![Figure : Sample max heaps
[4]
14
12 7
810 6
9
6 3
5
30
25
[1]
[2] [3]
[5] [6]
[1]
[2] [3]
[4]
[1]
[2]
Property:
The root of max heap (min heap) contains
the largest (smallest).](https://image.slidesharecdn.com/ds13-heap-180827163410/85/Data-Structure-and-Algorithms-Heaps-and-Trees-3-320.jpg)
![2
7 4
810 6
10
20 83
50
11
21
[1]
[2] [3]
[5] [6]
[1]
[2] [3]
[4]
[1]
[2]
[4]
Figure : Sample min heaps](https://image.slidesharecdn.com/ds13-heap-180827163410/85/Data-Structure-and-Algorithms-Heaps-and-Trees-4-320.jpg)
![Insertion into a Max Heap
INSHEAP(TREE, N, ITEM)
1. Set N=N+1 and PTR=N.
2. Repeat Steps 3 to 6 while PTR>1.
3. Set PAR= └PTR/2┘
4. If ITEM<= TREE[PAR], then:
Set TREE[PTR]=ITEM, and Return.
[End of If structure]
5. Set TREE[PTR]=TREE[PAR]
6. Set PTR=PAR.
[End of step 2]
7. Set TREE[1]=ITEM
8. Return.](https://image.slidesharecdn.com/ds13-heap-180827163410/85/Data-Structure-and-Algorithms-Heaps-and-Trees-6-320.jpg)
![Deletion from a Max Heap
DELHEAP(TREE, N, ITEM)
1. Set ITEM= TREE[1].
2. Set LAST= TREE[N] and N= N-1.
3. Set PTR=1, LEFT=2 and RIGHT=3.
4. Repeat Steps 5 to 7 while RIGHT<=N:
5. If LAST>=TREE[LEFT] and LAST>= TREE[RIGHT], then:
Set TREE[PTR]=LAST and Return.
[End of If structure.]
1. If TREE[RIGHT]<= TREE[LEFT], then:
Set TREE[PTR]= TREE[LEFT] and PTR=LEFT.
Else:
Set TREE[PTR]= TREE[RIGHT] and PTR=RIGHT.
[End of If structure.]
1. Set LEFT=2*PTR and RIGHT=LEFT+1.
[End of Step 4 loop.]
8. If LEFT=N and if LAST < TREE[LEFT], TREE[PTR]= TREE[LEFT]
then: Set PTR=LEFT.
9. Set TREE[PTR]= LAST.
10. Return.](https://image.slidesharecdn.com/ds13-heap-180827163410/85/Data-Structure-and-Algorithms-Heaps-and-Trees-8-320.jpg)
![HeapSort
HEAPSORT(A, N)
1. Repeat for J=1 to N-1:
Call INSHEAP(A, J, A[J+1]).
[End of loop.]
1. Repeat while N>1:
a) Call DELHEAP(A, N, ITEM).
b) A[N+1]=ITEM.
[End of loop.]
1. Exit.](https://image.slidesharecdn.com/ds13-heap-180827163410/85/Data-Structure-and-Algorithms-Heaps-and-Trees-9-320.jpg)
Data Structure and Algorithms Heaps and Trees
- 2. 2 Heap A max tree is a tree in which the key value in each node is no smaller than the key values in its children. A max heap is a complete binary tree that is also a max tree. A min tree is a tree in which the key value in each node is no larger than the key values in its children. A min heap is a complete binary tree that is also a min tree. Operations on heaps creation of an empty heap insertion of a new element into the heap; deletion of the largest element from the heap
- 3. Figure : Sample max heaps [4] 14 12 7 810 6 9 6 3 5 30 25 [1] [2] [3] [5] [6] [1] [2] [3] [4] [1] [2] Property: The root of max heap (min heap) contains the largest (smallest).
- 4. 2 7 4 810 6 10 20 83 50 11 21 [1] [2] [3] [5] [6] [1] [2] [3] [4] [1] [2] [4] Figure : Sample min heaps
- 5. 5 Example of Insertion to Max Heap 20 15 2 14 10 initial location of new node 21 15 20 14 10 2 insert 21 into heap 20 15 5 14 10 2 insert 5 into heap
- 6. Insertion into a Max Heap INSHEAP(TREE, N, ITEM) 1. Set N=N+1 and PTR=N. 2. Repeat Steps 3 to 6 while PTR>1. 3. Set PAR= └PTR/2┘ 4. If ITEM<= TREE[PAR], then: Set TREE[PTR]=ITEM, and Return. [End of If structure] 5. Set TREE[PTR]=TREE[PAR] 6. Set PTR=PAR. [End of step 2] 7. Set TREE[1]=ITEM 8. Return.
- 7. 7 Example of Deletion from Max Heap 20 remove 15 2 14 10 10 15 2 14 15 14 2 10
- 8. Deletion from a Max Heap DELHEAP(TREE, N, ITEM) 1. Set ITEM= TREE[1]. 2. Set LAST= TREE[N] and N= N-1. 3. Set PTR=1, LEFT=2 and RIGHT=3. 4. Repeat Steps 5 to 7 while RIGHT<=N: 5. If LAST>=TREE[LEFT] and LAST>= TREE[RIGHT], then: Set TREE[PTR]=LAST and Return. [End of If structure.] 1. If TREE[RIGHT]<= TREE[LEFT], then: Set TREE[PTR]= TREE[LEFT] and PTR=LEFT. Else: Set TREE[PTR]= TREE[RIGHT] and PTR=RIGHT. [End of If structure.] 1. Set LEFT=2*PTR and RIGHT=LEFT+1. [End of Step 4 loop.] 8. If LEFT=N and if LAST < TREE[LEFT], TREE[PTR]= TREE[LEFT] then: Set PTR=LEFT. 9. Set TREE[PTR]= LAST. 10. Return.
- 9. HeapSort HEAPSORT(A, N) 1. Repeat for J=1 to N-1: Call INSHEAP(A, J, A[J+1]). [End of loop.] 1. Repeat while N>1: a) Call DELHEAP(A, N, ITEM). b) A[N+1]=ITEM. [End of loop.] 1. Exit.