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 ith 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 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.
# 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# 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*/
<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>
Output1 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
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