1. CS 302 - Data Structures Chapter 17 Heaps Mehmet H Gunes
Modified from authors’ slides

2. The ADT Heap A heap is a complete binary tree that either is
Empty or …
Whose root contains a value  each of its children and has heaps as its subtrees
It is a special binary tree … different in that
It is ordered in a weaker sense
it will always be a complete binary tree
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3. What is a Heap? A heap is a binary tree that satisfies these
special SHAPE and ORDER properties:
Its shape must be a complete binary tree.
For each node in the heap, the value stored in that node is greater than or equal to the value in each of its children.

4. Are these Both Heaps? T A C
treePtr 50 20 18 30 10

5. Is this a Heap?
tree 70 60 12 40 30 8 10

6. Unique?

7. Where is the Largest Element in a Heap Always Found?
tree 70 60 12 40 30 8

8. We Can Number the Nodes Left to Right by Level This Way
tree 70 60 1 12 2 40 3 30 4 8 5

9. And use the Numbers as Array Indexes to Store the Heap
tree.nodes
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ] 70 60 12 40 30 8 70 60 1 40 3 30 4 12 2 8 5
tree

10. The ADT Heap (a) A maxheap and (b) a minheap
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11. With Array Representation
For any node tree.nodes[index]
its left child is in
tree.nodes[index*2 + 1]
right child is in
tree.nodes[index*2 + 2]
its parent is in
tree.nodes[(index – 1)/2]
Leaf nodes:
tree.nodes[numElements/2] to tree.nodes[numElements - 1]

12. The ADT Heap UML diagram for the class Heap
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13. The ADT Heap An interface for the ADT heap
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14. The ADT Heap An interface for the ADT heap
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15. The ADT Heap An interface for the ADT heap
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16. Array-Based Implementation of a Heap
A complete binary tree and its array-based implementation
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17. Algorithms for Array-Based Heap Operations
Assume following private data members
items: an array of heap items
itemCount: an integer equal to the number of items in the heap
maxItems: an integer equal to the maximum capacity of the heap
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18. Algorithms for Array-Based Heap Operations
Disjoint heaps after removing the heap’s root
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19. Algorithms for Array-Based Heap Operations
Recursive algorithm to transform semiheap to heap.
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20. Algorithms for Array-Based Heap Operations
Recursive algorithm to transform semiheap to heap.
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21. Algorithms for Array-Based Heap Operations
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22. Reheap Down
bottom

23. Algorithms for Array-Based Heap Operations
Recursive calls to heapRebuild
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24. Algorithms for Array-Based Heap Operations
Adding 15 to a heap
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25. Reheap Up
bottom

26. Algorithms for Array-Based Heap Operations
Pseudocode for add
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27. The Implementation The header file for the class ArrayMaxHeap
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28. The Implementation The header file for the class ArrayMaxHeap
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29. The Implementation The header file for the class ArrayMaxHeap
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30. The Implementation Definition of method getLeftChildIndex
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31. The Implementation Definition of the constructor
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33. The Implementation Array and its corresponding complete binary tree
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34. The Implementation Building a heap from an array of data
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35. The Implementation Transforming an array into a heap
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36. Building a Heap: i = 4 i = 3 i = 2 2 14 8 1 16 7 4 3 9 10 2 14 8 1 165 6 2 14 8 1 16 7 4 3 9 10 5 6 14 2 8 1 16 7 4 3 9 10 5 6
i = 1
i = 0 14 2 8 1 16 7 4 10 9 3 5 6 14 2 8 16 7 1 4 10 9 3 5 6 8 2 4 14 7 1 16 10 9 3 5 6

37. The Implementation C++ method heapCreate
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38. The Implementation C++ method peekTop which tests for an empty heap
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39. Heap Implementation of the ADT Priority Queue
A header file for the class HeapPriorityQueue
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40. Heap Implementation of the ADT Priority Queue
An implementation of the class HeapPriorityQueue
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41. Heap Implementation of the ADT Priority Queue
An implementation of the class HeapPriorityQueue
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42. Heap Implementation of the ADT Priority Queue
Heap versus a binary search tree
If you know maximum number of items in the priority queue, heap is the better implementation
Finite, distinct priority values
Many items likely have same priority value
Place in same order as encountered
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43. Heap Sort Heap sort partitions an array into two regions
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44. Heap Sort: Recall that . . .
A heap is a binary tree that satisfies these
special SHAPE and ORDER properties:
Its shape must be a complete binary tree.
For each node in the heap, the value stored in that node is greater than or equal to the value in each of its children.

45. The largest element in a heap
is always found in the root node
root 70 12 60 30 8 10 40

46. The heap can be stored in an array
values
root
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ] 70 60 12 40 30 8 10 70 60 1 12 2 30 4 10 6 40 3 8 5

47. Heap Sort Approach
First, make the unsorted array into a heap by satisfying the order property.
Then repeat the steps below until there are no more unsorted elements.
Take the root (maximum) element off the heap by swapping it into its correct place in the array at the end of the unsorted elements.
Reheap the remaining unsorted elements.
This puts the next-largest element into the root position.

48. After creating the original heap
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 70 60 12 40 30 8 10 70 60 1 12 2 30 4 10 6 40 3 8 5

49. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 70 60 12 40 30 8 10 70 60 1 12 2 30 4 10 6 40 3 8 5

50. After swapping root element into it place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 10 60 12 40 30 8 70 10 60 1 12 2 30 4 40 3 8 5
NO NEED TO CONSIDER AGAIN

51. After reheaping remaining unsorted elements
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 60 40 12 10 30 8 70 60 40 1 12 2 30 4 10 3 8 5

52. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 60 40 12 10 30 8 70 60 40 1 12 2 30 4 10 3 8 5

53. After swapping root element into its place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 8 40 12 10 30 60 70 8 40 1 12 2 30 4 10 3
NO NEED TO CONSIDER AGAIN

54. After reheaping remaining unsorted elements
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 40 30 12 10 6 60 70 40 30 1 12 2 6 4 10 3

55. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 40 30 12 10 6 60 70 40 30 1 12 2 6 4 10 3

56. After swapping root element into its place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 6 30 12 10 40 60 70 6 30 1 12 2 10 3
NO NEED TO CONSIDER AGAIN

57. After reheaping remaining unsorted elements
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 30 10 12 6 40 60 70 30 10 1 12 2 6 3

58. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 30 10 12 6 40 60 70 30 10 1 12 2 6 3

59. After swapping root element into its place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 6 10 12 30 40 60 70 6 10 1 12 2
NO NEED TO CONSIDER AGAIN

60. After reheaping remaining unsorted elements
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 12 10 6 30 40 60 70 12 10 1 6 2

61. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 12 10 6 30 40 60 70 12 10 1 6 2

62. After swapping root element into its place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 6 10 12 30 40 60 70 6 10 1
NO NEED TO CONSIDER AGAIN

63. After reheaping remaining unsorted elements
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 10 6 12 30 40 60 70 10 6 1

64. Swap root element into last place in unsorted array
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 10 6 12 30 40 60 70 10 6 1

65. After swapping root element into its place
values
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
root 6 10 12 30 40 60 70 6
ALL ELEMENTS ARE SORTED

66. template <class ItemType >
void HeapSort ( ItemType values[], int numValues )
// Post: Sorts array values[ numValues-1 ] into
// ascending order by key {
int index ;
// Convert array values[0..numValues-1] into a heap
for (index = numValues/2 - 1; index >= 0; index--)
ReheapDown ( values , index , numValues-1 ) ;
// Sort the array.
for (index = numValues-1; index >= 1; index--)
Swap (values[0], values[index]);
ReheapDown (values , 0 , index - 1); }

67. ReheapDown template< class ItemType >
void ReheapDown ( ItemType values[], int root, int bottom )
// Pre: root is the index of a node that may violate the
// heap order property
// Post: Heap order property is restored between root and
// bottom {
int maxChild ;
int rightChild ;
int leftChild ;
leftChild = root * ;
rightChild = root * ;

68. if (leftChild == bottom) maxChild = leftChild; else
if (leftChild <= bottom) // ReheapDown continued {
if (leftChild == bottom)
maxChild = leftChild;
else
if (values[leftChild] <= values [rightChild])
maxChild = rightChild ;
maxChild = leftChild ; }
if (values[ root ] < values[maxChild])
Swap (values[root], values[maxChild]);
ReheapDown ( maxChild, bottom ) ;

69. Heap Sort: How many comparisons?
In reheap down, an element
is compared with its 2 children
(and swapped with the larger).
But only one element at each
level makes this comparison,
and a complete binary tree
with N nodes has only O(log2N) levels.
root 24 60 1 12 2 30 3 40 4 8 5 10 6 15 7 6 8 18 9 70 10

70. Heap Sort of N elements: How many comparisons?
(N/2) * O(log N) compares to create original heap
(N-1) * O(log N) compares for the sorting loop
= O ( N * log N) compares total
(reheap takes O(log N) )

71. Heap Sort A trace of heap sort, beginning with the heap
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72. Heap Sort A trace of heap sort, beginning with the heap
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73. Heap Sort A trace of heap sort, beginning with the heap
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