Sorting Algorithms- 4 | Algorithms - Computer Science Engineering (CSE) PDF Download

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).

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 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);
      }
  }
  // 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);
  }
• 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);
      }
  }
• Python
  # 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]),
  # 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 = 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>

Output
Sorted array is
5 6 7 11 12 13
Notes:
Heap sort is an in-place algorithm.
Its typical implementation is not stable, but can be made stable

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.

Snapshots:

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