Question

Define the term Heap. Illustrate the tree changes in the bottom-up Make Heap algorithm applied to construct a max-Heap for the following list of values. [1,2,3,4,5,6,7]

Answer 1

Heap is a binary tree based data structure. It is an array object that can be viewed as a nearly complete binary tree. Each node of the tree corresponds to an element of the array that stores the value in the node. The heap tree is completely filled on all levels except possibly the lowest. Lowest level possibly filled from left up to a point.

Heap tree is categorized into max-heap and min-heap.

Max-heap: Every nodes of a max-heap except root node has the "less than or equal to" ( $\leq$ ) relation with its parent node.

Min-heap: Every nodes of a min-heap except root node has the "greater than or equal to" ( $\geq$ ) relation with its parent node.

A heap can be stored as an array. Here we need to build a max-heap tree for a given array [1,2,3,4,5,6,7]. The algorithm MakeHeap(A) we are using here contains a subroutine that is Max-Heapify(A,i).

Assume that given array is A and its elements stored in the location 1 through 7.

1

2

3

4

5

6

7

We can use MakeHeap(A) and Max-Heapify(A,i) to build a max-heap of the array A.

Length of array = length(A) = 7

Procedure for MakeHeap(A) and Max-Heapify(A,i) are given in the attachment. Illustration of the changes in the array and tree also mentioned in attachment.

aus el IF node curo - Procedure We can repousent biscury korea array. the parent localed at ith location of correy, then its lett child and child will be at aith and right (2i+15th location suspectively for Make Heap CA) Make HeapCA) 1. heap-size(A) = deng}CA) 2. for i= L long thLAJ/2] down to 1 do Max-Heapify CA, i) Procedure for Max Heapify CA, 1) Max-Heapify CA,1) 1.1= Left (1) 3. 2.91 = -l 4: 5 Right (i) 3. it is heap-size (A) and A[1] >ACIJ then largest else largest 6. if us heap-size CA) and ACAT > A[largest then largest if largest i then exchange ADI <> -A [largest Max-Heapify CA, borgest) =91 7. 8 9. 10.

the -Now We illustrabing changes array by using the maketteap() algorith of tree and rithm to constouct a max-beap au: Step 1: Initially and Tree Array 0 A 141567 1 2 3 4 5 6 7 Step 2: Figca) Initial call of Make Heap (A) makes the value of i=3 and call Max-Hecipity (A,3). Aftes first iteration, values location 3 and it will 7 interchange. in the Q 3 A574 1 2 3 4 5 6 7 Step 3 When i becomes 2 2 Fig (b) - in the second iteration, values in and and 5th location will interchange. These are shown ith lig (5) When & becomes 3 in the last iteration of Makeheap (A). algorith will call lax Heeipity CA, 1). This interchange values of 1 and 3. There is another recursive call Max Hecepity (A, 3). This call will interchange the values of the location Call the location 3 and 6. 7 2 6 7 <> A7564 G 3 ~ 2 3 4 5 6 7 (2 4 Figle)

In fig (c) we can see the final array and max-heap of given elements.

Note: MakeHeap() algorithm using a bottom-up approach to heapify the given array. In thi sexample the length of the array is 7. MakeHeap() procedure calls the Max-Heapify() for each value of i. ie, 3,2 and 1.

I hope I have answered your query.