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; } |
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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); } } |
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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 |
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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 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 (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; } |
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Output:
15 5 10 2 4 3
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