Submitted by:-- ABHISHEK

KUMAR ABHINAV KUMAR

0132cs0710060132cs071004

CS(4th Sem)

A Binary Heap is a priority Queue which can be store in an ARRAY and can be represented as a COMPLETE BINARY TREE. All levels of the tree except possibly the last one are fully filled.If the last level of tree is not complete then nodes are inserted from LEFT to RIGHT direction.

Let elements 75,19,40,15,10,25,30,2,1 are stored in an array then array A is:-

It can be represented as a HEAP data structure as:-

75

19 40

15 10 25 30

2 1

75

19

40

15

10

25 30 2

1

1.BINARY MAX HEAP

2.BINARY MIN HEAP

A binary maxheap is a heap tree in which the Root node is always greater than its child nodes & all subtree follow the same rule. 75

19 40

15 10 25 30

A Binary minheap is a heap tree in which the root nodeis always less than its child node and the same rule is followed by their subtree.

1

3 5

7 8 10 12

Operations Performed on A Heap:-1)Inserting an element to the Heap2)Deleting an element from the Heap

Insertion to the Heap:-Insertion is done in such a way that after inserting element this remains a maxheap.•Step for inserting element:--1)Add the element on the bottom level of heap.2)Compare the added element with its parent,if they are in the correct order,stop.3)If not,Swap the element with its parent and return to the previous step.

Time complexity of insert algorithm:--

If the value in the heap is less than the value of present node then this is the best case and the best case complexity is CONSTANT.

In worst case,the time complexity of algorithm is equal to the height of complete binary tree i.e log2n.

Insert 15 to the maxheap 11,5,8,3,4.

11

5 8

3 4

11

5 8

3 4 15

Insert 15

Since 15>8 then the Heap becomes--

11

5 15

3 4 8Since 15>11 then the Heap becomes--

15

5 11

3 4 8

2) Deletion from the Heap:-

Deletion is done from the root node either it is maxheap or minheap.It is done as follows:-

•Delete the root node from maxheap or minheap.• place the last element of heap on the root node and adjust the heap to get the maxheap or minheap.

Time complexity of Delete algorithm:-

The time complexity of delete algorithm for a maxheap or minheap is equal to the order of

log2n.

Delete 9 from the Maxheap

9,7,6,3,2,5,4

9

7 6

3 2 5 4

Delete 97 6

3 2 5 4

4

7 6

3 2 5

After Adjust procedure

7

4 6

3 2 5

Heap sort:- Heapsort is a comparision based sorting algorithm.In heapsort ,it begins by building a heap out of the data set and then removing the largest item and placing it at the end of sorted array.After removing the largest item,it reconstruct the heap and removes the largest remaining item and places it in the next open position from the end of the sorted array.This is repeated until there are no item left in the heap and the sorted array is full.

Time complexity of Heapsort algorithm:-The time complexity of Heapify algorithm is

equal to nlog2n.

Let us assume an array of elements 5,2,9,3,7,6,4 then call Heapify algorithm to convert the complete Binary tree to the Maxheap.Before Heapify:-

9

7 6

3 2 5 4

After Heapify:-

5

2 9

3 7 6 4

Pass 1: 4

7 6

3 2 5 99

7

4 6

3 2 5 9999

Pass 2: 5

4 6

3 2 7 9999

6

4 5

3 2 7 9999

Pass 3: 2

4 5

3 6 7 9999

5

4 2

3 6 7 9999

Pass 4: 3

4 2

5 6 7 9999

4

3 2

5 6 7 9999

Pass 5: 2

3 4

5 6 7 9999

3

2 4

5 6 7 9999

Pass 6:

2

3 4

5 6 7 9999

Required Sorted Tree

Comparision between diff. sorting techniques:-

Sorting Tech.

Av. Case time comp.

Worst case time comp.

Space comp.

Insertion Sort

O(n^2) O(n^2) No extra memory required

Bubble sort

O(n^2) O(n^2) No extra

memory required

Quick sort O(n^2)

O(nlogn)No extra memory required

Merge sort O(nlogn)

O(nlogn)

o(n)

Heap sort O(nlogn)

O(nlogn)

No extra memory required

From above Comparision it is clear that heap sort is the best sorting Technique among all acc. To the Time as well as Space complexity.