Heap Sort for decreasing order using min heap

Heap Sort ? Heap sort is a comparison-based algorithm that used binary tree data structures to sort a list of numbers in increasing or decreasing order. It heap sorts to create a heap data structure where the root is the smallest element and then removes the root and again sorting gives the second smallest number in the list at the root position.

Min Heap ? Min heap is a data structure where the parent node is always smaller than the child node, thus root node is the smallest element among all.

Problem Statement

Given an array of integers. Sort them in decreasing order using min heap.

Sample Example 1

Input: [2, 5, 1, 7, 0]

Output: [7, 5, 2, 1, 0]

Sample Example 2

Input: [55, 1, 23, 10, 1]

Output: [55, 23, 10, 1, 1]

Approach 1

To perform heap sort in decreasing order using min heap, we create a min heap of the elements and extract them one at a time to get an array in decreasing order by reversing the order.

Pseudocode

procedure heapSort (arr[], n)
   Initialize priority queue: minHeap
   for i = 1 to n
      add arr[i] to minHeap
   i = n - 1
   while minHeap is not empty
      arr[i-] = top element of minHeap
      Remove the top element of minHeap
end procedure

Example: C++ Implementation

In the following program, we use a min heap to sort an array and then reverse the order to get the result.

#include <bits/stdc++.h>
using namespace std;
// Function to heap sort in decreasing order using min heap
void heapSort(int arr[], int n){

   // Creating min heap using a priority queue
   priority_queue<int, vector<int>, greater<int> > minHeap;
   
   // Inserting input array to min heap
   for (int i = 0; i < n; i++){
      minHeap.push(arr[i]);
   }
   
   // Iterating backwards in the input array, where each element is replaced by the smallest element extracted from min heap
   int i = n - 1;
   while (!minHeap.empty()){
      arr[i--] = minHeap.top();
      minHeap.pop();
   }
}
int main(){
   int arr[6] = {5, 2, 9, 1, 5, 6};
   int n = 6;
   heapSort(arr, n);
   cout << "Sorted array : ";
   for (int i = 0; i < n; i++){
      cout << arr[i] << " ";
   }
   cout << endl;
   return 0;
}

Output

Sorted array : 9 6 5 5 2 1

Time Complexity ? O(nlogn)

Space Complexity ? O(n)

Approach 2

Another solution to the problem can be to build a min heap starting with the last non-leaf root mode and working backwards. Then we can sort the array by swapping the root node with the last leaf node and then restoring the min-heap property.

Pseudocode

procedure heapify (arr[], n , i)
   smallest = i
   l = 2i + 1
   r = 2i + 2
   if l < n and arr[l] < arr[smallest]
      smallest = l
   end if
   if r < n and arr[r] < arr[samllest]
      smallest = r
   end if
   if smallest is not i
      swap arr[i] to arr[smallest]
      heapify (arr, n, smallest)
   end if
end procedure

procedure heapSort (arr[], n)
   for i = n/2 - 1 to 0
      heapify(arr, n, i)
   for i = n-1 to 0
      swap arr[0] to arr[i]
      heapify (arr, i, 0)
end procedure

Example: C++ Implementation

In the following program, we use heapify() function that restores the min-heap property of the subtree rooted at index i and heapSort() builds min heap in reverse order.

#include <bits/stdc++.h>
using namespace std;

// Restores the min heap property of subtree rooted at index i
void heapify(int arr[], int n, int i){
   int smallest = i;
   int l = 2 * i + 1;
   int r = 2 * i + 2;
   if (l < n && arr[l] < arr[smallest]){
      smallest = l;
   }
   if (r < n && arr[r] < arr[smallest]){
      smallest = r;
   }
   if (smallest != i){
      swap(arr[i], arr[smallest]);
      heapify(arr, n, smallest);
   }
}
void heapSort(int arr[], int n){

   // Build the min heap in reverse order
   for (int i = n / 2 - 1; i >= 0; i--){
      heapify(arr, n, i);
   }
   
   // Sort the array by repeatedly swapping the root node with the last leaf node
   for (int i = n - 1; i >= 0; i--){
      swap(arr[0], arr[i]);
      heapify(arr, i, 0);
   }
}
int main(){
   int arr[6] = {5, 2, 9, 1, 5, 6};
   int n = 6;
   heapSort(arr, n);
   cout << "Sorted array : ";
   for (int i = 0; i < n; i++){
      cout << arr[i] << " ";
   }
   cout << endl;
   return 0;
}

Output

Sorted array : 9 6 5 5 2 1

Using the previous approach of using heapSort() function to create a min heap, we can use the same approach in this solution but rather than using heapify to restore property of min heap, we use a traditional hep sort algorithm to crete amin heap and sor telement sin increasing order which are further reversed to get the desired output.

Pseudocode

procedure heapSort (arr[], n)
   for i = n/2 - 1 to 0
   parent = i
      while parent *2+1 < n
         child = parent*2+1
         if child+1 < n and arr[child] >arr[child+1]
            child = child + 1
         end if
         if arr[parent] > arr[child]
            swap arr[parent] to arr[child]
            parent = child
         else
            break
         end if
   for i = n-1 to 0
      swap arr[0] to arr[i]
      parent = 0
      while parent*2+1 < i
            child = parent*2+1
         if child+1 < n and arr[child] >arr[child+1]
            child = child + 1
         end if
         if arr[parent] > arr[child]
            swap arr[parent] to arr[child]
            parent = child
         else
            break
         end if
end procedure

Example: C++ Implentation

In the following program, we modify heap sort algorithm to sort array in decreasing order.

#include <bits/stdc++.h>
using namespace std;

void heapSort(int arr[], int n){
   // Building min heap in reverse order
   for (int i = n / 2 - 1; i >= 0; i--) {
   
      // Starting from last parent node, apply heapify operation
      int parent = i;
      while (parent * 2 + 1 < n) {
         int child = parent * 2 + 1;
         if (child + 1 < n && arr[child] > arr[child + 1]){
            child++;
         }
         if (arr[parent] > arr[child]){
            swap(arr[parent], arr[child]);
            parent = child;
         }
         else{
            break;
         }
      }
   }
   
   // Extarct elekemnhts form min heap in decreasing order
   for (int i = n - 1; i > 0; i--){
      swap(arr[0], arr[i]);
      int parent = 0;
      
      // Perform heapify operation at new root node
      while (parent * 2 + 1 < i){
         int child = parent * 2 + 1;
         if (child + 1 < i && arr[child] > arr[child + 1]){
            child++;
         }
         if (arr[parent] > arr[child]){
            swap(arr[parent], arr[child]);
            parent = child;
         }
         else {
            break;
         }
      }
   }
}
int main(){
   int arr[6] = {5, 2, 9, 1, 5, 6};
   int n = 6;
   heapSort(arr, n);
   cout << "Sorted array : ";
   for (int i = 0; i < n; i++) {
      cout << arr[i] << " ";
   }
   cout << endl;
   return 0;
}

Output

Sorted array : 9 6 5 5 2 1

Conclusion

In conclusion, for performing heap sort for decreasing order using min heap, we can use several approaches, a few of which are mentione dabove having time complexities of O(nlogn) and varying space complexity for each approach.