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A Heap Implementation
Chapter 26
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Contents
Reprise: The ADT Heap
Using an Array to Represent a Heap
Adding an Entry
Removing the Root
Creating a Heap
Heap Sort
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Objectives
Use an array to represent a heap
Add an entry to an array-based heap
Remove the root of an array-based heap
Create a heap from given entries
Sort an array by using a heap sort
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Reprise: The ADT Heap
A heap:
Complete binary tree
Nodes contain Comparable objects
A maxheap:
Object in each node greater than or equal to the objects in node’s descendants
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Reprise: The ADT Heap
Interface considered in Chapter 23
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6. Array to Represent a Heap
Begin by using array to represent complete binary tree.
Complete tree is full to its next-to-last level
Leaves on last level are filled from left to right
Locate either children or parent of any node by performing simple computation on node’s number
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Figure 26-1 (a) A complete binary tree with its nodes numbered in level order; (b) its representation as an array
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Question 1 If an array contains the entries of a heap in level order beginning at index 0, what array entries represent a node’s parent, left child, and right child?
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Question 1 If an array contains the entries of a heap in level order beginning at index 0, what array entries represent a node’s parent, left child, and right child?
The node at array index i
• Has a parent at index (i - 1)/2, unless the node is the root (i is 0).
• Has any children at indices 2i + 1 and 2i + 2.
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10. Array to Represent a Heap
View code of class MaxHeap, Listing 26-1
Data fields:
Array of Comparable heap entries
Index of last entry in the array
A constant for default initial capacity of heap
Note: Code listing files
must be in same folder
as PowerPoint files
for links to work
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11. Figure 26-2 The steps in adding 85 to the maxheap in Figure 26-1a
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12. Figure 26-2 The steps in adding 85 to the maxheap in Figure 26-1a
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13. Figure 26-2 The steps in adding 85 to the maxheap in Figure 26-1a
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Question 2 What steps are necessary to add 100 to the heap in Figure 26-2c?
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Question 2 What steps are necessary to add 100 to the heap in Figure 26-2c?
Place 100 as a right child of 80. Then swap 100 with 80, swap 100 with 85, and finally swap 100 with 90.
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Figure 26-3 A revision of the steps shown in Figure 26-2, to avoid swaps
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Figure 26-3 A revision of the steps shown in Figure 26-2, to avoid swaps
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18. Figure 26-4 An array representation of the steps in Figure 26-3
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19. Figure 26-4 An array representation of the steps in Figure 26-3
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Figure 26-5 The steps to remove the entry in the root of the maxheap in Figure 26-3d
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Figure 26-5 The steps to remove the entry in the root of the maxheap in Figure 26-3d
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Q 6 What steps are necessary to remove the root from the heap in Figure 26-5d?
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Q 6 What steps are necessary to remove the root from the heap in Figure 26-5d?
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Figure 26-6 The steps that transform a semiheap into a heap without swaps
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Figure 26-6 The steps that transform a semiheap into a heap without swaps
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Figure 26-7 The steps in adding 20, 40, 30, 10, 90, and 70 to an initially empty heap
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Figure 26-7 The steps in adding 20, 40, 30, 10, 90, and 70 to an initially empty heap
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Figure 26-7 The steps in adding 20, 40, 30, 10, 90, and 70 to an initially empty heap
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Question 8 If an array represents a heap, and lastIndex is the index of the heap’s last leaf, show why the index of the first nonleaf closest to the end of the array is lastIndex/2 .
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Question 8 If an array represents a heap, and lastIndex is the index of the heap’s last leaf, show why the index of the first nonleaf closest to the end of the array is lastIndex/2 .
If a node at index i has children, they are at indices 2 i and 2 i + 1. The node at lastIndex/2 then has a child at lastIndex. Since this child is the last leaf, any nodes beyond the one at lastIndex/2 cannot have children and so
must be leaves. Thus, the node at lastIndex/2 must be the nonleaf closest to the end of the array.
Alternatively, examine some complete trees and notice that the desired nonleaf is the parent of the last child. This child is at index lastIndex of the array representation, so its parent has index lastIndex/2 .
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Figure 26-8 the steps in creating a heap of the entries 20, 40, 30, 10, 90, and 70 by using reheap
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Figure 26-8 the steps in creating a heap of the entries 20, 40, 30, 10, 90, and 70 by using reheap
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Figure 26-8 the steps in creating a heap of the entries 20, 40, 30, 10, 90, and 70 by using reheap Complexity
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34. Figure 26-9 A trace of heap sort
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35. Figure 26-9 A trace of heap sort
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36. Figure 26-9 A trace of heap sort
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Efficiency
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Question 9Trace the steps that the method heapSort takes when sorting the following array into ascending order:
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Question 9Trace the steps that the method heapSort takes when sorting the following array into ascending order:
Original array
After repeated calls to reheap
After swap
After reheap
After swap
After reheap
After swap
After reheap
After swap
After reheap
After swap
After reheap
After swap
After reheap
After swap
Done
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End
Chapter 26
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