Generate permutation of [1, N] having bitwise XOR of adjacent differences as 0

Given an integer N, the task is to generate a permutation from 1 to N such that the bitwise XOR of differences between adjacent elements is 0 i.e., | A[1]− A[2] | ^ | A[2]− A[3] | ^ . . . ^ | A[N −1] − A[N] | = 0, where |X - Y| represents absolute difference between X and Y.

Examples:

Input: N = 4
Output: 2 3 1 4
Explanation:  |2 -3| ^ |3 -1| ^ |1-4| = 1 ^ 2 ^ 3 = 0

Input: N = 3
Output: 1 2 3

Approach: This problem can be solved based on the following observation:

• The XOR of even number of same elements is 0. So if odd number of elements are there (which implies even number of adjacent differences) then arrange them in a  way such that the difference between any two adjacent elements is same. 
• Else if N is even (which implies odd number of adjacent differences) then arrange the first four in such a way that the XOR of first three differences is 0. Then the remaining elements in the above mentioned away.

Follow the steps mentioned below to implement the above observation:

• If N is odd, arrange all the N elements in a sorted manner because the difference between any two adjacent elements will be 1 and the number of adjacent differences are even.
• If N is even:
  • Keep 2, 3, 1, 4 as the first four elements because the 3 differences have XOR 0.
  • Now start from 5 and print the remaining elements in sorted order, which will give the difference as 1 for all the remaining even number of differences.

Below is the implementation of the above approach.

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;

// Function to print shuffle array
vector<int> shuffleArray(int n)
{
    vector<int> res;

    // Base case
    if (n < 3)
        cout << -1 << endl;

    // When n is odd print array in
    // increasing order
    else if (n % 2 != 0) {
        for (int i = 1; i <= n; i++)
            res.push_back(i);
    }

    // When n is even print first 2 3 1 4
    // rest element in increasing order
    else {
        res = { 2, 3, 1, 4 };
        for (int i = 5; i <= n; i++)
            res.push_back(i);
    }
    return res;
}

// Driver code
int main()
{
    int N = 4;

    vector<int> ans = shuffleArray(N);
    for (int x : ans)
        cout << x << " ";
    return 0;
}

// Java implementation of above approach
import java.util.*;
public class GFG {

  // Function to print shuffle array
  static ArrayList<Integer> shuffleArray(int n)
  {
    ArrayList<Integer> res = new ArrayList<Integer>();

    // Base case
    if (n < 3)
      System.out.println(-1);

    // When n is odd print array in
    // increasing order
    else if (n % 2 != 0) {
      for (int i = 1; i <= n; i++)
        res.add(i);
    }

    // When n is even print first 2 3 1 4
    // rest element in increasing order
    else {
      res.clear();

      res.add(2);
      res.add(3);
      res.add(1);
      res.add(4);

      for (int i = 5; i <= n; i++)
        res.add(i);
    }
    return res;
  }

  // Driver code
  public static void main(String args[])
  {
    int N = 4;

    ArrayList<Integer> ans = shuffleArray(N);
    for (int i = 0; i < ans.size(); i++)
      System.out.print(ans.get(i) + " ");
  }
}

// This code is contributed by Samim Hossain Mondal.

# Python3 implementation of above approach

# Function to print shuffle array
def shuffleArray(n):

    res = []

    # Base case
    if (n < 3):
        print(-1)

    # When n is odd print array in
    # increasing order
    elif (n % 2 != 0):
        for i in range(1, n):
            res.append(i)

    # When n is even print first 2 3 1 4
    # rest element in increasing order
    else:
        res = [2, 3, 1, 4]
        for i in range(5, n):
            res.append(i)

    return res

# Driver code
if __name__ == '__main__':
    n = 4
    res = shuffleArray(n)
    for i in res:
        print(i, end=' ')

        # This code is contributed by richasalan57.

// C# implementation of above approach
using System;
using System.Collections;

class GFG
{

// Function to print shuffle array
static ArrayList shuffleArray(int n)
{
    ArrayList res = new ArrayList();

    // Base case
    if (n < 3)
        Console.WriteLine(-1);

    // When n is odd print array in
    // increasing order
    else if (n % 2 != 0) {
        for (int i = 1; i <= n; i++)
            res.Add(i);
    }

    // When n is even print first 2 3 1 4
    // rest element in increasing order
    else {
        res.Clear();
        
        res.Add(2);
        res.Add(3);
        res.Add(1);
        res.Add(4);
        
        for (int i = 5; i <= n; i++)
            res.Add(i);
    }
    return res;
}

// Driver code
public static void Main()
{
    int N = 4;

    ArrayList ans = shuffleArray(N);
    foreach (int x in ans)
        Console.Write(x + " ");
}
}

// This code is contributed by Samim Hossain Mondal.

<script>

// JavaScript implementation of above approach

// Function to print shuffle array
function shuffleArray(n)
{
    let res = [];

    // Base case
    if (n < 3)
        document.write(-1,"</br>");

    // When n is odd print array in
    // increasing order
    else if (n % 2 != 0) {
        for (let i = 1; i <= n; i++)
            res.push(i);
    }

    // When n is even print first 2 3 1 4
    // rest element in increasing order
    else {
        res = [ 2, 3, 1, 4 ];
        for (let i = 5; i <= n; i++)
            res.push(i);
    }
    return res;
}

// Driver code
let N = 4;

let ans = shuffleArray(N);
for(let x of ans)
    document.write(x," ");

// This code is contributed by shinjanpatra

</script>

2 3 1 4 

Time Complexity: O(N)
Auxiliary space: O(N) because it is using extra space for vector res