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Quicksort is a fast divide-and-cOnquer algorithm, when properly implemented In practice it is the fastest comparisonbased sorting algorithm
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As its name implies, quicksort is a fast divide-and-conquer algorithm; in practice it is the fastest sorting algorithm known Its average running time is O(N log m Its speed is mainly due to a very tight and highly optimized inner loop It has quadratic worst-case performance, which can be made statistically unlikely to occur with a little effort On the one hand, the quicksort algorithm is relatively simple to understand and prove correct because it relies on recursion On the other hand, it is a tricky algorithm to implement because minute changes in the code can make significant differences in running time We first describe the algorithm in broad terms We then provide
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an analysis that shows its best-, worst-, and average-case running times We use this analysis to make decisions about how to implement certain details in C++, such as the handling of duplicate items
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The basic quicksort algorithm is recursive Details include choosing the pivot, deciding how to partition, and dealing with duplicates Wrong decisions give common inputs The pivotdivides array elements two groups: those smaIIer than the pivot and those larger than the Pivot
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The basic algorithm Quicksort(S) consists of the following four steps I If the number of elements in S is 0 or 1, then return 2 Pick anv element v in S It is called the ~ i v o t 3 Partition S - {v} (the remaining elements in S) into two disjoint groups:L= { Y E S - { v ) J u I v } a n d R = { X E S - { v } l x L v } 4 Return the result of Quicksort(L) followed by v followed by Quicksort(R) Several points stand out when we look at these steps First, the base case of the recursion includes the possibility that S might be an empty set This provision is needed because the recursive calls could generate empty subsets Second, the algorithm allows any element to be used as the pivot The pivot divides array elements into two groups: elements that are smaller than the pivot and elements that are larger than the pivot The analysis performed here shows that some choices for the pivot are better than others Thus, when we provide an actual implementation, we do not use just any pivot Instead we try to make an educated choice In the partition step, every element in S, except for the pivot, is placed in either L (which stands for the left-hand part of the array) or R (which stands for the right-hand part of the array) The intent is that elements that are smaller than the pivot go to L and that elements larger than the pivot go to R The description in the algorithm, however, ambiguously describes what to do with elements equal to the pivot It allows each instance of a duplicate to go into either subset, specifying only that it must go to one or the other Part of a good C++ implementation is handling this case as efficiently as possible Again the analysis allows us to make an informed decision Figure 99 shows the action of quicksort on a set of numbers The pivot is chosen (by chance) to be 65 The remaining elements in the set are partitioned into two smaller subsets Each group is then sorted recursively Recall that, by the third rule of recursion we can assume that this step works The sorted arrangement of the entire group is then trivially obtained In a C++ implementation, the items would be stored in a part of an array delimited by l o w and high After the partitioning step, the pivot would wind up in some
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