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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 131 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 148 The Heap Sort (Precondition: s = {s0 sn 1} is a sequence of n ordinal values.

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and, therefore, r I = -0.3793. Thus, the structure is as follows:

CHAP. 14]

This filter structure has three delays and requires five multiplications and five additions to evaluate each value of the output, y(n). A direct-form structure also requires three delays but only three multiplications and three additions.

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) (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 149 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.

Without factoring any polynomials, determine whether o r not the following causal filter is stable:

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Theorem 1415 The heap sort is correct See Problem 1431 on page 277 Theorem 1416 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.

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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 1410 The Heap Sort.

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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.

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We may easily check the stability of this filter using the Schur-Cohn stability test, which involves checking to see whether or not the magnitudes of the reflection coefficients corresponding to the denominator polynomial are bounded by 1 in magnitude. With

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