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EXAMPLE 14.9 The Quick Sort
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public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; sort(a, 0, a.length); } private static void sort(int[] a, int p, int q) { // PRECONDITION: 0 <= p <= q <= a.length // POSTCONDITION: a[p..q) is sorted; if (q - p < 2) { return; } int m = partition(a, p, q); // step 2 sort(a, p, m); // step 4 sort(a, m+1, q); // step 5 } private static int partition(int[] a, int p, int q) { // RETURNS: index m of pivot element a[m]; // POSTCONDITION: a[i] <= a[m] <= a[j] for p <= i <= m <= j < q; int pivot = a[p], i = p, j = q; // steps 1-2 while (i < j) { // step 3 while (i < j && a[--j] >= pivot) ; // step 4 if (i < j) { a[i] = a[j]; // step 5 } while (i < j && a[++i] <= pivot) ; // step 6 if (i < j) { a[j] = a[i]; // step 7 } } a[j] = pivot; // step 8 return j; }
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Note the empty loop at line 22 and line 26. All the action is managed within the loop condition, so no statements are in its body.
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Algorithm 14.7 selects the pivot element to be the last element in the sequence. The algorithm works just as well if it is selected to be the first element or the middle element. Slightly better performance is obtained by selecting the median of those three elements. The Java Arrays.sort() method implements the quick sort, selecting the pivot as the median of the three elements {s0, sn/2, sn 1} when n 40, and the median of 9 equally spaced elements when n > 40. It also switches to the insertion sort (Algorithm 14.3 on page 258) when n < 7.
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SORTING
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Theorem 14.11 The quick sort runs in O(n lgn) time in the best case. The best case is when the sequence values are uniformly randomly distributed so that each call to the quick partition algorithm will result in balanced split of the sequence. In that case, each recursive call to the quick sort algorithm divides the sequence into two subsequences of nearly equal length. As with the binary search and the merge sort (Algorithm 14.5 on page 261), this repeated subdivision takes lgn steps to get down to size 1 subsequences, as illustrated in the diagram in Figure 14.2 on page 262. So there are O(lgn) calls made to the quick partition algorithm which runs in O(n) time, so the total running time for the quick sort algorithm is O(n lgn). Theorem 14.12 The quick sort runs in O(n2) time in the worst case. The worst case is when the sequence is already sorted (or sorted in reverse order). In that case, the quick partition algorithm will always select the last element (or the first element, if the sequence is sorted in reverse order), resulting in the most unbalanced split possible: One piece has n 2 elements, and the other piece has 1 element. Repeated division of this type will occur O(n) times before both pieces get down to size 1. So there are O(n) calls made to the quick partition algorithm which runs in O(n) time, so the total running time for the quick sort algorithm is O(n2). Note that in the worst case, the quick sort reverts to the selection sort (Algorithm 14.2 on page 257) because each call to quick partition amounts to selecting the largest element from the subsequence passed to it. So actually, Theorem 14.12 is a corollary to Theorem 14.4 on page 258. Theorem 14.13 The quick sort runs in O(n lgn) time in the average case. The proof of this fact is beyond the scope of this outline. Theorem 14.14 The quick sort is correct. The invariant inside the while loop proof claims that all the elements to the left of a[i] are less than or equal to the pivot element and that all the elements to the right of a[j] are greater than or equal to the pivot. This is true because every element to the left of a[i] that is greater than the pivot was swapped with some element to the right of a[j] that is less than the pivot, and conversely (every element to the right of a[j] that is less than the pivot was swapped with some element to the left of a[i] that is greater than the pivot. When that loop terminates, j i, so at that point all the elements that are greater than the pivot have been moved to the right of a[i], and all the elements that are less than the pivot have been moved to the left of a[i]. This is the invariant in step 7 of the quick partition algorithm. So after the swap in step 8, all the elements that are greater than the a[i] are to the right of a[i], and all the elements that are less than the a[i] are to the left of a[i]. This is the invariant in step 7 of the quick partition algorithm, which is the same as the invariant in step 3 of the quick sort algorithm. So then sorting the left segment and the right segment independently will render the entire sequence sorted. THE HEAP SORT A heap is by definition partially sorted, because each linear string from root to leaf is sorted. (See 13.) This leads to an efficient general sorting algorithm called the heap sort. As
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