Heap’s Algorithm for generating permutations

Last Updated : 14 Dec, 2022

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:

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.
If n is odd, swap the first and last element and if n is even, then swap the i^th element (i is the counter starting from 0) and the last element and repeat the above algorithm till i is less than n.
In each iteration, the algorithm will produce all the permutations that end with the current last element.

Implementation:

C++

   // 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 0th i.e (first) and 
        // (size-1)th i.e (last) element
        if (size % 2 == 1)
            swap(a[0], a[size - 1]);
 
        // If size is even, swap ith and 
        // (size-1)th i.e (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;
}
 
  

Java

   // Java program to print all permutations using
// Heap's algorithm
import java.io.*;
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 0th i.e (first) and
            // (size-1)th i.e (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 (size-1)th i.e 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.
 
  

Python3

   # Python program to print all permutations using
# Heap's algorithm
 
# Generating permutation using Heap Algorithm
def heapPermutation(a, size):
 
    # if size becomes 1 then prints the obtained
    # permutation
    if size == 1:
        print(a)
        return
 
    for i in range(size):
        heapPermutation(a, size-1)
 
        # if size is odd, swap 0th i.e (first)
        # and (size-1)th i.e (last) element
        # else If size is even, swap ith
        # and (size-1)th i.e (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)
 
# This code is contributed by ankush_953
# This code was cleaned up to by more pythonic by glubs9
 
  

C#

   // 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 0th i.e (first) and
            // (size-1)th i.e (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
            // (size-1)th i.e (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*/
 
  

Javascript

   <script>
 
// JavaScript program to print all permutations using
// Heap's algorithm
 
// Prints the array
function printArr(a,n)
{
    document.write(a.join(" ")+"<br>");
     
}
 
// Generating permutation using Heap Algorithm
function heapPermutation(a,size,n)
{
        // if size becomes 1 then prints the obtained
        // permutation
        if (size == 1)
            printArr(a, n);
  
        for (let i = 0; i < size; i++) {
            heapPermutation(a, size - 1, n);
  
            // if size is odd, swap 0th i.e (first) and
            // (size-1)th i.e (last) element
            if (size % 2 == 1) {
                let temp = a[0];
                a[0] = a[size - 1];
                a[size - 1] = temp;
            }
  
            // If size is even, swap ith
            // and (size-1)th i.e last element
            else {
                let temp = a[i];
                a[i] = a[size - 1];
                a[size - 1] = temp;
            }
        }
}
 
// Driver code
let a=[1, 2, 3];
heapPermutation(a, a.length, a.length);
 
 
// This code is contributed by rag2127
 
</script>
 
  

Output

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

Time Complexity: O(N*N!), where N is the size of the given array.
Auxiliary Space: O(N), for recursive stack space of size N.

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

Generate all permutations of a string that follow given constraints

Rahul Agrawal

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Mathematical

Heap’s Algorithm for generating permutations

C++

Java

Python3

C#

Javascript

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