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Problems
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13.1 Determine which of the binary trees in Figure 13.7 is a heap.
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Figure 13.7 Binary trees
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13.2 13.3 13.4
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Determine which of the arrays in Figure 13.8 on page 251 has the heap property. Show the heap after inserting each of these keys in this order: 44, 66, 33, 88, 77, 77, 22. Show the array obtained from the natural map of each of the heaps obtained in Problem 13.3.
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HEAPS AND PRIORITY QUEUES
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Figure 13.8 Arrays
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boolean isHeap(int[] a) // returns true if and only if the specified array // has the heap property
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13.6 13.7
Prove that every subtree of a heap is a heap. Show the heap after inserting each of these keys in this order: 50, 95, 70, 30, 90, 25, 35, 80, 60, 40, 20, 10, 75, 45, 35.
Answers to Review Questions
13.1 13.2 13.3 13.4 Heaps are used to implement priority queues and the heap sort. (See page 266.) Insertions into and removals from a heap are very efficient; they run in O(lgn). A priority queue is a best-in-first-out container, that is, the element with the highest priority comes out first. Elements are removed from a queue in the same order in which they are inserted: first-in-first-out. Elements in a priority queue must have an ordinal key field which determines the priority order in which they are to be removed. Heaps are used to implement priority queues because they allow O(lgn) insertions and removals. This is because both the add() and the remove() methods are implemented by traversing a root-to-leaf path through the heap. Such paths are no longer than the height of the tree which is at most lgn. The natural mapping starts at a[1] instead of a[0] to facilitate navigation up and down the heap tree. By numbering the root 1 and continuing sequentially with a level order traversal, the number of the parent of any node numbered k will be k/2, and the numbers of its child nodes will be 2k and 2k+1. If it takes an average of 3ms to remove an element from a priority queue with 1,000 elements, then it should take about 6ms to remove an element from a priority queue with 1,000,000 elements. The run time for a method that uses a priority queue to sort an array would be O(2n lgn) because it will make n insertions and n removals, each running in O(lgn) time.
13.7 13.8
Solutions to Problems
13.1 a. This is not a heap because the root-to-leaf path {88, 44, 77} is not descending (44 < 77). b. This is a heap. c. This is not a heap because the root-to-leaf path {55, 33, 44} is not descending (33 < 44) and the root-to-leaf path {55, 77, 88} is not descending (55 < 77 < 88). d. This is not a heap because the binary tree is not complete. e. This is a heap. f. This is not a heap because the tree is not binary.
HEAPS AND PRIORITY QUEUES
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a. This array does not have the heap property because the root-to-leaf path {a[1], a[3], a[6]} = {88, 44, 77} is not descending (44 < 77). b. This array does have the heap property. c. This array does have the heap property. d. This array does not have the heap property because its data elements are not contiguous: It does not represent a complete binary tree. e. This array does have the heap property. f. This array does not have the heap property because the root-to-leaf path {a[1], a[3], a[6]} = {88, 22, 55} is not descending (22 < 55) and the root-to-leaf path {a[1], a[3], a[7]} = {88, 22, 66} is not descending (22 < 66). Figure 13.9 shows a trace of the insertion of the keys 44, 66, 33, 88, 77, 55, 22 into a heap.
Figure 13.9 Trace of insertions into a heap
13.4 13.5
Figure 13.10 on page 253 shows the arrays for the heaps in Problem 13.3.
boolean isHeap(int[] a) { // returns true if and only if the specified array // has the heap property int n = a.length; for (int i = n/2; i < n; i++) { for (int j = i; j > 1; j /=2) { if (a[j/2] < a[j]) { return false; } } } return true; }
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