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

2. Contents The ADT Heap An Array-Based Implementation of a Heap
A Heap Implementation of the ADT Priority Queue
Heap Sort
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

3. The ADT Heap Definition A heap is a complete binary tree that either
is empty … or …
it’s root
Contains a value greater than or equal to the value in each of its children, and
Has heaps as its subtrees
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

4. Complete Binary Tree
(1) A binary tree that is either full or full through the next-to-last level
(2) The last level is full from left to right (i.e., leaves are as far to the left as possible)
Complete Binary Tree

5. 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.

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

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

8. Unique?

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

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

11. 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

12. 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]

13. The ADT Heap (a) A maxheap and (b) a minheap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

14. View interface, Listing 17-1
The ADT Heap
UML diagram for the class Heap
View interface, Listing 17-1
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

15. Array-Based Implementation of a Heap
(a) Level-by-level numbering of a complete binary tree; (b) its array-based implementation
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

16. Algorithms for Array-Based Heap Operations
(a) Disjoint heaps after removing the heap’s root;
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

17. Algorithms for Array-Based Heap Operations
(b) a semiheap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

18. Algorithms for Array-Based Heap Operations
(c) the restored heap
View algorithm to make this conversion, Listing 17-A
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

19. Reheap Down
bottom

20. Algorithms for Array-Based Heap Operations
The array representation the heap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

21. Algorithms for Array-Based Heap Operations
The array representation of the semiheap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

22. Algorithms for Array-Based Heap Operations
The array representation of the restored heap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

23. Algorithms for Array-Based Heap Operations
Recursive calls to heapRebuild
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

24. Algorithms for Array-Based Heap Operations
Insertion into a heap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

25. Reheap Up
bottom

26. Algorithms for Array-Based Heap Operations
Pseudocode for add
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

27. The Implementation The header file for class ArrayMaxHeap
An array-based implementation of the ADT heap
View Listing 17-2.
This heap is a maxheap.
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

28. The Implementation Definition of constructor
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

29. The Implementation
(a) The initial contents of an array; (b) the array’s corresponding complete binary tree
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

30. The Implementation Transforming an array into a heap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

31. The Implementation Transforming an array into a heap
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

32. 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

33. The Implementation Method heapCreate
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

34. The Implementation Method peekTop
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

35. Heap Implementation of the ADT Priority Queue
Using a heap to define a priority queue results in a more time-efficient implementation
Listing 17-3 contains a header file for a class of priority queues.
View implementation, Listing 17-4
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

36. Heap Sort Heap sort partitions an array into two regions
View heap sort algorithm, Listing 17-B
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

37. Heap Sort A trace of heap sort, beginning with the heap in Figure 17-9
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

38. Heap Sort A trace of heap sort, beginning with the heap in Figure 17-9
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

39. Heap Sort A trace of heap sort, beginning with the heap in Figure 17-9
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

40. Heap Sort A trace of heap sort, beginning with the heap in Figure 17-9
Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013