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Generate a circular permutation with number of mismatching bits between pairs of adjacent elements exactly 1
  • Last Updated : 04 Mar, 2021
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Given two integers N and S, the task is to find a circular permutation of numbers from the range [0, 2(N – 1)], starting with S such that the count of mismatching bits between any pair of adjacent numbers is one.

Examples:  

Input: N = 2, S = 3
Output: [3, 2, 0, 1]
Explanation: 
The binary representation of numbers 3, 2, 0 and 1 are “11”, “10”, “01” and “00” respectively.
Therefore, arranging them in the order [3, 2, 0, 1] ensures that the number of bit differences between each pair of adjacent elements (circular) is 1.

Input: N = 3, S = 2
Output: [2, 6, 7, 5, 4, 0, 1, 3]

Approach: The given problem can be solved based on the following observations: 



  • A simple observation is that the numbers in the range [2i, 2i + 1 – 1] can be obtained in their natural order by placing ‘1’s before each number in the range [0, 2i – 1].
  • Therefore, the problem can be solved recursively by adding ‘1’ before each number before 2i – 1th index and reverse it before appending the new numbers to permutation.

Follow the steps below to solve the problem: 

  • Initialize a list, say res, to stores the required permutation.
  • Initialize an integer, say index, to store the position of S in the permutation starting with 0.
  • Iterate over the range [0, N – 1] and traverse the array res[] in reverse order and check if the sum of the current number and 2i is S or not. If found to be true, then update index with the current index of res and append the current number + 2i in the list res.
  • Rotate the list res[] by index positions.
  • After completing the above steps, print the list res[] as the answer.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the permutation of
// integers from a given range such that
// number of mismatching bits between
// pairs of adjacent elements is 1
vector<int> circularPermutation(int n, int start)
{
     
    // Initialize an arrayList to
    // store the resultant permutation
    vector<int> res = {0};
    vector<int> ret;
     
    // Store the index of rotation
    int index = -1;
     
    // Iterate over the range [0, N - 1]
    for(int k = 0, add = 1 << k; k < n;
            k++, add = 1 << k)
    {
     
        // Traverse all the array elements
        // up to (2 ^ k)-th index in reverse
        for (int i = res.size() - 1;
                 i >= 0; i--)
        {
         
            // If current element is S
            if (res[i] + add == start)
                index = res.size();
                 
            res.push_back(res[i] + add);
        }
    }
     
    // Check if S is zero
    if (start == 0)
        return res;
     
    // Rotate the array by index
    // value to the left
    while (ret.size() < res.size())
    {
        ret.push_back(res[index]);
        index = (index + 1) % res.size();
    }
    return ret;
}
 
// Driver Code
int main()
{
    int N = 2, S = 3;
    vector<int> print = circularPermutation(N, S);
    cout << "[";
    for(int i = 0; i < print.size() - 1; i++ )
    {
        cout << print[i] << ", ";
    }
    cout << print[print.size() - 1] << "]";
     
    return 0;
}
 
// This code is contributed by susmitakundugoaldanga

Java




// Java program for the above approach
 
import java.io.*;
import java.util.*;
 
class GFG {
 
    // Function to find the permutation of
    // integers from a given range such that
    // number of mismatching bits between
    // pairs of adjacent elements is 1
    public static List<Integer> circularPermutation(
        int n, int start)
    {
        // Initialize an arrayList to
        // store the resultant permutation
        List<Integer> res = new ArrayList<>(List.of(0)),
                      ret = new ArrayList<>();
 
        // Store the index of rotation
        int index = -1;
 
        // Iterate over the range [0, N - 1]
        for (int k = 0, add = 1 << k; k < n;
             k++, add = 1 << k) {
 
            // Traverse all the array elements
            // up to (2 ^ k)-th index in reverse
            for (int i = res.size() - 1;
                 i >= 0; i--) {
 
                // If current element is S
                if (res.get(i) + add == start)
                    index = res.size();
 
                res.add(res.get(i) + add);
            }
        }
 
        // Check if S is zero
        if (start == 0)
            return res;
 
        // Rotate the array by index
        // value to the left
        while (ret.size() < res.size()) {
            ret.add(res.get(index));
            index = (index + 1) % res.size();
        }
 
        return ret;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int N = 2, S = 3;
 
        System.out.println(
            circularPermutation(N, S));
    }
}

Python3




# Python3 program for the above approach
 
# Function to find the permutation of
# integers from a given range such that
# number of mismatching bits between
# pairs of adjacent elements is 1
def circularPermutation(n, start):
   
    # Initialize an arrayList to
    # store the resultant permutation
    res = [0]
    ret = []
 
    # Store the index of rotation
    index, add = -1, 1
 
    # Iterate over the range [0, N - 1]
    for k in range(n):
        add = 1<<k
 
        # Traverse all the array elements
        # up to (2 ^ k)-th index in reverse
        for i in range(len(res) - 1, -1, -1):
 
            # If current element is S
            if (res[i] + add == start):
                index = len(res)
            res.append(res[i] + add)
        add  = 1 << k
 
    # Check if S is zero
    if (start == 0):
        return res
 
    # Rotate the array by index
    # value to the left
    while (len(ret) < len(res)):
        ret.append(res[index])
        index = (index + 1) % len(res)
    return ret
 
# Driver Code
if __name__ == '__main__':
    N,S = 2, 3
 
    print (circularPermutation(N, S))
 
    # This code is contributed by mohit kumar 29.

C#




// C# program for the above approach
using System;
using System.Collections.Generic;
 
public class GFG
{
 
  // Function to find the permutation of
  // integers from a given range such that
  // number of mismatching bits between
  // pairs of adjacent elements is 1
  public static List<int> circularPermutation(
    int n, int start)
  {
 
    // Initialize an arrayList to
    // store the resultant permutation
    List<int> res = new List<int>(){0};
    List<int> ret = new List<int>();
 
    // Store the index of rotation
    int index = -1;
 
    // Iterate over the range [0, N - 1]
    for (int k = 0, add = 1 << k; k < n;
         k++, add = 1 << k)
    {
 
      // Traverse all the array elements
      // up to (2 ^ k)-th index in reverse
      for (int i = res.Count - 1;
           i >= 0; i--)
      {
 
        // If current element is S
        if (res[i] + add == start)
          index = res.Count;
        res.Add(res[i] + add);
      }
    }
 
    // Check if S is zero
    if (start == 0)
      return res;
 
    // Rotate the array by index
    // value to the left
    while (ret.Count < res.Count)
    {
      ret.Add(res[index]);
      index = (index + 1) % res.Count;
    }
    return ret;
  }
 
  // Driver Code
  static public void Main ()
  {
    int N = 2, S = 3;
    List<int> print = circularPermutation(N, S);
    Console.Write("[");
    for(int i = 0; i < print.Count - 1; i++ )
    {
      Console.Write(print[i] + ", ");
    }
    Console.Write(print[print.Count-1] + "]");
  }
}
 
// This code is contributed by avanitrachhadiya2155
Output: 
[3, 2, 0, 1]

 

Time Complexity: O(N2)
Auxiliary Space: O(N)

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