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vb.net code to generate barcode Problems in Java
Problems Recognizing EAN13 In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. GS1  13 Drawer In Java Using Barcode generator for Java Control to generate, create UPC  13 image in Java applications. 13.1 Determine which of the binary trees in Figure 13.7 is a heap.
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Recognizing Barcode In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Encode European Article Number 13 In C#.NET Using Barcode generator for VS .NET Control to generate, create European Article Number 13 image in Visual Studio .NET applications. 13.2 13.3 13.4 EAN13 Creator In .NET Using Barcode drawer for ASP.NET Control to generate, create EAN 13 image in ASP.NET applications. GS1  13 Creator In .NET Using Barcode drawer for .NET framework Control to generate, create EAN13 Supplement 5 image in VS .NET applications. 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. Encoding GTIN  13 In VB.NET Using Barcode drawer for .NET framework Control to generate, create GTIN  13 image in VS .NET applications. Paint GS1 128 In Java Using Barcode drawer for Java Control to generate, create EAN 128 image in Java applications. CHAP. 13] Generate Code 128C In Java Using Barcode drawer for Java Control to generate, create Code 128 Code Set C image in Java applications. Draw Bar Code In Java Using Barcode encoder for Java Control to generate, create barcode image in Java applications. HEAPS AND PRIORITY QUEUES
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Code 3/9 Reader In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Creating GTIN  12 In .NET Framework Using Barcode creator for .NET framework Control to generate, create UPCA image in Visual Studio .NET applications. boolean isHeap(int[] a) // returns true if and only if the specified array // has the heap property
Generate Code 3/9 In VS .NET Using Barcode maker for Reporting Service Control to generate, create Code39 image in Reporting Service applications. Creating Data Matrix In None Using Barcode printer for Word Control to generate, create DataMatrix image in Office Word applications. 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 bestinfirstout 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: firstinfirstout. 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 roottoleaf 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 roottoleaf path {88, 44, 77} is not descending (44 < 77). b. This is a heap. c. This is not a heap because the roottoleaf path {55, 33, 44} is not descending (33 < 44) and the roottoleaf 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
[CHAP. 13
a. This array does not have the heap property because the roottoleaf 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 roottoleaf path {a[1], a[3], a[6]} = {88, 22, 55} is not descending (22 < 55) and the roottoleaf 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; } CHAP. 13]

