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Iterative approach to print all permutations of an Array

  • Difficulty Level : Hard
  • Last Updated : 17 Jun, 2019

Given an array arr[] of size N, the task is to generate and print all permutations of the given array.

Examples:

Input: arr[] = {1, 2}
Output:
1 2
2 1

Input: {0, 1, 2}
Output:
0 1 2
1 0 2
0 2 1
2 0 1
1 2 0
2 1 0

Approach: The recursive methods to solve the above problems are discussed here and here. In this post, an iterative method to output all permutations for a given array will be discussed.
The iterative method acts as a state machine. When the machine is called, it outputs a permutation and move to the next one.



To begin, we need an integer array Indexes to store all the indexes of the input array, and values in array Indexes are initialized to be 0 to n – 1. What we need to do is to permute the Indexes array.

During the iteration, we find the smallest index Increase in the Indexes array such that Indexes[Increase] < Indexes[Increase + 1], which is the first “value increase”. Then, we have Indexes[0] > Indexes[1] > Indexes[2] > … > Indexes[Increase], which is a tract of decreasing values from index[0]. The next steps will be:

  1. Find the index mid such that Indexes[mid] is the greatest with the constraints that 0 ≤ mid ≤ Increase and Indexes[mid] < Indexes[Increase + 1]; since array Indexes is reversely sorted from 0 to Increase, this step can use binary search.
  2. Swap Indexes[Increase + 1] and Indexes[mid].
  3. Reverse Indexes[0] to Indexes[Increase].

When the values in Indexes become n – 1 to 0, there is no “value increase”, and the algorithm terminates.

To output the combination, we loop through the index array and the values of the integer array are the indexes of the input array.

The following image illustrates the iteration in the algorithm.

Below is the implementation of the above approach:

C++




// C++ implementation of the approach
#include <iostream>
using namespace std;
  
template <typename T>
class AllPermutation {
private:
    // The input array for permutation
    const T* Arr;
  
    // Length of the input array
    const int Length;
  
    // Index array to store indexes of input array
    int* Indexes;
  
    // The index of the first "increase"
    // in the Index array which is the smallest
    // i such that Indexes[i] < Indexes[i + 1]
    int Increase;
  
public:
    // Constructor
    AllPermutation(T* arr, int length)
        : Arr(arr), Length(length)
    {
        this->Indexes = nullptr;
        this->Increase = -1;
    }
  
    // Destructor
    ~AllPermutation()
    {
        if (this->Indexes != nullptr) {
            delete[] this->Indexes;
        }
    }
  
    // Initialize and output
    // the first permutation
    void GetFirst()
    {
  
        // Allocate memory for Indexes array
        this->Indexes = new int[this->Length];
  
        // Initialize the values in Index array
        // from 0 to n - 1
        for (int i = 0; i < this->Length; ++i) {
            this->Indexes[i] = i;
        }
  
        // Set the Increase to 0
        // since Indexes[0] = 0 < Indexes[1] = 1
        this->Increase = 0;
  
        // Output the first permutation
        this->Output();
    }
  
    // Function that returns true if it is
    // possible to generate the next permutation
    bool HasNext()
    {
  
        // When Increase is in the end of the array,
        // it is not possible to have next one
        return this->Increase != (this->Length - 1);
    }
  
    // Output the next permutation
    void GetNext()
    {
  
        // Increase is at the very beginning
        if (this->Increase == 0) {
  
            // Swap Index[0] and Index[1]
            this->Swap(this->Increase, this->Increase + 1);
  
            // Update Increase
            this->Increase += 1;
            while (this->Increase < this->Length - 1
                   && this->Indexes[this->Increase]
                          > this->Indexes[this->Increase + 1]) {
                ++this->Increase;
            }
        }
        else {
  
            // Value at Indexes[Increase + 1] is greater than Indexes[0]
            // no need for binary search,
            // just swap Indexes[Increase + 1] and Indexes[0]
            if (this->Indexes[this->Increase + 1] > this->Indexes[0]) {
                this->Swap(this->Increase + 1, 0);
            }
            else {
  
                // Binary search to find the greatest value
                // which is less than Indexes[Increase + 1]
                int start = 0;
                int end = this->Increase;
                int mid = (start + end) / 2;
                int tVal = this->Indexes[this->Increase + 1];
                while (!(this->Indexes[mid] < tVal
                         && this->Indexes[mid - 1] > tVal)) {
                    if (this->Indexes[mid] < tVal) {
                        end = mid - 1;
                    }
                    else {
                        start = mid + 1;
                    }
                    mid = (start + end) / 2;
                }
  
                // Swap
                this->Swap(this->Increase + 1, mid);
            }
  
            // Invert 0 to Increase
            for (int i = 0; i <= this->Increase / 2; ++i) {
                this->Swap(i, this->Increase - i);
            }
  
            // Reset Increase
            this->Increase = 0;
        }
        this->Output();
    }
  
private:
    // Function to output the input array
    void Output()
    {
        for (int i = 0; i < this->Length; ++i) {
  
            // Indexes of the input array
            // are at the Indexes array
            cout << (this->Arr[this->Indexes[i]]) << " ";
        }
        cout << endl;
    }
  
    // Swap two values in the Indexes array
    void Swap(int p, int q)
    {
        int tmp = this->Indexes[p];
        this->Indexes[p] = this->Indexes[q];
        this->Indexes[q] = tmp;
    }
};
  
// Driver code
int main()
{
    int arr[] = { 0, 1, 2 };
    AllPermutation<int> perm(arr, sizeof(arr) / sizeof(int));
    perm.GetFirst();
    while (perm.HasNext()) {
        perm.GetNext();
    }
  
    return 0;
}

Java




// Java implementation of the approach
class AllPermutation 
{
  
    // The input array for permutation
    private final int Arr[];
  
    // Index array to store indexes of input array
    private int Indexes[];
  
    // The index of the first "increase"
    // in the Index array which is the smallest
    // i such that Indexes[i] < Indexes[i + 1]
    private int Increase;
  
    // Constructor
    public AllPermutation(int arr[])
    {
        this.Arr = arr;
        this.Increase = -1;
        this.Indexes = new int[this.Arr.length];
    }
  
    // Initialize and output
    // the first permutation
    public void GetFirst()
    {
  
        // Allocate memory for Indexes array
        this.Indexes = new int[this.Arr.length];
  
        // Initialize the values in Index array
        // from 0 to n - 1
        for (int i = 0; i < Indexes.length; ++i) 
        {
            this.Indexes[i] = i;
        }
  
        // Set the Increase to 0
        // since Indexes[0] = 0 < Indexes[1] = 1
        this.Increase = 0;
  
        // Output the first permutation
        this.Output();
    }
  
    // Function that returns true if it is
    // possible to generate the next permutation
    public boolean HasNext()
    {
  
        // When Increase is in the end of the array,
        // it is not possible to have next one
        return this.Increase != (this.Indexes.length - 1);
    }
  
    // Output the next permutation
    public void GetNext()
    {
  
        // Increase is at the very beginning
        if (this.Increase == 0
        {
  
            // Swap Index[0] and Index[1]
            this.Swap(this.Increase, this.Increase + 1);
  
            // Update Increase
            this.Increase += 1;
            while (this.Increase < this.Indexes.length - 1
                && this.Indexes[this.Increase]
                        > this.Indexes[this.Increase + 1]) 
            {
                ++this.Increase;
            }
        }
        else
        {
  
            // Value at Indexes[Increase + 1] is greater than Indexes[0]
            // no need for binary search,
            // just swap Indexes[Increase + 1] and Indexes[0]
            if (this.Indexes[this.Increase + 1] > this.Indexes[0]) 
            {
                this.Swap(this.Increase + 1, 0);
            }
            else
            {
  
                // Binary search to find the greatest value
                // which is less than Indexes[Increase + 1]
                int start = 0;
                int end = this.Increase;
                int mid = (start + end) / 2;
                int tVal = this.Indexes[this.Increase + 1];
                while (!(this.Indexes[mid]<tVal&& this.Indexes[mid - 1]> tVal)) 
                {
                    if (this.Indexes[mid] < tVal)
                    {
                        end = mid - 1;
                    }
                    else 
                    {
                        start = mid + 1;
                    }
                    mid = (start + end) / 2;
                }
  
                // Swap
                this.Swap(this.Increase + 1, mid);
            }
  
            // Invert 0 to Increase
            for (int i = 0; i <= this.Increase / 2; ++i)
            {
                this.Swap(i, this.Increase - i);
            }
  
            // Reset Increase
            this.Increase = 0;
        }
        this.Output();
    }
  
    // Function to output the input array
    private void Output()
    {
        for (int i = 0; i < this.Indexes.length; ++i) 
        {
  
            // Indexes of the input array
            // are at the Indexes array
            System.out.print(this.Arr[this.Indexes[i]]);
            System.out.print(" ");
        }
        System.out.println();
    }
  
    // Swap two values in the Indexes array
    private void Swap(int p, int q)
    {
        int tmp = this.Indexes[p];
        this.Indexes[p] = this.Indexes[q];
        this.Indexes[q] = tmp;
    }
}
  
// Driver code
class AppDriver 
{
    public static void main(String args[])
    {
        int[] arr = { 0, 1, 2 };
          
        AllPermutation perm = new AllPermutation(arr);
        perm.GetFirst();
        while (perm.HasNext())
        {
            perm.GetNext();
        }
    }
}
  
// This code is contributed by ghanshyampandey

C#




// C# implementation of the approach
using System;
namespace Permutation {
  
class AllPermutation<T> {
  
    // The input array for permutation
    private readonly T[] Arr;
  
    // Index array to store indexes of input array
    private int[] Indexes;
  
    // The index of the first "increase"
    // in the Index array which is the smallest
    // i such that Indexes[i] < Indexes[i + 1]
    private int Increase;
  
    // Constructor
    public AllPermutation(T[] arr)
    {
        this.Arr = arr;
        this.Increase = -1;
    }
  
    // Initialize and output
    // the first permutation
    public void GetFirst()
    {
  
        // Allocate memory for Indexes array
        this.Indexes = new int[this.Arr.Length];
  
        // Initialize the values in Index array
        // from 0 to n - 1
        for (int i = 0; i < Indexes.Length; ++i) {
            this.Indexes[i] = i;
        }
  
        // Set the Increase to 0
        // since Indexes[0] = 0 < Indexes[1] = 1
        this.Increase = 0;
  
        // Output the first permutation
        this.Output();
    }
  
    // Function that returns true if it is
    // possible to generate the next permutation
    public bool HasNext()
    {
  
        // When Increase is in the end of the array,
        // it is not possible to have next one
        return this.Increase != (this.Indexes.Length - 1);
    }
  
    // Output the next permutation
    public void GetNext()
    {
  
        // Increase is at the very beginning
        if (this.Increase == 0) {
  
            // Swap Index[0] and Index[1]
            this.Swap(this.Increase, this.Increase + 1);
  
            // Update Increase
            this.Increase += 1;
            while (this.Increase < this.Indexes.Length - 1
                   && this.Indexes[this.Increase]
                          > this.Indexes[this.Increase + 1]) {
                ++this.Increase;
            }
        }
        else {
  
            // Value at Indexes[Increase + 1] is greater than Indexes[0]
            // no need for binary search,
            // just swap Indexes[Increase + 1] and Indexes[0]
            if (this.Indexes[this.Increase + 1] > this.Indexes[0]) {
                this.Swap(this.Increase + 1, 0);
            }
            else {
  
                // Binary search to find the greatest value
                // which is less than Indexes[Increase + 1]
                int start = 0;
                int end = this.Increase;
                int mid = (start + end) / 2;
                int tVal = this.Indexes[this.Increase + 1];
                while (!(this.Indexes[mid]<tVal&& this.Indexes[mid - 1]> tVal)) {
                    if (this.Indexes[mid] < tVal) {
                        end = mid - 1;
                    }
                    else {
                        start = mid + 1;
                    }
                    mid = (start + end) / 2;
                }
  
                // Swap
                this.Swap(this.Increase + 1, mid);
            }
  
            // Invert 0 to Increase
            for (int i = 0; i <= this.Increase / 2; ++i) {
                this.Swap(i, this.Increase - i);
            }
  
            // Reset Increase
            this.Increase = 0;
        }
        this.Output();
    }
  
    // Function to output the input array
    private void Output()
    {
        for (int i = 0; i < this.Indexes.Length; ++i) {
  
            // Indexes of the input array
            // are at the Indexes array
            Console.Write(this.Arr[this.Indexes[i]]);
            Console.Write(" ");
        }
        Console.WriteLine();
    }
  
    // Swap two values in the Indexes array
    private void Swap(int p, int q)
    {
        int tmp = this.Indexes[p];
        this.Indexes[p] = this.Indexes[q];
        this.Indexes[q] = tmp;
    }
}
  
// Driver code
class AppDriver {
    static void Main()
    {
        int[] arr = { 0, 1, 2 };
        AllPermutation<int> perm = new AllPermutation<int>(arr);
        perm.GetFirst();
        while (perm.HasNext()) {
            perm.GetNext();
        }
    }
}
}
Output:
0 1 2 
1 0 2 
0 2 1 
2 0 1 
1 2 0 
2 1 0

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