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 the implementation of this solution.

C++

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// 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)-1;
  
    isHeap(arr, 0, n)? printf("Yes"): printf("No");
  
    return 0;
}

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Java

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// Java program to check whether a given array 
// represents a max-heap or not
  
class GFG {
  
// Returns true if arr[i..n-1] represents a 
// max-heap 
    static boolean 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 
    public static void main(String[] args) {
        int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
        int n = arr.length-1;
        if (isHeap(arr, 0, n)) {
            System.out.println("Yes");
        } else {
            System.out.println("No");
        }
    }
}
  
//This code contributed by 29AjayKumar 

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C#

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// C# program to check whether a given  
// array represents a max-heap or not
using System;
  
class GFG
{
  
// Returns true if arr[i..n-1] represents a 
// max-heap 
static 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 Code 
public static void Main(String[] args)
{
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = arr.Length-1;
    if (isHeap(arr, 0, n))
    {
        Console.Write("Yes");
    
      
    else
    {
        Console.Write("No");
    }
}
}

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PHP

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<?php
// PHP program to check whether a given 
// array represents a max-heap or not
  
// Returns true if arr[i..n-1] 
// represents a max-heap
function isHeap($arr, $i, $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 Code
$arr = array(90, 15, 10, 7, 12, 2, 7, 3);
$n = sizeof($arr);
  
if(isHeap($arr, 0, $n))
    echo "Yes";
else
    echo "No";
  
// This code is contributed 
// by Akanksha Rai
?>

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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++

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// 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 (2*i+2 < n && 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;
}

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Java

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// Java program to check whether a given array 
// represents a max-heap or not
  
class GFG {
  
// Returns true if arr[i..n-1] represents a 
// max-heap 
    static boolean 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 (2 * i + 2 < n && arr[2 * i + 2] > arr[i]) {
                return false;
            }
        }
        return true;
    }
  
// Driver program 
    public static void main(String[] args) {
        int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
        int n = arr.length;
        if (isHeap(arr, n)) {
            System.out.println("Yes");
        } else {
            System.out.println("No");
        }
    }
}
// This code is contributed by 29AjayKumar

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C#

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// C# program to check whether a given array 
// represents a max-heap or not 
using System;
  
class GFG 
{
  
// Returns true if arr[i..n-1] 
// represents a max-heap 
static 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 (2 * i + 2 < n && arr[2 * i + 2] > arr[i]) 
        
            return false
        
    
    return true
  
// Driver Code 
public static void Main() 
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3}; 
    int n = arr.Length; 
    if (isHeap(arr, n))
    
        Console.Write("Yes"); 
    
    else 
    
        Console.Write("No"); 
    
  
// This code is contributed 
// by 29AjayKumar

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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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