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 in arr[]. N is 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 first 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); 
    } 
} 

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 

Output:

5 4 3 2

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 isnerted 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 less than its parent 3, swap them.
       10
    /    \
   5      15
  / \    /
 2   4  3

-> 15 is again less than its parent 10, swap them.
       15
    /    \
   5      10
  / \    /
 2   4  3

Therefore, the final heap after insertion is:
-> 15 is less than its parent 3, swap them.
       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 (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; 
} 

Output:

15 5 10 2 4 3

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