Introduction to Max-Heap – Data Structure and Algorithm Tutorials

A Max-Heap is defined as a type of Heap Data Structure in which each internal node is greater than or equal to its children. 

The heap data structure is a type of binary tree that is commonly used in computer science for various purposes, including sorting, searching, and organizing data.

Introduction to Max-Heap Data Structure

Purpose and Use Cases of Max-Heap:

• Priority Queue: One of the primary uses of the heap data structure is for implementing priority queues. 
• Heap Sort: The heap data structure is also used in sorting algorithms.  
• Memory Management: The heap data structure is also used in memory management. When a program needs to allocate memory dynamically, it uses the heap data structure to keep track of the available memory. 
• Graph Algorithms: The heap data structure is used in various graph algorithms. For example, Dijkstra’s shortest path algorithm uses a heap data structure to keep track of the vertices with the shortest path from the source vertex. 

Max-Heap Data structure in Different languages:

1. Max-Heap in C++

A max heap can be implemented using the priority_queue container from the Standard Template Library (STL). The priority_queue container is a type of container adapter that provides a way to store elements in a queue-like data structure in which each element has a priority associated with it.

Syntax: priority_queue<int>maxH;

2. Max-Heap in Java

In Java, a max heap can be implemented using the PriorityQueue class from java.util package. The PriorityQueue class is a priority queue that provides a way to store elements in a queue-like data structure in which each element has a priority associated with it.

Syntax: PriorityQueue maxHeap= new PriorityQueue<>(Comparator.reverseOrder());

3. Max-Heap in Python 

In Python, a max heap can be implemented using the heapq module, which provides functions for implementing heaps. Specifically, the heapq module provides a way to create and manipulate heap data structures.

Syntax:  heap = []
              heapify(heap)

4. Max-Heap in C#

In C#, a max heap can be implemented using the PriorityQueue<T> class from the System.Collections.Generic namespace. The PriorityQueue<T> class is a priority queue that provides a way to store elements in a queue-like data structure in which each element has a priority associated with it.

Syntax: var maxHeap = new PriorityQueue<int>((a, b) => b - a);

5. Max-Heap in JavaScript

A max heap is a binary tree where every node has a value greater than or equal to its children. In JavaScript, you can implement a max heap using an array, where the first element represents the root node, and the children of a node at index i are located at indices 2i+1 and 2i+2.

Syntax: const miaxHeap = new MaxHeap();

Difference between Max and Min Heap

	Min Heap	Max Heap
1.	In a Min-Heap the key present at the root node must be less than or equal to among the keys present at all of its children.	In a Max-Heap the key present at the root node must be greater than or equal to among the keys present at all of its children.
2.	In a Min-Heap the minimum key element present at the root.	In a Max-Heap the maximum key element present at the root.
3.	A Min-Heap uses the ascending priority.	A Max-Heap uses the descending priority.
4.	In the construction of a Min-Heap, the smallest element has priority.	In the construction of a Max-Heap, the largest element has priority.
5.	In a Min-Heap, the smallest element is the first to be popped from the heap.	In a Max-Heap, the largest element is the first to be popped from the heap.

Internal Implementation of Max-Heap Data Structure:

A Min heap is typically represented as an array. 

• The root element will be at Arr[0]. 
• For any ith node Arr[i].
  • left child is stored at index 2i+1
  • Right child is stored at index 2i+2
  • Parent is stored at index floor((i-1)/2)

The Internal Implementation of the Max-Heap require 3 major steps:

1. Insertion: To insert a new element into the heap, it is added to the end of the array and then “bubbled up” until it satisfies the heap property.
2. Deletion: To delete the maximum element (the root of the heap), the last element in the array is swapped with the root, and the new root is “bubbled down” until it satisfies the heap property.
3. Heapify: A heapify operation can be used to create a max heap from an unsorted array. 

Operations on Max-heap Data Structure and their Implementation:

Here are some common operations that can be performed on a Heap Data Structure data structure,

1. Insertion in Max-Heap Data Structure:  

Elements can be inserted to the heap following a similar approach as discussed above for deletion. The idea is to: 

• First increase the heap size by 1, so that it can store the new element.
• Insert the new element at the end of the Heap.
• This newly inserted element may distort the properties of Heap for its parents. So, in order to keep the properties of Heap, heapify this newly inserted element following a bottom-up approach.

Illustration:

Suppose the Heap is a Max-Heap as:

Insertion in Max-Heap

Implementation of insertion operation in Max-Heap:

C++

// C++ program to insert new element to Heap

#include <iostream>
using namespace std;

#define MAX 1000 // Max size of Heap

// Function to heapify ith node in a Heap
// of size n following a Bottom-up approach
void heapify(int arr[], int n, int i)
{
    // Find parent
    int parent = (i - 1) / 2;

    if (arr[parent] > 0) {
        // For Max-Heap
        // If current node is greater than its parent
        // Swap both of them and call heapify again
        // for the parent
        if (arr[i] > arr[parent]) {
            swap(arr[i], arr[parent]);

            // Recursively heapify the parent node
            heapify(arr, n, parent);
        }
    }
}

// Function to insert a new node to the Heap
void insertNode(int arr[], int& n, int Key)
{
    // Increase the size of Heap by 1
    n = n + 1;

    // Insert the element at end of Heap
    arr[n - 1] = Key;

    // Heapify the new node following a
    // Bottom-up approach
    heapify(arr, n, n - 1);
}

// A utility function to print array of size n
void printArray(int arr[], int n)
{
    for (int i = 0; i < n; ++i)
        cout << arr[i] << " ";

    cout << "\n";
}

// Driver Code
int main()
{
    // Array representation of Max-Heap
    // 10
    // / \
    // 5 3
    // / \
    // 2 4
    int arr[MAX] = { 10, 5, 3, 2, 4 };

    int n = 5;

    int key = 15;

    insertNode(arr, n, key);

    printArray(arr, n);
    // Final Heap will be:
    // 15
    // / \
    // 5     10
    // / \ /
    // 2 4 3
    return 0;
}

Java

// Java program for implementing insertion in Heaps
public class insertionHeap {

    // Function to heapify ith node in a Heap
    // of size n following a Bottom-up approach
    static void heapify(int[] arr, int n, int i)
    {
        // Find parent
        int parent = (i - 1) / 2;

        if (arr[parent] > 0) {
            // For Max-Heap
            // If current node is greater than its parent
            // Swap both of them and call heapify again
            // for the parent
            if (arr[i] > arr[parent]) {

                // swap arr[i] and arr[parent]
                int temp = arr[i];
                arr[i] = arr[parent];
                arr[parent] = temp;

                // Recursively heapify the parent node
                heapify(arr, n, parent);
            }
        }
    }

    // Function to insert a new node to the heap.
    static int insertNode(int[] arr, int n, int Key)
    {
        // Increase the size of Heap by 1
        n = n + 1;

        // Insert the element at end of Heap
        arr[n - 1] = Key;

        // Heapify the new node following a
        // Bottom-up approach
        heapify(arr, n, n - 1);

        // return new size of Heap
        return n;
    }

    /* A utility function to print array of size n */
    static void printArray(int[] arr, int n)
    {
        for (int i = 0; i < n; ++i)
            System.out.println(arr[i] + " ");

        System.out.println();
    }

    // Driver Code
    public static void main(String args[])
    {
        // Array representation of Max-Heap
        // 10
        // / \
        // 5 3
        // / \
        // 2 4

        // maximum size of the array
        int MAX = 1000;
        int[] arr = new int[MAX];

        // initializing some values
        arr[0] = 10;
        arr[1] = 5;
        arr[2] = 3;
        arr[3] = 2;
        arr[4] = 4;

        // Current size of the array
        int n = 5;

        // the element to be inserted
        int Key = 15;

        // The function inserts the new element to the heap and
        // returns the new size of the array
        n = insertNode(arr, n, Key);

        printArray(arr, n);
        // Final Heap will be:
        // 15
        // / \
        // 5     10
        // / \ /
        // 2 4 3
    }
}

// The code is contributed by Gautam goel

Python3

# program to insert new element to Heap

# Function to heapify ith node in a Heap
# of size n following a Bottom-up approach


def heapify(arr, n, i):
    parent = int(((i-1)/2))
    # For Max-Heap
    # If current node is greater than its parent
    # Swap both of them and call heapify again
    # for the parent
    if arr[parent] > 0:
        if arr[i] > arr[parent]:
            arr[i], arr[parent] = arr[parent], arr[i]
            # Recursively heapify the parent node
            heapify(arr, n, parent)
# Function to insert a new node to the Heap


def insertNode(arr, key):
    global n
    # Increase the size of Heap by 1
    n += 1
    # Insert the element at end of Heap
    arr.append(key)
    # Heapify the new node following a
    # Bottom-up approach
    heapify(arr, n, n-1)
# A utility function to print array of size n


def printArr(arr, n):
    for i in range(n):
        print(arr[i], end=" ")


# Driver Code
# Array representation of Max-Heap
'''
        10
    / \
    5 3
    / \
    2 4
'''
arr = [10, 5, 3, 2, 4, 1, 7]
n = 7
key = 15
insertNode(arr, key)
printArr(arr, n)
# Final Heap will be:
'''
    15
    / \
5     10
/ \ /
2 4 3

Code is written by Rajat Kumar....
'''

C#

// C# program for implementing insertion in Heaps

using System;

public class insertionHeap {

    // Function to heapify ith node in a Heap of size n following a Bottom-up approach
    static void heapify(int[] arr, int n, int i) {
        // Find parent
        int parent = (i - 1) / 2;

        if (arr[parent] > 0) {
            // For Max-Heap
            // If current node is greater than its parent
            // Swap both of them and call heapify again
            // for the parent
            if (arr[i] > arr[parent]) {

                // swap arr[i] and arr[parent]
                int temp = arr[i];
                arr[i] = arr[parent];
                arr[parent] = temp;

                // Recursively heapify the parent node
                heapify(arr, n, parent);
            }
        }
    }

    // Function to insert a new node to the heap.
    static int insertNode(int[] arr, int n, int Key) {
        // Increase the size of Heap by 1
        n = n + 1;

        // Insert the element at end of Heap
        arr[n - 1] = Key;

        // Heapify the new node following a
        // Bottom-up approach
        heapify(arr, n, n - 1);

        // return new size of Heap
        return n;
    }

    /* A utility function to print array of size n */
    static void printArray(int[] arr, int n) {
        for (int i = 0; i < n; ++i)
            Console.WriteLine(arr[i] + " ");

        Console.WriteLine("");
    }

    public static void Main(string[] args) {
        // Array representation of Max-Heap
        //     10
        // / \
        // 5 3
        // / \
        // 2 4

        // maximum size of the array
        int MAX = 1000;
        int[] arr = new int[MAX];

        // initializing some values
        arr[0] = 10;
        arr[1] = 5;
        arr[2] = 3;
        arr[3] = 2;
        arr[4] = 4;

        // Current size of the array
        int n = 5;

        // the element to be inserted
        int Key = 15;

        // The function inserts the new element to the heap and
        // returns the new size of the array
        n = insertNode(arr, n, Key);

        printArray(arr, n);
        // Final Heap will be:
        //     15
        // / \
        // 5     10
        // / \ /
        // 2 4 3
    }
}

// This code is contributed by ajaymakvana.

Javascript

// Javascript program for implement insertion in Heaps

// To heapify a subtree rooted with node i which is
// an index in arr[].Nn is size of heap

let MAX = 1000;

// Function to heapify ith node in a Heap of size n following a Bottom-up approach
function heapify(arr, n, i)
{
    // Find parent
    let parent = Math.floor((i-1)/2);

    if (arr[parent] >= 0) {
        // For Max-Heap
        // If current node is greater than its parent
        // Swap both of them and call heapify again
        // for the parent
        if (arr[i] > arr[parent]) {
            let temp = arr[i];
            arr[i] = arr[parent];
            arr[parent] = temp;

            // Recursively heapify the parent node
            heapify(arr, n, parent);
        }
    }
}

// Function to insert a new node to the Heap
function insertNode(arr, n, Key)
{
    // Increase the size of Heap by 1
    n = n + 1;

    // Insert the element at end of Heap
    arr[n - 1] = Key;

    // Heapify the new node following a
    // Bottom-up approach
    heapify(arr, n, n - 1);

    return n;
}

/* A utility function to print array of size N */
function printArray(arr, n)
{
    for (let i = 0; i < n; ++i)
        console.log(arr[i] + " ");

    console.log("</br>");
}

let arr = [ 10, 5, 3, 2, 4 ];

let n = arr.length;

let key = 15;

n = insertNode(arr, n, key);

printArray(arr, n);

// This code is contributed by ajaymakvana

Output

15 5 10 2 4 3 

Time Complexity:  O(log(n)) (where n is no of elements in the heap)
Auxiliary Space: O(n)

2. Deletion in Max-Heap Data Structure:

Deleting an element at any intermediary position in the heap can be costly, so we can simply replace the element to be deleted with the last element and delete the last element of the Heap. 

• Replace the root or element to be deleted with the last element.
• Delete the last element from the Heap.
• Since the last element is now placed at the position of the root node. So, it may not follow the heap property. Therefore, heapify the last node placed at the position of the root.

Illustration:  

Suppose the Heap is a Max-Heap as:

Deletion in Max-Heap 

The element to be deleted is root, i.e. 10.

Process:

The last element is 4.

Step 1: Replace the last element with root, and delete it.

Deletion in Max-Heap 

Step 2: Heapify root.

Final Heap:

Deletion in Max-Heap 

Implementation of Deletion operation in Max-Heap:

C++

// C++ program for implement deletion in Heaps

#include <iostream>

using namespace std;

// To heapify a subtree rooted with node i which is
// an index of arr[] and n is the size of heap
void heapify(int arr[], int n, int i)
{
    int largest = i; // Initialize largest as root
    int l = 2 * i + 1; // left = 2*i + 1
    int r = 2 * i + 2; // right = 2*i + 2

    // If left child is larger than root
    if (l < n && arr[l] > arr[largest])
        largest = l;

    // If right child is larger than largest so far
    if (r < n && arr[r] > arr[largest])
        largest = r;

    // If largest is not root
    if (largest != i) {
        swap(arr[i], arr[largest]);

        // Recursively heapify the affected sub-tree
        heapify(arr, n, largest);
    }
}

// Function to delete the root from Heap
void deleteRoot(int arr[], int& n)
{
    // Get the last element
    int lastElement = arr[n - 1];

    // Replace root with last element
    arr[0] = lastElement;

    // Decrease size of heap by 1
    n = n - 1;

    // heapify the root node
    heapify(arr, n, 0);
}

/* A utility function to print array of size n */
void printArray(int arr[], int n)
{
    for (int i = 0; i < n; ++i)
        cout << arr[i] << " ";
    cout << "\n";
}

// Driver Code
int main()
{
    // Array representation of Max-Heap
    //     10
    // / \
    // 5 3
    // / \
    // 2 4
    int arr[] = { 10, 5, 3, 2, 4 };

    int n = sizeof(arr) / sizeof(arr[0]);

    deleteRoot(arr, n);

    printArray(arr, n);

    return 0;
}

Java

// Java program for implement deletion in Heaps
public class deletionHeap {

    // To heapify a subtree rooted with node i which is
    // an index in arr[].Nn is size of heap
    static void heapify(int arr[], int n, int i)
    {
        int largest = i; // Initialize largest as root
        int l = 2 * i + 1; // left = 2*i + 1
        int r = 2 * i + 2; // right = 2*i + 2

        // If left child is larger than root
        if (l < n && arr[l] > arr[largest])
            largest = l;

        // If right child is larger than largest so far
        if (r < n && arr[r] > arr[largest])
            largest = r;

        // If largest is not root
        if (largest != i) {
            int swap = arr[i];
            arr[i] = arr[largest];
            arr[largest] = swap;

            // Recursively heapify the affected sub-tree
            heapify(arr, n, largest);
        }
    }

    // Function to delete the root from Heap
    static int deleteRoot(int arr[], int n)
    {
        // Get the last element
        int lastElement = arr[n - 1];

        // Replace root with first element
        arr[0] = lastElement;

        // Decrease size of heap by 1
        n = n - 1;

        // heapify the root node
        heapify(arr, n, 0);

        // return new size of Heap
        return n;
    }

    /* A utility function to print array of size N */
    static void printArray(int arr[], int n)
    {
        for (int i = 0; i < n; ++i)
            System.out.print(arr[i] + " ");

        System.out.println();
    }

    // Driver Code
    public static void main(String args[])
    {
        // Array representation of Max-Heap
        // 10
        // / \
        // 5 3
        // / \
        // 2 4
        int arr[] = { 10, 5, 3, 2, 4 };

        int n = arr.length;

        n = deleteRoot(arr, n);

        printArray(arr, n);
    }
}

Python3

# Python 3 program for implement deletion in Heaps

# To heapify a subtree rooted with node i which is
# an index of arr[] and n is the size of heap
def heapify(arr, n, i):

    largest = i #Initialize largest as root
    l = 2 * i + 1 # left = 2*i + 1
    r = 2 * i + 2 # right = 2*i + 2

    #If left child is larger than root
    if (l < n and arr[l] > arr[largest]):
        largest = l

    #If right child is larger than largest so far
    if (r < n and arr[r] > arr[largest]):
        largest = r

    # If largest is not root
    if (largest != i):
        arr[i],arr[largest]=arr[largest],arr[i]

        #Recursively heapify the affected sub-tree
        heapify(arr, n, largest)

#Function to delete the root from Heap
def deleteRoot(arr):
    global n

    # Get the last element
    lastElement = arr[n - 1]

    # Replace root with last element
    arr[0] = lastElement

    # Decrease size of heap by 1
    n = n - 1

    # heapify the root node
    heapify(arr, n, 0)

# A utility function to print array of size n
def printArray(arr, n):

    for i in range(n):
        print(arr[i],end=" ")
    print()

# Driver Code
if __name__ == '__main__':

    # Array representation of Max-Heap
    #     10
    #     / \
    # 5 3
    # / \
    # 2 4
    arr = [ 10, 5, 3, 2, 4 ]

    n = len(arr)

    deleteRoot(arr)

    printArray(arr, n)

    # This code is contributed by Rajat Kumar.

C#

// C# program for implement deletion in Heaps
using System;

public class deletionHeap
{

    // To heapify a subtree rooted with node i which is
    // an index in arr[].Nn is size of heap
    static void heapify(int []arr, int n, int i)
    {
        int largest = i; // Initialize largest as root
        int l = 2 * i + 1; // left = 2*i + 1
        int r = 2 * i + 2; // right = 2*i + 2

        // If left child is larger than root
        if (l < n && arr[l] > arr[largest])
            largest = l;

        // If right child is larger than largest so far
        if (r < n && arr[r] > arr[largest])
            largest = r;

        // If largest is not root
        if (largest != i)
        {
            int swap = arr[i];
            arr[i] = arr[largest];
            arr[largest] = swap;

            // Recursively heapify the affected sub-tree
            heapify(arr, n, largest);
        }
    }

    // Function to delete the root from Heap
    static int deleteRoot(int []arr, int n)
    {
        // Get the last element
        int lastElement = arr[n - 1];

        // Replace root with first element
        arr[0] = lastElement;

        // Decrease size of heap by 1
        n = n - 1;

        // heapify the root node
        heapify(arr, n, 0);

        // return new size of Heap
        return n;
    }

    /* A utility function to print array of size N */
    static void printArray(int []arr, int n)
    {
        for (int i = 0; i < n; ++i)
            Console.Write(arr[i] + " ");

        Console.WriteLine();
    }

    // Driver Code
    public static void Main()
    {
        // Array representation of Max-Heap
        // 10
        // / \
        // 5 3
        // / \
        // 2 4
        int []arr = { 10, 5, 3, 2, 4 };
        int n = arr.Length;
        n = deleteRoot(arr, n);
        printArray(arr, n);
    }
}

// This code is contributed by Ryuga

Javascript

<script>
    // Javascript program for implement deletion in Heaps

    // To heapify a subtree rooted with node i which is
    // an index in arr[].Nn is size of heap
    function heapify(arr, n, i)
    {
        let largest = i; // Initialize largest as root
        let l = 2 * i + 1; // left = 2*i + 1
        let r = 2 * i + 2; // right = 2*i + 2

        // If left child is larger than root
        if (l < n && arr[l] > arr[largest])
            largest = l;

        // If right child is larger than largest so far
        if (r < n && arr[r] > arr[largest])
            largest = r;

        // If largest is not root
        if (largest != i)
        {
            let swap = arr[i];
            arr[i] = arr[largest];
            arr[largest] = swap;

            // Recursively heapify the affected sub-tree
            heapify(arr, n, largest);
        }
    }

    // Function to delete the root from Heap
    function deleteRoot(arr, n)
    {
        // Get the last element
        let lastElement = arr[n - 1];

        // Replace root with first element
        arr[0] = lastElement;

        // Decrease size of heap by 1
        n = n - 1;

        // heapify the root node
        heapify(arr, n, 0);

        // return new size of Heap
        return n;
    }

    /* A utility function to print array of size N */
    function printArray(arr, n)
    {
        for (let i = 0; i < n; ++i)
            document.write(arr[i] + " ");

        document.write("</br>");
    }

    let arr = [ 10, 5, 3, 2, 4 ];
    let n = arr.length;
    n = deleteRoot(arr, n);
    printArray(arr, n);

// This code is contributed by divyeshrabdiya07.
</script>

Output

5 4 3 2 

Time complexity: O(log n) where n is no of elements in the heap
Auxiliary Space: O(n)

3. Peek operation on Max-heap Data Structure:

 To access the maximum element (i.e., the root of the heap), the value of the root node is returned. The time complexity of peek in a max heap is O(1).

Peek Element

Implementation of Peek operation in Max-Heap:

C++

#include <iostream>
#include <queue>

int main() {
    // Create a max heap with some elements using a priority_queue
    std::priority_queue<int> maxHeap;
    maxHeap.push(9);
    maxHeap.push(8);
    maxHeap.push(7);
    maxHeap.push(6);
    maxHeap.push(5);
    maxHeap.push(4);
    maxHeap.push(3);
    maxHeap.push(2);
    maxHeap.push(1);

    // Get the peak element (i.e., the largest element)
    int peakElement = maxHeap.top();

    // Print the peak element
    std::cout << "Peak element: " << peakElement << std::endl;

    return 0;
}

Java

import java.util.PriorityQueue;

public class GFG {
    public static void main(String[] args) {
        // Create a max heap with some elements using a PriorityQueue
        PriorityQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);
        maxHeap.add(9);
        maxHeap.add(8);
        maxHeap.add(7);
        maxHeap.add(6);
        maxHeap.add(5);
        maxHeap.add(4);
        maxHeap.add(3);
        maxHeap.add(2);
        maxHeap.add(1);

        // Get the peak element (i.e., the largest element)
        int peakElement = maxHeap.peek();

        // Print the peak element
        System.out.println("Peak element: " + peakElement);
    }
}

Python3

import heapq

# Create a max heap with some elements using a list
max_heap = [1,2,3,4,5,6,7,8,9]
heapq.heapify(max_heap)

# Get the peak element (i.e., the largest element)
peak_element = heapq.nlargest(1, max_heap)[0]

# Print the peak element
print("Peak element:", peak_element)

C#

using System;
using System.Collections.Generic;

public class GFG {
    public static void Main() {
        // Create a min heap with some elements using a PriorityQueue
        var maxHeap = new PriorityQueue<int>();
        maxHeap.Enqueue(9);
        maxHeap.Enqueue(8);
        maxHeap.Enqueue(7);
        maxHeap.Enqueue(6);
        maxHeap.Enqueue(5);
        maxHeap.Enqueue(4);
        maxHeap.Enqueue(3);
        maxHeap.Enqueue(2);
        maxHeap.Enqueue(1);

        // Get the peak element (i.e., the smallest element)
        int peakElement = maxHeap.Peek();

        // Print the peak element
        Console.WriteLine("Peak element: " + peakElement);
    }
}
// Define a PriorityQueue class that uses a max heap
class PriorityQueue<T> where T : IComparable<T> {
    private List<T> heap;

    public PriorityQueue() {
        this.heap = new List<T>();
    }

    public int Count {
        get { return this.heap.Count; }
    }

    public void Enqueue(T item) {
        this.heap.Add(item);
        this.BubbleUp(this.heap.Count - 1);
    }

    public T Dequeue() {
        T item = this.heap[0];
        int lastIndex = this.heap.Count - 1;
        this.heap[0] = this.heap[lastIndex];
        this.heap.RemoveAt(lastIndex);
        this.BubbleDown(0);
        return item;
    }

    public T Peek() {
        return this.heap[0];
    }

    private void BubbleUp(int index) {
        while (index > 0) {
            int parentIndex = (index - 1) / 2;
            if (this.heap[parentIndex].CompareTo(this.heap[index]) >= 0) {
                break;
            }
            Swap(parentIndex, index);
            index = parentIndex;
        }
    }

    private void BubbleDown(int index) {
        while (index < this.heap.Count) {
            int leftChildIndex = index * 2 + 1;
            int rightChildIndex = index * 2 + 2;
            int largestChildIndex = index;
            if (leftChildIndex < this.heap.Count && this.heap[leftChildIndex].CompareTo(this.heap[largestChildIndex]) > 0) {
                largestChildIndex = leftChildIndex;
            }
            if (rightChildIndex < this.heap.Count && this.heap[rightChildIndex].CompareTo(this.heap[largestChildIndex]) > 0) {
                largestChildIndex = rightChildIndex;
            }
            if (largestChildIndex == index) {
                break;
            }
            Swap(largestChildIndex, index);
            index = largestChildIndex;
        }
    }

    private void Swap(int i, int j) {
        T temp = this.heap[i];
        this.heap[i] = this.heap[j];
        this.heap[j] = temp;
    }
}

Javascript

// Define a MaxHeap class that uses an array
class MaxHeap {
    constructor() {
        this.heap = [];
    }

    push(item) {
        this.heap.push(item);
        this.bubbleUp(this.heap.length - 1);
    }

    pop() {
        let item = this.heap[0];
        let lastIndex = this.heap.length - 1;
        this.heap[0] = this.heap[lastIndex];
        this.heap.pop();
        this.bubbleDown(0);
        return item;
    }

    peak() {
        return this.heap[0];
    }

    bubbleUp(index) {
        while (index > 0) {
            let parentIndex = Math.floor((index - 1) / 2);
            if (this.heap[parentIndex] >= this.heap[index]) {
                break;
            }
            this.swap(parentIndex, index);
            index = parentIndex;
        }
    }

    bubbleDown(index) {
        while (index < this.heap.length) {
            let leftChildIndex = index * 2 + 1;
            let rightChildIndex = index * 2 + 2;
            let largestChildIndex = index;
            if (leftChildIndex < this.heap.length && this.heap[leftChildIndex] > this.heap[largestChildIndex]) {
                largestChildIndex = leftChildIndex;
            }
            if (rightChildIndex < this.heap.length && this.heap[rightChildIndex] > this.heap[largestChildIndex]) {
                largestChildIndex = rightChildIndex;
            }
            if (largestChildIndex === index) {
                break;
            }
            this.swap(largestChildIndex, index);
            index = largestChildIndex;
        }
    }

    swap(i, j) {
        let temp = this.heap[i];
        this.heap[i] = this.heap[j];
        this.heap[j] = temp;
    }
}

// Create a max heap with some elements using an array
let maxHeap = new MaxHeap();
maxHeap.push(9);
maxHeap.push(8);
maxHeap.push(7);
maxHeap.push(6);
maxHeap.push(5);
maxHeap.push(4);
maxHeap.push(3);
maxHeap.push(2);
maxHeap.push(1);

// Get the peak element (i.e., the largest element)
let peakElement = maxHeap.peak();

// Print the peak element
console.log("Peak element: " + peakElement);

Output

Peak element: 9

Time complexity: 

• In a max heap implemented using an array or a list, the peak element can be accessed in constant time, O(1), as it is always located at the root of the heap.
• In a max heap implemented using a binary tree, the peak element can also be accessed in O(1) time, as it is always located at the root of the tree.

Auxiliary Space: O(n)

4. Heapify operation on Max-heap Data Structure:

Heapify operation

A heapify operation can be used to create a max heap from an unsorted array. This is done by starting at the last non-leaf node and repeatedly performing the “bubble down” operation until all nodes satisfy the heap property. The time complexity of heapify in a max heap is O(n).

5. Search operation on Max-heap Data Structure:: 

To search for an element in the max heap, a linear search can be performed over the array that represents the heap. However, the time complexity of a linear search is O(n), which is not efficient. Therefore, searching is not a commonly used operation in a max heap.

Here’s an example code that shows how to search for an element in a max heap using std::find():

C++

#include <iostream>
#include <queue> // for std::priority_queue
using namespace std;

int main() {
    std::priority_queue<int> max_heap;
  // example max heap

  max_heap.push(10);
  max_heap.push(9);
  max_heap.push(8);
  max_heap.push(6);
  max_heap.push(4);

    int element = 6; // element to search for
    bool found = false;

    // Copy the max heap to a temporary queue and search for the element
    std::priority_queue<int> temp = max_heap;
    while (!temp.empty()) {
        if (temp.top() == element) {
            found = true;
            break;
        }
        temp.pop();
    }

    if (found) {
        std::cout << "Element found in the max heap." << std::endl;
    } else {
        std::cout << "Element not found in the max heap." << std::endl;
    }

    return 0;
}

Java

import java.util.PriorityQueue;

public class GFG {
    public static void main(String[] args) {
        PriorityQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);
        maxHeap.add(3); // insert elements into the priority queue
        maxHeap.offer(1);
        maxHeap.offer(4);
        maxHeap.offer(1);
        maxHeap.offer(6);

        int element = 6; // element to search for
        boolean found = false;

        // Copy the max heap to a temporary queue and search for the element
        PriorityQueue<Integer> temp = new PriorityQueue<>(maxHeap);
        while (!temp.isEmpty()) {
            if (temp.poll() == element) {
                found = true;
                break;
            }
        }

        if (found) {
            System.out.println("Element found in the max heap.");
        } else {
            System.out.println("Element not found in the max heap.");
        }
    }
}

Python3

import heapq

max_heap = [10, 8, 7, 6, 5, 3, 2, 1] # example max heap
heapq._heapify_max(max_heap)

element = 6 # element to search for
found = False

# Copy the max heap to a temporary list and search for the element
temp = list(max_heap)
while temp:
    if heapq._heappop_max(temp) == element:
        found = True
        break

if found:
    print("Element found in the max heap.")
else:
    print("Element not found in the max heap.")

C#

using System;
using System.Collections.Generic;

class Program {
    static void Main(string[] args) {
        // Create a max heap with some elements using a PriorityQueue
        PriorityQueue<int> maxHeap = new PriorityQueue<int>();
        maxHeap.Enqueue(10);
        maxHeap.Enqueue(9);
        maxHeap.Enqueue(8);
        maxHeap.Enqueue(6);
        maxHeap.Enqueue(4);

        int element = 6; // element to search for
        bool found = false;

        // Copy the max heap to a temporary queue and search for the element
        PriorityQueue<int> temp = new PriorityQueue<int>(maxHeap);
        while (temp.Count > 0) {
            if (temp.Peek() == element) {
                found = true;
                break;
            }
            temp.Dequeue();
        }

        if (found) {
            Console.WriteLine("Element found in the max heap.");
        } else {
            Console.WriteLine("Element not found in the max heap.");
        }
    }
}

// PriorityQueue class
class PriorityQueue<T> where T : IComparable<T> {
    private List<T> heap = new List<T>();

    public void Enqueue(T item) {
        heap.Add(item);
        int child = heap.Count - 1;
        while (child > 0) {
            int parent = (child - 1) / 2;
            if (heap[child].CompareTo(heap[parent]) > 0) {
                T tmp = heap[child];
                heap[child] = heap[parent];
                heap[parent] = tmp;
                child = parent;
            } else {
                break;
            }
        }
    }

    public T Dequeue() {
        int last = heap.Count - 1;
        T frontItem = heap[0];
        heap[0] = heap[last];
        heap.RemoveAt(last);

        last--;

        int parent = 0;
        while (true) {
            int leftChild = parent * 2 + 1;
            if (leftChild > last) {
                break;
            }
            int rightChild = leftChild + 1;
            if (rightChild <= last && heap[leftChild].CompareTo(heap[rightChild]) < 0) {
                leftChild = rightChild;
            }
            if (heap[parent].CompareTo(heap[leftChild]) < 0) {
                T tmp = heap[parent];
                heap[parent] = heap[leftChild];
                heap[leftChild] = tmp;
                parent = leftChild;
            } else {
                break;
            }
        }

        return frontItem;
    }

    public T Peek() {
        return heap[0];
    }

    public int Count {
        get {
            return heap.Count;
        }
    }
}

Javascript

const maxHeap = new PriorityQueue((a, b) => b - a);
maxHeap.add(3); // insert elements into the priority queue
maxHeap.add(1);
maxHeap.add(4);
maxHeap.add(1);
maxHeap.add(6);

const element = 6; // element to search for
let found = false;

// Copy the max heap to a temporary queue and search for the element
const temp = new PriorityQueue(maxHeap);
while (!temp.isEmpty()) {
if (temp.poll() === element) {
found = true;
break;
}
}

if (found) {
console.log("Element found in the max heap.");
} else {
console.log("Element not found in the max heap.");
}

Output

Element found in the max heap.

Time complexity: O(n), where n is the size of the heap.
Auxiliary Space: O(n),

Applications of Max-Heap Data Structure:

• Heapsort algorithm: The heap data structure is the basis for the heapsort algorithm, which is an efficient sorting algorithm with a worst-case time complexity of O(n log n). The heapsort algorithm is used in various applications, including database indexing and numerical analysis.
• Memory management: The heap data structure is used in memory management systems to allocate and deallocate memory dynamically. The heap is used to store the memory blocks, and the heap data structure is used to efficiently manage the memory blocks and allocate them to programs as needed.
• Graph algorithms: The heap data structure is used in various graph algorithms, including Dijkstra’s algorithm, Prim’s algorithm, and Kruskal’s algorithm. These algorithms require efficient priority queue implementation, which can be achieved using the heap data structure.
• Job scheduling: The heap data structure is used in job scheduling algorithms, where tasks are scheduled based on their priority or deadline. The heap data structure allows efficient access to the highest-priority task, making it a useful data structure for job scheduling applications.

Advantages of Max-Heap Data Structure:

• Efficiently maintain the maximum value: A max heap allows constant-time access to the maximum element in the heap, which makes it useful in applications where the maximum element needs to be found quickly.
• Efficient insert and delete operations: The insert and delete operations in a max heap have a time complexity of O(log n), which makes them efficient for large collections of elements.
• Priority Queues: A max heap can be used to implement a priority queue, which is useful in many applications such as job scheduling, task prioritization, and event-driven simulation.
• Sorting: A max heap can be used to implement heapsort, which is an efficient sorting algorithm that has a worst-case time complexity of O(n log n).
• Space efficiency: A max heap can be implemented as an array, which requires less memory compared to other data structures such as a binary search tree or a linked list.

Max heap data structure is a useful and efficient tool for maintaining and manipulating collections of elements, particularly when the maximum element needs to be accessed quickly or when elements need to be sorted or prioritized.