How to check if a given array represents a Binary Heap?

Given an array, how to check if the given array represents a Binary Max-Heap.

Examples:

Input:  arr[] = {90, 15, 10, 7, 12, 2} 
Output: True
The given array represents below tree
       90
     /    \
   15      10
  /  \     /
 7    12  2 
The tree follows max-heap property as every
node is greater than all of its descendants.

Input:  arr[] = {9, 15, 10, 7, 12, 11} 
Output: False
The given array represents below tree
       9
     /    \
   15      10
  /  \     /
 7    12  11
The tree doesn't follows max-heap property 9 is 
smaller than 15 and 10, and 10 is smaller than 11. 

A Simple Solution is to first check root, if it’s greater than all of its descendants. Then check for children of root. Time complexity of this solution is O(n2)

An Efficient Solution is to compare root only with its children (not all descendants), if root is greater than its children and same is true for for all nodes, then tree is max-heap (This conclusion is based on transitive property of > operator, i.e., if x > y and y > z, then x > z).

The last internal node is present at index (2n-2)/2 assuming that indexing begins with 0.

Below is C++ implementation of this solution.

// C program to check whether a given array
// represents a max-heap or not
#include <stdio.h>
#include <limits.h>

// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[], int i, int n)
{
   // If a leaf node
   if (i > (n - 2)/2)
       return true;

   // If an internal node and is greater than its children, and
   // same is recursively true for the children
   if (arr[i] >= arr[2*i + 1]  &&  arr[i] >= arr[2*i + 2] &&
       isHeap(arr, 2*i + 1, n) && isHeap(arr, 2*i + 2, n))
       return true;

   return false;
}

// Driver program
int main()
{
    int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = sizeof(arr) / sizeof(int);

    isHeap(arr, 0, n)? printf("Yes"): printf("No");

    return 0;
}

Output:

Yes

Time complexity of this solution is O(n). The solution is similar to preorder traversal of Binary Tree.

Thanks to Utkarsh Trivedi for suggesting the above solution.

An Iterative Solution is to traverse all internal nodes and check id node is greater than its children or not.

// C program to check whether a given array
// represents a max-heap or not
#include <stdio.h>
#include <limits.h>

// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[],  int n)
{
    // Start from root and go till the last internal
    // node
    for (int i=0; i<=(n-2)/2; i++)
    {
        // If left child is greater, return false
        if (arr[2*i +1] > arr[i])
                return false;

        // If right child is greater, return false
        if (arr[2*i+2] > arr[i])
                return false;
    }
    return true;
}

// Driver program
int main()
{
    int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = sizeof(arr) / sizeof(int);

    isHeap(arr, n)? printf("Yes"): printf("No");

    return 0;
}

Output:

Yes

Thanks to Himanshu for suggesting this solution.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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