Heap’s Algorithm for generating permutations

Heap’s algorithm is used to generate all permutations of n objects. The idea is to generate each permutation from the previous permutation by choosing a pair of elements to interchange, without disturbing the other n-2 elements.
Following is the illustration of generating all the permutations of n given numbers.

Example:

Input: 1 2 3
Output: 1 2 3
    2 1 3
    3 1 2
     1 3 2
    2 3 1
    3 2 1

Algorithm:

  1. The algorithm generates (n-1)! permutations of the first n-1 elements, adjoining the last element to each of these. This will generate all of the permutations that end with the last element.
  2. If n is odd, swap the first and last element and if n is even, then swap the ith element (i is the counter starting from 0) and the last element and repeat the above algorithm till i is less than n.
  3. In each iteration, the algorithm will produce all the permutations that end with the current last element.

Implementation:

C++

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// C++ program to print all permutations using
// Heap's algorithm
#include <bits/stdc++.h>
using namespace std;
  
//Prints the array
void printArr(int a[],int n)
{
    for (int i=0; i<n; i++)
        cout << a[i] << " ";
    printf("\n");
}
  
// Generating permutation using Heap Algorithm
void heapPermutation(int a[], int size, int n)
{
    // if size becomes 1 then prints the obtained
    // permutation
    if (size == 1)
    {
        printArr(a, n);
        return;
    }
  
    for (int i=0; i<size; i++)
    {
        heapPermutation(a,size-1,n);
  
        // if size is odd, swap first and last
        // element
        if (size%2==1)
            swap(a[0], a[size-1]);
  
        // If size is even, swap ith and last
        // element
        else
            swap(a[i], a[size-1]);
    }
}
  
// Driver code
int main()
{
    int a[] = {1, 2, 3};
    int n = sizeof a/sizeof a[0];
    heapPermutation(a, n, n);
    return 0;
}

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Java

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// Java program to print all permutations using
// Heap's algorithm
class HeapAlgo
{
    //Prints the array
    void printArr(int a[], int n)
    {
        for (int i=0; i<n; i++)
            System.out.print(a[i] + " ");
        System.out.println();
    }
  
    //Generating permutation using Heap Algorithm
    void heapPermutation(int a[], int size, int n)
    {
        // if size becomes 1 then prints the obtained
        // permutation
        if (size == 1)
            printArr(a,n);
  
        for (int i=0; i<size; i++)
        {
            heapPermutation(a, size-1, n);
  
            // if size is odd, swap first and last
            // element
            if (size % 2 == 1)
            {
                int temp = a[0];
                a[0] = a[size-1];
                a[size-1] = temp;
            }
  
            // If size is even, swap ith and last
            // element
            else
            {
                int temp = a[i];
                a[i] = a[size-1];
                a[size-1] = temp;
            }
        }
    }
  
    // Driver code
    public static void main(String args[])
    {
        HeapAlgo obj = new HeapAlgo();
        int a[] = {1,2,3};
        obj.heapPermutation(a, a.length, a.length);
    }
}
  
// This code has been contributed by Amit Khandelwal.

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Python3

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# Python program to print all permutations using
# Heap's algorithm
  
#Prints the array
def printArr(a, n):
    for i in range(n):
        print(a[i],end=" ")
    print()
  
# Generating permutation using Heap Algorithm
def heapPermutation(a, size, n):
      
    # if size becomes 1 then prints the obtained
    # permutation
    if (size == 1):
        printArr(a, n)
        return
  
    for i in range(size):
        heapPermutation(a,size-1,n);
  
        # if size is odd, swap first and last
        # element
        # else If size is even, swap ith and last element
        if size&1:
            a[0], a[size-1] = a[size-1],a[0]
        else:
            a[i], a[size-1] = a[size-1],a[i]
          
  
# Driver code
a = [1, 2, 3]
n = len(a)
heapPermutation(a, n, n)
  
# This code is contributed by ankush_953

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C#

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// C# program to print all permutations using
// Heap's algorithm
using System;
                      
  
public class GFG {
//Prints the array
    static void printArr(int []a, int n)
    {
        for (int i=0; i<n; i++)
            Console.Write(a[i] + " ");
        Console.WriteLine();
    }
   
    //Generating permutation using Heap Algorithm
    static void heapPermutation(int []a, int size, int n)
    {
        // if size becomes 1 then prints the obtained
        // permutation
        if (size == 1)
            printArr(a,n);
   
        for (int i=0; i<size; i++)
        {
            heapPermutation(a, size-1, n);
   
            // if size is odd, swap first and last
            // element
            if (size % 2 == 1)
            {
                int temp = a[0];
                a[0] = a[size-1];
                a[size-1] = temp;
            }
   
            // If size is even, swap ith and last
            // element
            else
            {
                int temp = a[i];
                a[i] = a[size-1];
                a[size-1] = temp;
            }
        }
    }
   
    // Driver code
    public static void Main()
    {
         
        int []a = {1,2,3};
        heapPermutation(a, a.Length, a.Length);
    }
}
   
/* This Java code is contributed by 29AjayKumar*/

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Output:



1 2 3
2 1 3
3 1 2
1 3 2
2 3 1
3 2 1

References:
1. “https://en.wikipedia.org/wiki/Heap%27s_algorithm#cite_note-3

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