Data Structures and Algorithm Analysis - Computer Science at ...

240 Chap. 7 Internal Sorting72 6 57 88 60 42 83 73 48 85Pivot = 6048 6 57 42 60 88 83 73 72 85Pivot = 6 Pivot = 736 42 57 4872 73 85 88 83Pivot = 57Pivot = 88Pivot = 42 42 48 5785 83 88 Pivot = 8542 4883856 42 48 57 60 72 73 83 85 88Final Sorted ArrayFigure 7.14 An illustration of Quicksort.how much work is done by the nested while loops. The do loop as a whole isguaranteed to move both l and r inward at least one position on each first pass.Each while loop moves its variable at least once (except in the special case wherer is at the left edge of the array, but this can happen only once). Thus, we see thatthe do loop can be executed at most s times, the total amount of work done movingl and r is s, and each while loop can fail its test at most s times. The total workfor the entire partition function is therefore Θ(s).Knowing the cost of findpivot and partition, we can determine thecost of Quicksort. We begin with a worst-case analysis. The worst case will occurwhen the pivot does a poor job of breaking the array, that is, when there are noelements in one partition, and n − 1 elements in the other. In this case, the divideand conquer strategy has done a poor job of dividing, so the conquer phase willwork on a subproblem only one less than the size of the original problem. If thishappens at each partition step, then the total cost of the algorithm will ben∑k = Θ(n 2 ).k=1In the worst case, Quicksort is Θ(n 2 ). This is terrible, no better than BubbleSort. 2 When will this worst case occur? Only when each pivot yields a bad partitioningof the array. If the pivot values are selected at random, then this is extremelyunlikely to happen. When selecting the middle position of the current subarray, it2 The worst insult that I can think of for a sorting algorithm.