HeapSort
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).
How to “heapify” a tree?
The process of reshaping a binary tree into a Heap data structure is known as ‘heapify’. A binary tree is a tree data structure that has two child nodes at max. If a node’s children nodes are ‘heapified’, then only ‘heapify’ process can be applied over that node. A heap should always be a complete binary tree.
Starting from a complete binary tree, we can modify it to become a Max-Heap by running a function called ‘heapify’ on all the non-leaf elements of the heap. i.e. ‘heapify’ uses recursion.
Algorithm for “heapify”:
heapify(array)
Root = array[0]Largest = largest( array[0] , array [2 * 0 + 1]. array[2 * 0 + 2])
if(Root != Largest)
Swap(Root, Largest)
Example of “heapify”:
30(0)
/ \
70(1) 50(2)Child (70(1)) is greater than the parent (30(0))
Swap Child (70(1)) with the parent (30(0))
70(0)
/ \
30(1) 50(2)
Heap Sort Algorithm for sorting in increasing order:
- Build a max heap from the input data.
- 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.
- 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:
How heap sort works?
- To understand heap sort more clearly, let’s take an unsorted array and try to sort it using heap sort.
- After that, the task is to construct a tree from that unsorted array and try to convert it into max heap.
- To transform a heap into a max-heap, the parent node should always be greater than or equal to the child nodes
- Here, in this example, as the parent node 4 is smaller than the child node 10, thus, swap them to build a max-heap.
- Now, as seen, 4 as a parent is smaller than the child 5, thus swap both of these again and the resulted heap and array should be like this:
- Next, simply delete the root element (10) from the max heap. In order to delete this node, try to swap it with the last node, i.e. (1). After removing the root element, again heapify it to convert it into max heap.
- Resulted heap and array should look like this:
- After repeating the above steps, at last swap first and last node and delete the last node from heap, and thus the resulted array seems to be the sorted one as shown in figure:
Illustration:
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.
Below is the implementation:
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 heap void heapify(int arr[], int n, int i) { // Initialize largest as root int largest = i; // left = 2*i + 1 int l = 2 * i + 1; // right = 2*i + 2 int r = 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 sort void 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 code int 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); } |
C
// Heap Sort in C #include <stdio.h> // Function to swap the position of two elements void swap(int *a, int *b) { int temp = *a; *a = *b; *b = temp; } // 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) { // Find largest among root, left child and right child // Initialize largest as root int largest = i; // left = 2*i + 1 int left = 2 * i + 1; // right = 2*i + 2 int right = 2 * i + 2; // If left child is larger than root if (left < n && arr[left] > arr[largest]) largest = left; // If right child is larger than largest // so far if (right < n && arr[right] > arr[largest]) largest = right; // Swap and continue heapifying if root is not largest // 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 sort void heapSort(int arr[], int n) { // Build max heap for (int i = n / 2 - 1; i >= 0; i--) heapify(arr, n, i); // Heap sort for (int i = n - 1; i >= 0; i--) { swap(&arr[0], &arr[i]); // Heapify root element to get highest element at root again 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++) printf("%d ", arr[i]); printf("\n"); } // Driver code int main() { int arr[] = {12, 11, 13, 5, 6, 7}; int n = sizeof(arr) / sizeof(arr[0]); heapSort(arr, n); printf("Sorted array is given in the following way \n"); printArray(arr, n); } // This code is contributed by _i_plus_plus_. |
Java
// Java program for implementation of Heap Sort 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) 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); } } |
Python3
# Python program for implementation of heap Sort # To heapify subtree rooted at index i. # n is 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 # 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 size def 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 code arr = [12, 11, 13, 5, 6, 7] heapSort(arr) n = len(arr) print("Sorted array is") for i in range(n): print("%d" % arr[i],end=" ") # This code is contributed by Mohit Kumra |
C#
// C# program for implementation of Heap Sort using 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 heap function 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, $largest); } } // main function to do heap sort function heapSort(&$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 Sort function sort( arr) { var n = arr.length; // Build heap (rearrange array) for (var i = Math.floor(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 = [ 5, 12, 11, 13, 4, 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
Time Complexity: O(n logn),
- Time complexity of heapify is O(Logn).
- Time complexity of createAndBuildHeap() is O(n)
- And, hence the overall time complexity of Heap Sort is O(nLogn).
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)
Advantages of heapsort –
- Efficiency – The time required to perform Heap sort increases logarithmically while other algorithms may grow exponentially slower as the number of items to sort increases. This sorting algorithm is very efficient.
- Memory Usage – Memory usage is minimal because apart from what is necessary to hold the initial list of items to be sorted, it needs no additional memory space to work
- Simplicity – It is simpler to understand than other equally efficient sorting algorithms because it does not use advanced computer science concepts such as recursion
Applications of HeapSort
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
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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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