Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to selection sort where we first find the minimum element and place the minimum element at the beginning. We repeat the same process for the remaining elements.
What is Binary Heap?
Let us first define a Complete Binary Tree. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible (Source Wikipedia)
A Binary Heap is a Complete Binary Tree where items are stored in a special order such that the value in a parent node is greater(or smaller) than the values in its two children nodes. The former is called max heap and the latter is called min-heap. The heap can be represented by a binary tree or array.
Why array based representation for Binary Heap?
Since a Binary Heap is a Complete Binary Tree, it can be easily represented as an array and the array-based representation is space-efficient. If the parent node is stored at index I, the left child can be calculated by 2 * I + 1 and the right child by 2 * I + 2 (assuming the indexing starts at 0).
Heap Sort Algorithm for sorting in increasing order:
1. Build a max heap from the input data.
2. At this point, the largest item is stored at the root of the heap. Replace it with the last item of the heap followed by reducing the size of heap by 1. Finally, heapify the root of the tree.
3. Repeat step 2 while the size of the heap is greater than 1.
How to build the heap?
Heapify procedure can be applied to a node only if its children nodes are heapified. So the heapification must be performed in the bottom-up order.
Lets understand with the help of an example:
Input data: 4, 10, 3, 5, 1
4(0)
/ \
10(1) 3(2)
/ \
5(3) 1(4)
The numbers in bracket represent the indices in the array
representation of data.
Applying heapify procedure to index 1:
4(0)
/ \
10(1) 3(2)
/ \
5(3) 1(4)
Applying heapify procedure to index 0:
10(0)
/ \
5(1) 3(2)
/ \
4(3) 1(4)
The heapify procedure calls itself recursively to build heap
in top down manner.C++
// C++ program for implementation of Heap Sort#include <iostream>using namespace std;// To heapify a subtree rooted with node i which is// an index in arr[]. n is size of heapvoid 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); }}// main function to do heap sortvoid heapSort(int arr[], int n){ // Build heap (rearrange array) for (int i = n / 2 - 1; i >= 0; i--) heapify(arr, n, i); // One by one extract an element from heap for (int i = n - 1; i > 0; i--) { // Move current root to end swap(arr[0], arr[i]); // call max heapify on the reduced heap heapify(arr, i, 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 codeint main(){ int arr[] = { 12, 11, 13, 5, 6, 7 }; int n = sizeof(arr) / sizeof(arr[0]); heapSort(arr, n); cout << "Sorted array is \n"; printArray(arr, n);} |
Java
// Java program for implementation of Heap Sortpublic class HeapSort { public void sort(int arr[]) { int n = arr.length; // Build heap (rearrange array) for (int i = n / 2 - 1; i >= 0; i--) heapify(arr, n, i); // One by one extract an element from heap for (int i = n - 1; i > 0; i--) { // Move current root to end int temp = arr[0]; arr[0] = arr[i]; arr[i] = temp; // call max heapify on the reduced heap heapify(arr, i, 0); } } // 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) { int swap = arr[i]; arr[i] = arr[largest]; arr[largest] = swap; // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } /* A utility function to print array of size n */ static void printArray(int arr[]) { int n = arr.length; for (int i = 0; i < n; ++i) System.out.print(arr[i] + " "); System.out.println(); } // Driver code public static void main(String args[]) { int arr[] = { 12, 11, 13, 5, 6, 7 }; int n = arr.length; HeapSort ob = new HeapSort(); ob.sort(arr); System.out.println("Sorted array is"); printArray(arr); }} |
Python
# Python program for implementation of heap Sort# To heapify subtree rooted at index i.# n is size of heapdef 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 # See if left child of root exists and is # greater than root if l < n and arr[largest] < arr[l]: largest = l # See if right child of root exists and is # greater than root if r < n and arr[largest] < arr[r]: largest = r # Change root, if needed if largest != i: arr[i], arr[largest] = arr[largest], arr[i] # swap # Heapify the root. heapify(arr, n, largest)# The main function to sort an array of given sizedef heapSort(arr): n = len(arr) # Build a maxheap. for i in range(n//2 - 1, -1, -1): heapify(arr, n, i) # One by one extract elements for i in range(n-1, 0, -1): arr[i], arr[0] = arr[0], arr[i] # swap heapify(arr, i, 0)# Driver codearr = [12, 11, 13, 5, 6, 7]heapSort(arr)n = len(arr)print("Sorted array is")for i in range(n): print("%d" % arr[i]),# This code is contributed by Mohit Kumra |
C#
// C# program for implementation of Heap Sortusing System;public class HeapSort { public void sort(int[] arr) { int n = arr.Length; // Build heap (rearrange array) for (int i = n / 2 - 1; i >= 0; i--) heapify(arr, n, i); // One by one extract an element from heap for (int i = n - 1; i > 0; i--) { // Move current root to end int temp = arr[0]; arr[0] = arr[i]; arr[i] = temp; // call max heapify on the reduced heap heapify(arr, i, 0); } } // 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) { int swap = arr[i]; arr[i] = arr[largest]; arr[largest] = swap; // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } /* A utility function to print array of size n */ static void printArray(int[] arr) { int n = arr.Length; for (int i = 0; i < n; ++i) Console.Write(arr[i] + " "); Console.Read(); } // Driver code public static void Main() { int[] arr = { 12, 11, 13, 5, 6, 7 }; int n = arr.Length; HeapSort ob = new HeapSort(); ob.sort(arr); Console.WriteLine("Sorted array is"); printArray(arr); }}// This code is contributed// by Akanksha Rai(Abby_akku) |
PHP
<?php// Php program for implementation of Heap Sort// To heapify a subtree rooted with node i which is// an index in arr[]. n is size of heapfunction 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 && $arr[$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]; $arr[$i] = $arr[$largest]; $arr[$largest] = $swap; // Recursively heapify the affected sub-tree heapify($arr, $n, $largest); }}// main function to do heap sortfunction heapSort(&$arr, $n){ // Build heap (rearrange array) for ($i = $n / 2 - 1; $i >= 0; $i--) heapify($arr, $n, $i); // One by one extract an element from heap for ($i = $n-1; $i > 0; $i--) { // Move current root to end $temp = $arr[0]; $arr[0] = $arr[$i]; $arr[$i] = $temp; // call max heapify on the reduced heap heapify($arr, $i, 0); }}/* A utility function to print array of size n */function printArray(&$arr, $n){ for ($i = 0; $i < $n; ++$i) echo ($arr[$i]." ") ; }// Driver program $arr = array(12, 11, 13, 5, 6, 7); $n = sizeof($arr)/sizeof($arr[0]); heapSort($arr, $n); echo 'Sorted array is ' . "\n"; printArray($arr , $n);// This code is contributed by Shivi_Aggarwal?> |
Javascript
<script>// JavaScript program for implementation// of Heap Sortfunction sort( arr) { var n = arr.length; // Build heap (rearrange array) for (var i = n / 2 - 1; i >= 0; i--) heapify(arr, n, i); // One by one extract an element from heap for (var i = n - 1; i > 0; i--) { // Move current root to end var temp = arr[0]; arr[0] = arr[i]; arr[i] = temp; // call max heapify on the reduced heap heapify(arr, i, 0); } } // To heapify a subtree rooted with node i which is // an index in arr[]. n is size of heap function heapify(arr, n, i) { var largest = i; // Initialize largest as root var l = 2 * i + 1; // left = 2*i + 1 var 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) { var swap = arr[i]; arr[i] = arr[largest]; arr[largest] = swap; // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } /* A utility function to print array of size n */ function printArray(arr) { var n = arr.length; for (var i = 0; i < n; ++i) document.write(arr[i] + " "); } var arr = [ 12, 11, 13, 5, 6, 7 ]; var n = arr.length; sort(arr); document.write( "Sorted array is <br>"); printArray(arr, n);// This code is contributed by SoumikMondal</script> |
Sorted array is 5 6 7 11 12 13
Here is previous C code for reference.
Notes:
Heap sort is an in-place algorithm.
Its typical implementation is not stable, but can be made stable (See this)
Time Complexity: Time complexity of heapify is O(Logn). Time complexity of createAndBuildHeap() is O(n) and the overall time complexity of Heap Sort is O(nLogn).
Applications of HeapSort
1. Sort a nearly sorted (or K sorted) array
2. k largest(or smallest) elements in an array
Heap sort algorithm has limited uses because Quicksort and Mergesort are better in practice. Nevertheless, the Heap data structure itself is enormously used. See Applications of Heap Data Structure
https://youtu.be/MtQL_ll5KhQ
Snapshots:
Quiz on Heap Sort
Other Sorting Algorithms on GeeksforGeeks/GeeksQuiz:
QuickSort, Selection Sort, Bubble Sort, Insertion Sort, Merge Sort, Heap Sort, QuickSort, Radix Sort, Counting Sort, Bucket Sort, ShellSort, Comb Sort, Pigeonhole Sort
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