1. Heaps A heap is a binary tree.
A heap is best implemented in sequential representation (using an array).
Two important uses of heaps are:
(i) efficient implementation of priority queues
(ii) sorting -- Heapsort.

2. A Heap Any node’s key value is less than its children’s. 10 20 30 4050 25 55 52 42
Any node’s key value is
less than its children’s.

3. Sequential Representation of Trees
There are three methods of representing a binary tree using array representation.
1. Using index values to represent edges:
class Node {
Data elt;
int left;
int right;
} Node;
Node[] BinaryTree[TreeSize];

4. Method 1: Example Index Element Left Right 1 A 2 3 B 4 6 C D 5 I E 7 GD 5 I E 7 G

5. Method 2 2. Store the nodes in one of the natural traversals:
class Node {
Data elt;
boolean left;
boolean right; };
Node[] BinaryTree[TreeSize];

6. Method 2: Example Index Element Left Right 1 A T 2 B 3 D F 4 E 5 G 6 IC E G I
Index
Element
Left
Right 1 A T 2 B 3 D F 4 E 5 G 6 I 7 C
Elements stored in Pre-Order traversal

7. Method 3
3. Store the nodes in fixed positions: (i) root goes into first index, (ii) in general left child of tree[i] is stored in tree[2i] and right child in tree[2i+1].

8. Method 3: Example A D B C E G A B C D E - G 1 2 3 4 5 6 7 8 9 10 11 12

9. Heaps Heaps are represented sequentially using the third method.
Heap is a complete binary tree: shortest-path length tree with nodes on the lowest level in their leftmost positions.
Complete Binary Tree: let h be the height of the heap
for i = 0, … , h - 1, there are 2i nodes of depth i
at depth h - 1, the internal nodes are to the left of the external nodes

10. Heaps (Cont.)
Max-Heap has max element as root. Min-Heap has min element as root.
The elements in a heap satisfy heap conditions: for Min-Heap: key[parent] < key[left-child] or key[right-child].
The last node of a heap is the rightmost node of maximum depth 2 5 6
last node 9 7

11. Heap: An example [1] 10 [2] 20 30 40 [3] 25 [4] 50 [5] 42 [6] [7] 55
[8] 52
[9] 10 25 20 30 40 42 50 52 55
All the three arrangements
satisfy min heap conditions

12. Heap: An example [1] 10 [2] 20 30 40 [3] 25 [4] 50 [5] 42 [6] [7] 55
[8] 52
[9] 10 20 30 40 50 25 55 52 42
All the three arrangements
satisfy min heap conditions

13. Heap: An example [1] 10 [2] 20 30 40 [3] 25 [4] 50 [5] 42 [6] [7] 55
[8] 52
[9] 10 20 40 50 42 30 25 52 55
All the three arrangements
satisfy min heap conditions

14. Constructing Heaps There are two methods of constructing heaps:
Using SiftDown operation.
Using SiftUp operation.
SiftDown operation inserts a new element into the Heap from the top.
SiftUp operation inserts a new element into the Heap from the bottom.

15. ADT Heap Elements: The elements are called HeapElements.
Structure: The elements of the heap satisfy the heap conditions.
Domain: Bounded. Type name: Heap.

16. ADT Heap Operations: Method SiftUp (int n)
requires: Elements H[1],H[2],…,H[n-1] satisfy heap conditions.
results: Elements H[1],H[2],…,H[n] satisfy heap conditions.
Method SiftDown (int m,n)
requires: Elements H[m+1],H[m+2],…,H[n] satisfy the heap conditions.
results: Elements H[m],H[m+1],…,H[n] satisfy the heap conditions.
Method Heap (int n) // Constructor
results: Elements H[1],H[2],….H[n] satisfy the heap conditions.

17. ADT Heap: Element
public class HeapElement<T> { T data; Priority p; public HeapElement(T e, Priority pty) { data = e; p = pty; } // Setters/Getters...

18. Insertion into a Heap
Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap
The insertion algorithm consists of three steps
Find the insertion node z (the new last node)
Store k at z
Restore the heap-order property (discussed next)
© 2010 Goodrich, Tamassia
Heaps

19. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 5 6 z 9 7
insertion node
© 2010 Goodrich, Tamassia
Heaps

20. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 5 6 9 7 1
© 2010 Goodrich, Tamassia
Heaps

21. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 5 1 9 7 6
© 2010 Goodrich, Tamassia
Heaps

22. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 9 7 6
© 2010 Goodrich, Tamassia
Heaps

23. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 z 9 7 6
insertion node
© 2010 Goodrich, Tamassia
Heaps

24. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 9 7 6 10
© 2010 Goodrich, Tamassia
Heaps

25. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 9 7 6 10 z
© 2010 Goodrich, Tamassia
Heaps
insertion node

26. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 9 7 6 10 4
© 2010 Goodrich, Tamassia
Heaps

27. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 5 2 4 7 6 10 9
© 2010 Goodrich, Tamassia
Heaps

28. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 1 4 2 5 7 6 10 9
© 2010 Goodrich, Tamassia
Heaps

29. Removal from a Heap (§ 7.3.3)
Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap
The removal algorithm consists of three steps
Replace the root key with the key of the last node w
Remove w
Restore the heap-order property (discussed next) 2 5 6 w 9 7
last node 7 5 6 w 9
new last node
© 2010 Goodrich, Tamassia
Heaps

30. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 2 5 6 w 9 7
last node
© 2010 Goodrich, Tamassia
Heaps

31. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 w 9 7
last node
© 2010 Goodrich, Tamassia
Heaps

32. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 w 9
delete last node
© 2010 Goodrich, Tamassia
Heaps

33. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 9
DownHeap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

34. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 5 7 6 9
© 2010 Goodrich, Tamassia
Heaps

35. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 5 7 6 w 9
last node
© 2010 Goodrich, Tamassia
Heaps

36. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 7 6 w 9
last node
© 2010 Goodrich, Tamassia
Heaps

37. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 7 6 w
delete last node
© 2010 Goodrich, Tamassia
Heaps

38. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 7 6
DownHeap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

39. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 6 7 9
© 2010 Goodrich, Tamassia
Heaps

40. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 6 w 7 9
last node
© 2010 Goodrich, Tamassia
Heaps

41. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 w 7 9
last node
© 2010 Goodrich, Tamassia
Heaps

42. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 w 7
delete last node
© 2010 Goodrich, Tamassia
Heaps

43. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 9 7
DownHeap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

44. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 9
© 2010 Goodrich, Tamassia
Heaps

45. Updating the Last Node
The insertion node can be found by traversing a path of O(log n) nodes
Go up until a left child or the root is reached
If a left child is reached, go to the right child
Go down left until a leaf is reached
Similar algorithm for updating the last node after a removal
© 2010 Goodrich, Tamassia
Heaps

46. Heap-Sort
Consider a priority queue with n items implemented by means of a heap
the space used is O(n)
methods insert and removeMin take O(log n) time
methods size, isEmpty, and min take time O(1) time
Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time
The resulting algorithm is called heap-sort
Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort
© 2010 Goodrich, Tamassia
Heaps

47. Vector-based Heap Implementation
We can represent a heap with n keys by means of a vector of length n + 1
For the node at rank i
the left child is at rank 2i
the right child is at rank 2i + 1
Links between nodes are not explicitly stored
The cell of at rank 0 is not used
Operation insert corresponds to inserting at rank n + 1
Operation removeMin corresponds to removing at rank n
Yields in-place heap-sort 2 5 6 9 7 2 5 6 9 7 1 3 4
© 2010 Goodrich, Tamassia
Heaps

48. Bottom-up Heap Construction
We can construct a heap storing n given keys in using a bottom-up construction with log n phases
In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys
2i -1
2i+1-1
© 2010 Goodrich, Tamassia
Heaps

49. Example 16 15 4 12 6 7 23 20 25 5 11 27 16 15 4 12 6 7 23 20
© 2010 Goodrich, Tamassia
Heaps

50. Example (contd.) 25 5 11 27 16 15 4 12 6 9 23 20 15 4 6 23 16 25 5 12 11 9 27 20
© 2010 Goodrich, Tamassia
Heaps

51. Example (contd.) 7 8 15 4 6 23 16 25 5 12 11 9 27 20 4 6 15 5 8 23 16 25 7 12 11 9 27 20
© 2010 Goodrich, Tamassia
Heaps

52. Example (end) 10 4 6 15 5 8 23 16 25 7 12 11 9 27 20 4 5 6 15 7 8 23 16 25 10 12 11 9 27 20
© 2010 Goodrich, Tamassia
Heaps

53. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 3 2 8 5 4 6
© 2010 Goodrich, Tamassia
Heaps

54. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property k 7 3 2 8 5 4 6
© 2010 Goodrich, Tamassia
Heaps

55. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property k 7 3 2 8 5 4 6
Merge
© 2010 Goodrich, Tamassia
Heaps

56. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 7 3 2 8 5 4 6
Downheap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

57. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 2 3 7 8 5 4 6
Downheap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

58. Merging Two Heaps We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 2 3 4 8 5 7 6
Downheap/SiftDown
© 2010 Goodrich, Tamassia
Heaps

59. Heaps and Priority Queues
We can use a heap to implement a priority queue
We store a (key, element) item at each internal node
We keep track of the position of the last node
(2, Sue)
(5, Pat)
(6, Mark)
(9, Jeff)
(7, Anna)
© 2010 Goodrich, Tamassia
Heaps

60. Priority Queue as Heap Representation as a Heap
public class HeapPQ<T> {
Heap<T> pq;
/** Creates a new instance of HeapPQ */
public HeapPQ() {
pq = new Heap(10); }

61. Priority Queue as Heap Representation as a Heap
public class HeapPQ<T> {
Heap<T> pq;
/** Creates a new instance of HeapPQ */
public HeapPQ() {
pq = new Heap(10); }
Create new heap (maxsize=10)

62. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;

63. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Create new heap element (data/priority)

64. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Insert the element at the end of heap, and SiftUp

65. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Each enqueue is inserted in the heap and Sifted Up.
This ensures the element with the lowest priority (min-heap)
or the highest priority (max-heap) is at the top of the heap (index 1).

66. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Get data of the root of the heap (index 1) and store it in e

67. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Get priority of the root of the heap (index 1)

68. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Set “pty” value to the root’s priority value

69. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Copy the last node into the root (delete root)

70. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Update the size to delete last node (delete duplicate node)

71. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
SiftDown the copied last element from the root
down the tree until it satisfies the heap conditions

72. Priority Queue as Heap public void enqueue(T e, Priority pty){
HeapElement<T> x = new HeapElement<T>(e, pty);
pq.SiftUp(x); }
public T serve(Priority pty){
T e;
Priority p;
e = pq.heap[1].get_data();
p = pq.heap[1].get_priority();
pty.set_value(p.get_value());
pq.heap[1] = pq.heap[pq.size];
pq.size--;
pq.SiftDown(1);
return e;
Return e (the deleted root)

73. Priority Queue as Heap public int length(){ return pq.size; }
public boolean full(){
return false;

74. HeapSort Heap can be used for sorting. Two step process:
Step 1: the data is put in a heap.
Step 2: the data are extracted from the heap in sorted order.
HeapSort based on the idea that heap always has the smallest or largest element at the root.

75. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());

76. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
Loop the number of elements in the array

77. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
Swap first element (root) with last element
In the heap array

78. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
Each time reduce the size
(size define last element which will be used with swap/SiftDown)

79. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
SiftDown the swapped element (last element) from the root down the heap until it satisfies heap conditions.

80. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
Repeat until the entire heap array is sorted
If it is min-heap, it will be sorted in descending order
If it is max-heap, it will be sorted in ascending order

81. ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void HeapSort(){
while(size > 1){
swap(heap, 1, size);
size--;
SiftDown(1); }
//Display the sorted elements.
for(int i = 1; i <= maxsize; i++){
System.out.println(heap[i].get_priority().get_value());
Print the sorted array
(print priorities values)