose median as thepivot item each iteration. Figure 4 shows a example of the bestcase of Quicksort. Each partition decreases by almost half thearray size each iteration. From line 7 to line8, after 6 timescomparison, 6 is in the right place and we get partition {5,3,4}and {9,8,7}. It is the same with line 8-9. Line 10, when there aretwo elements left, the array is sorted.Let T(N) be the number of comparison needed to sort an arrayof N elements. In the best case, we need N-1 comparisons tosort the pivot in the first iteration. We need T((N-1)/2)comparisons to sort each two partition.T(N)=N-1 + 2T((N-1)/2) [1]We need 0 comparisons to sort 1 element.T(1)=0 [2]TN‐1N‐2TN‐2TN‐2N‐3TN‐3…TN‐N‐1N‐N‐1TN‐N‐11T01T(N)=(N-1)+T(N-1)=(N-1)+(N-2)+ TN‐2(N-1)+(N-2)+(N-3)+ TN‐3(N-1)+(N-2)+(N-3)+…+T(N-(N-1))=(N-1)+(N-2)+(N-3)+…+1=(N 2 +N)/2=O(N 2 )Thus, we have a O(N 2 ) time complexity in the worst case.345T((N-1)/2)=2T((N-3)/4)+(N-1)/2-1 [3]Equation [3] replacing T((N-1)/2)) in equation [1] yieldsT(N)=2[(2T((N-3)/4)+(N-1)/2-1)]+N-1=2 2 T((N-3)/2 2 )+N-3+N-1 [4]T((N-3)/2 2 )=2T((N-7)/2 3 )+(N-3)/4-1 [5]Equation [5] replacing T((N-3)/2 2 ) in equation [4] yieldsT(N)=2 3 T((N-7)/2 3 )+N-7+N-3+N-1 [6]T(N)=2 k T(N-1+N-3+N-7+…+N-(2 k -1))5.Empirical EvaluationThe efficiency of the Quicksort algorithm will be measured inCPU time which is measured using the system clock on amachine with minimal background processes running, withrespect to the size of the input array, and compared to theMergesort algorithm. Quick sort algorithm will be run with thearray size parameter set to: 1k, 2k,…10k. To ensurereproducibility, all datasets and algorithms used in this
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Quick Sort Algorithm - Florida Institute of Technology

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