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1. Data Structures
Topic: Heap
By
Chandani Bhasin
Asst. Professor,
Lovely Professional University, Punjab

2. Contents
• Introduction
• Insertion in Heap
• Deleting the Root of Heap
• Heap Sort
• Heap Sort Complexity
• Review Questions

3. Introduction
• Heap (MaxHeap): A complete binary tree H with
n elements is called a Heap if each node N of H has the
following property:
“ The value at N is greater than or equal to the value
at each of the children of N.”
• If the value at N is less than or equal to the value at each of the
children of N, then it is called MinHeap.
• Heaps are maintained in memory by using linear array TREE.

4. Work Space
Create a Max Heap - 14,24,12,11,25,8,35

5. Create Min Heap
145,40,25,65,12,48,18,1,100,27,7,3,45,9,30

6. HEAPIFY TOTAL NUMBER
OF SWAPS(S)

7. Insertion in a Heap
• INSERT_HEAP(TREE, N, ITEM)
1.Set N = N+1, and PTR = N.
2.Repeat Step 3 to 6 while PTR > 1
3. Set PAR =
4. If ITEM <= TREE [PAR], then:
Set TREE[PTR] = ITEM and Return.
5. Else: Set TREE[PTR] = TREE[PAR].
6. Set PTR = PAR.
7.Set TREE[1] = ITEM.
8.Return.
/ 2
PTR
 
 

8. Work Space
200,150,350,100,70,110,250,50

9. Example
• Create a Heap from the following list of numbers:
40, 30, 50, 20, 60, 55, 70, 60, 65, 50

10. Deletion of Root of a Heap
•Can we directly delete any element from the heap?

11. • You can delete either from the root or the
rightmost element from last level.
•Assign Root R to some variable ITEM.
• Replace the deleted node R by the last node
L of Heap H so that H is still complete tree,
but not a Heap.
• Call MaxHeapify/MinHeapify to maintain
the heap property.

12. Complexity to delete an element
• Best Case – O(n)
• Worst Case – O(n log n)

13. Consider the following keys
• 1,2,3,4,5,6,7
• Perform the deletion operation.
• Step 1-Delete from the root
• Step 2- Replace it with rightmost element
from the last level(Heapify)

14. Work Space

15. Complexity
• To delete one element- O(1) and then to
arrange it – log n
• For n elements:-
Best Case-O(n)
Worst Case-O(n log n)

16. 16
Example of Deletion from Max Heap
20
remove
15 2
14 10
10
15 2
14
15
14 2
10

17. 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.]
6. 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.]
7. Set LEFT=2*PTR and RIGHT=LEFT+1.
[End of Step 4 loop.]
8. If LEFT=N and if LAST < TREE[LEFT] set TREE[PTR]= TREE[LEFT]
Set PTR=LEFT.
9. Set TREE[PTR]= LAST.
10. Return.

18. HEAP SORT(INPLACE,UNSTABLE,O(N LOG N)
TO PERFORM HEAP SORT:-
STEP 1- INSERTION IN HEAP
STEP 2 – DELETION IN HEAP

19. Work Space
• CONSIDER THE FOLLOWING KEYS:-
• 4,6,10,9,2
• PRINT IT IN ASCENDING ORDER

20. COMPLEXITY
• BUILDINGO(n)
• DELETION O(n log n)
• HEAP SORTO(n) + O(n log n)
• - O(n log n)

21. Build MaxHeap
BUILD_MAX-HEAP(A)
1.heapsize[A] = length[A]
2.Repeat for i = └ length[A]/2 ┘ to 1
3. Call MAX_HEAPIFY(A, i)
4.Exit

22. Maintaining Heap Property
MAX_HEAPIFY(A, i)
1.Set: l = LEFT (i)
2.Set: r = RIGHT (i)
3.If l <= heapsize [A] and A[l] > A[i], then:
4. largest = l.
5.Else: largest = i.
6.If r <= heapsize [A] and A[r] > A[largest], then:
7. largest = r.
8.If largest != i, then:
9. Exchange A[i] A[largest].
10. MAX_HEAPIFY (A, largest).

23. Heap Sort
HEAP_SORT (A)
1.BUILD_MAXHEAP (A)
2.Repeat for i = length[A] to 2
3. Exchange A[1] A[i].
4. Heapsize[A] = Heapsize [A] ─ 1.
5. Call MAX_HEAPIFY(A, 1).
6.Exit.

24. Complexity of Heap Sort
Average and Worst case Complexity of Heap sort = O(nlogn).

25. Questions

26. Review Questions
• What do you mean by Heap?
• What is the complexity of Heap sort?
• Calculate the complexity of Heap sort.
• How will you insert an element in a heap?
• What is the difference between Maxheap and Minheap?