Insertion and Deletion in Heaps

Deletion in Heap:

Given a Binary Heap and an element present in the given Heap. The task is to delete an element from this Heap. 

The standard deletion operation on Heap is to delete the element present at the root node of the Heap. That is if it is a Max Heap, the standard deletion operation will delete the maximum element and if it is a Min heap, it will delete the minimum element.

Process of Deletion: 
Since deleting an element at any intermediary position in the heap can be costly, so we can simply replace the element to be deleted by the last element and delete the last element of the Heap. 

• Replace the root or element to be deleted by 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 root.

Illustration:  

Suppose the Heap is a Max-Heap as:
      10
    /    \
   5      3
  / \
 2   4
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.
      4
    /    \
   5      3
  / 
 2   
Step 2: Heapify root.
Final Heap:
      5
    /    \
   4      3
  / 
 2   

Implementation:  

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(logn) where n is no of elements in the heap
Auxiliary Space: O(n)

Insertion in Heaps:

The insertion operation is also similar to that of the deletion process. 

Given a Binary Heap and a new element to be added to this Heap. The task is to insert the new element to the Heap maintaining the properties of Heap.

Process of Insertion: 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:
      10
    /    \
   5      3
  / \
 2   4
The new element to be inserted is 15.
Process:
Step 1: Insert the new element at the end.
      10
    /    \
   5      3
  / \    /
 2   4  15
Step 2: Heapify the new element following bottom-up 
        approach.
-> 15 is more than its parent 3, swap them.
       10
    /    \
   5      15
  / \    /
 2   4  3
-> 15 is again more than its parent 10, swap them.
       15
    /    \
   5      10
  / \    /
 2   4  3
Therefore, the final heap after insertion is:
       15
    /    \
   5      10
  / \    /
 2   4  3

Implementation: 

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 (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 (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 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 (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 (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)