barcode in ssrs 2008 Figure 14.3 The natural mapping for the heap sort in Java

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Figure 14.3 The natural mapping for the heap sort
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Now the main (second) for loop progresses through n-1 iterations. Each iteration does two things: it swaps the root element with element a[i], and then it applies the heapify() function to the subtree of elements a[0..i). That subtree consists of the part of the array that is still unsorted. Before the swap() executes on each iteration, the subarray a[0..i] has the heap property, so a[i] is the largest element in that subarray. That means that the swap() puts element a[i] in its correct position. The first seven iterations of the main for loop have the effect shown by the seven pictures in Figure 14.4 on page 268. The array (and its corresponding imaginary binary tree) is partitioned into two parts: The first part is the subarray a[0..i) that has the heap property, and the second part is the remaining a[i..n) whose elements are in their correct positions. The second part is shaded in each of the seven pictures in Figure 14.4 on page 268. Each iteration of the main for loop decrements the size of the first part and increments the size of the second part. So when the loop has finished, the first part is empty and the second (sorted) part constitutes the entire array. This analysis verifies the following theorem.
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Figure 14.4 Tracing the heap sort
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Theorem 14.15 The heap sort is correct. See Problem 14.31 on page 277. Theorem 14.16 The heap sort runs in O(n lgn) time. Each call to the heapify() function takes at most lg n steps because it iterates only along a path from the current element down to a leaf. The longest such path for a complete binary tree of n elements is lg n. The heapify() function is called n/2 times in the first for loop and n 1 times in the second for loop. That comes to less than (3n/2) lg n, which is proportional to n lg n. If we regard a sorting algorithm as a stream process wherein elements stream into an array in random order and then stream out in sorted order, then the heap sort can be regarded as an efficient mean between the extremes of the selection sort and the insertion sort. The selection sort does all its sorting during the removal stage of the process, having stored the elements in the unsorted order in which they arrived. The insertion sort does all its sorting during the insertion stage of the process so that the elements can stream out of the array in the sorted order in which they were stored. But the heap sort does a partial sorting by inserting the elements into a heap and then finishes the sorting as the elements are removed from the heap. The payoff from this mean between the extremes is greater efficiency: O(n lg n) instead of O(n2). THE SPEED LIMIT FOR COMPARISON SORTS Theorem 14.17 No sorting algorithm that rearranges the array by comparing its elements can have a worst-case complexity function better than O(n lg n). Consider the decision tree that covers all possible outcomes of the algorithm on an array of size n. Since the algorithm rearranges the array by comparing its elements, each node in the decision tree represents a condition of the form (a[i] < a[j]). Each such condition has two possible outcomes (true or false), so the decision tree is a binary tree. And since the tree must cover all possible arrangements, it must have at least n! leaves. Therefore, by Corollary 11.3 on page 203, the height of the decision tree must be at least lg(n!). In the worst case, the number of comparisons that the algorithm makes is the same as the height of the decision tree. Therefore, the algorithm s worst-case complexity function must be O(lg(n!)). Now by Stirling s Formula (outlined on page 325), n! so log n! log 2n n -e
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