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with all sorting algorithms, it applies to an array (or vector). But the underlying heap structure (a binary tree) that the array represents is used to define this algorithm. Like the merge sort and the quick sort, the heap sort uses an auxiliary function that is called from the sort() function. And also like the merge sort and the quick sort, the heap sort has complexity function O(n lg n). But unlike the merge sort and the quick sort, the heap sort is not recursive. The heap sort essentially loads n elements into a heap and then unloads them. By Theorem 13.1 on page 247, each element takes O(lgn) time to load and O(lgn) time to unload, so the entire process on n element runs in O(n lgn) time. Algorithm 14.8 The Heap Sort (Precondition: s = {s0 . . . sn 1} is a sequence of n ordinal values.) (Postcondition: The entire sequence s is sorted.) 1. Do steps 2 3 for i = n/2 1 down to 0. 2. Apply the heapify algorithm to the subsequence {si . . . sn 1}. 3. (Invariant: every root-to-leaf path in s is nonincreasing.) 4. Do steps 5 7 for i = n 1 down to 1. 5. Swap si with s0 . 6. (Invariant: The subsequence {si . . . sn 1} is sorted.) 7. Apply the heapify algorithm to the subsequence {s0 . . . si 1}. Algorithm 14.9 The Heapify (Preconditions: ss = {si . . . sj 1} is a subsequence of j i ordinal values, and both subsequences {si+1 . . . sj 1} and {si+2 . . . sj 1} have the heap property.) (Postcondition: ss itself has the heap property.) 1. Let t = s2i+1. 2. Let sk = max{s2i+1, s2i+2}, so k = 2i+1 or 2i+2, the index of the larger child. 3. If t < sk , do steps 4 6. 4. Set si = sk . 5. Set i = k. 6. If i < n/2 and si < max{s2i+1, s2i+2}, repeat steps 1 4. 7. Set sk = t. There are two aspects to these algorithms that distinguish them from the methods outlined in 12. The heaps here are in the reverse order, so each root-to-leaf path is descending. And these algorithms use 0-based indexing. The reverse order guarantees that heapify will always leave the largest element at the root of the subsequence. Using 0-base indexing instead of 1based indexing renders the sort() method consistent with all the other sort() methods at the expense of making the code a little more complicated. EXAMPLE 14.10 The Heap Sort
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public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; int n = a.length; for (int i = n/2 - 1; i >= 0; i--) { // step 1 heapify(a, i, n); // step 2 }
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// step 4 // step 5 // step 7
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for (int i = n - 1; i > 0; i--) { swap(a, 0, i); heapify(a, 0, i); } }
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private static void heapify(int[] a, int i, int j) { int ai = a[i]; // step 1 48 while (2*i+1 < j) { 49 int k = 2*i + 1; 50 if (k + 1 < j && a[k+1] > a[k]) { 51 ++k; // a[k] is the larger child 52 } 53 if (ai >= a[k]) { 54 break; // step 3 55 } 56 a[i] = a[k]; // step 4 57 i = k; // step 5 58 } 59 a[i] = ai; // step 7 60 } The sort() function first converts the array so that its underlying complete binary tree is transformed into a heap. This is done by applying the heapify() function to each nontrivial subtree. The nontrivial
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subtrees (i.e., those having more than one element) are the subtrees that are rooted above the leaf level. In the array, the leaves are stored at positions a[n/2] through a[n]. So the first for loop in the sort() function applies the heapify() function to elements a[n/2-1] back through a[0] (which is the root of the underlying tree). The result is an array whose corresponding tree has the heap property, illustrated in Figure 14.3.
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