The document discusses heaps and heapsort. It defines max heaps and min heaps as complete binary trees where each node's key value is greater than or less than its children's key values, respectively. It describes operations on heaps like insertion, deletion of the max/min element, and creation of an empty heap. Algorithms for insertion and deletion into max heaps are provided, with examples. Heapsort is described as using insertion to build a max heap of the input array and then deleting elements to sort the array.

1. insertion,
deletion,
sorting

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
8
10 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
8
10 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.