Data structures

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.