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Find first K characters in Nth term of the Thue-Morse sequence

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  • Last Updated : 23 Feb, 2022

Given two integers N and K, the task is to print the first K bits of the Nth term of the Thue-Morse sequence. The Thue-Morse sequence is a binary sequence. It starts with a “0” as its first term. And then after the next term is generated by replacing “0” with “01” and “1” with “10”.

Examples:

Input: N = 3, K = 2
Output: 01
Explanation: The 1st term is “0”.
The 2nd term is obtained by replacing “0” with “01” i.e. 2nd term is “01”.
The 3rd term in the sequence is obtained by replacing “0” with “01” and “1” with “10”.
So the 3rd term becomes “0110”. Hence, the 1st 2 characters of the 3rd term is “01”.

Input: N = 4, K = 7
Output: 0110100

 

Approach: The basic approach to solve this problem is to generate the Nth term of the sequence and print the first K characters of that term. This can be done using the algorithm discussed here.

Time Complexity: O(N * 2N)
Auxiliary Space: O(1) 

Efficient Approach: The above approach can be optimized by observing that the ith term is the concatenation of (i – 1)th term and the inverse of (i – 1)th term where inverse means changing the polarity of all bits in a binary integer. Hence, xth term, Ai[x] = Ai-1[x – 1] if (x < 2i-1), otherwise Ai[x] = !Ai-1[x – 2i-1]. Therefore, using this relation, a recursive function can be created to calculate the value of each bit in the Nth term.

Below is the implementation of the above approach:

C++




#include <bits/stdc++.h>
using namespace std;
 
// Recursive function to find the
// value of the kth bit in Nth term
int findDig(int N, long K,  int curr)
{
   
  // Base Case
  if (N == 0) {
    return curr;
  }
 
  // Stores the middle index
  long middle = (long)pow(2, N) / 2;
 
  // If K lies in 1st part
  if (K <= middle) {
 
    // Recursive Call
    return findDig(N - 1, K, curr);
  }
 
  // If K lies in 2nd part
  // having inverted value
  else {
    if (curr == 0) {
      curr = 1;
    }
    else {
      curr = 0;
    }
 
    // Recursive Call
    return findDig(N - 1,
                   K - middle, curr);
  }
}
 
// Function to find first K characters
// in Nth term of Thue-Morse sequence
void firstKTerms(int N, int K)
{
   
  // Loop to iterate all K bits
  for (int i = 1; i <= K; ++i)
  {
 
    // Print value of ith bit
    cout << (findDig(N, i, 0));
  }
}
 
// Driver Code
int main() {
 
  int N = 4;
  int K = 7;
 
  firstKTerms(N, K);
  return 0;
}
 
// This code is contributed by hrithikgarg03188.


Java




// Java Implementation of the above approach
import java.io.*;
import java.util.*;
class GFG {
 
    // Recursive function to find the
    // value of the kth bit in Nth term
    public static int findDig(int N, long K,
                              int curr)
    {
        // Base Case
        if (N == 0) {
            return curr;
        }
 
        // Stores the middle index
        long middle = (long)Math.pow(2, N) / 2;
 
        // If K lies in 1st part
        if (K <= middle) {
 
            // Recursive Call
            return findDig(N - 1, K, curr);
        }
 
        // If K lies in 2nd part
        // having inverted value
        else {
            if (curr == 0) {
                curr = 1;
            }
            else {
                curr = 0;
            }
 
            // Recursive Call
            return findDig(N - 1,
                           K - middle, curr);
        }
    }
 
    // Function to find first K characters
    // in Nth term of Thue-Morse sequence
    public static void firstKTerms(int N, int K)
    {
        // Loop to iterate all K bits
        for (int i = 1; i <= K; ++i) {
 
            // Print value of ith bit
            System.out.print(findDig(N, i, 0));
        }
    }
 
    // Driver Code
    public static void main(String args[])
    {
        int N = 4;
        int K = 7;
 
        firstKTerms(N, K);
    }
}


Python3




# Recursive function to find the
# value of the kth bit in Nth term
def findDig(N, K, curr):
 
    # Base Case
    if (N == 0):
        return curr
 
    # Stores the middle index
    middle = pow(2, N) // 2
 
    # If K lies in 1st part
    if (K <= middle):
 
        # Recursive Call
        return findDig(N - 1, K, curr)
 
    # If K lies in 2nd part
    # having inverted value
    else:
        if (curr == 0):
            curr = 1
 
        else:
            curr = 0
 
        # Recursive Call
        return findDig(N - 1, K - middle, curr)
 
# Function to find first K characters
# in Nth term of Thue-Morse sequence
def firstKTerms(N, K):
 
    # Loop to iterate all K bits
    for i in range(1, K+1):
 
     # Print value of ith bit
        print(findDig(N, i, 0), end="")
 
# Driver Code
if __name__ == "__main__":
 
    N = 4
    K = 7
 
    firstKTerms(N, K)
 
    # This code is contributed by rakeshsahni


C#




// C# Implementation of the above approach
using System;
class GFG
{
 
  // Recursive function to find the
  // value of the kth bit in Nth term
  public static int findDig(int N, long K, int curr)
  {
     
    // Base Case
    if (N == 0)
    {
      return curr;
    }
 
    // Stores the middle index
    long middle = (long)Math.Pow(2, N) / 2;
 
    // If K lies in 1st part
    if (K <= middle)
    {
 
      // Recursive Call
      return findDig(N - 1, K, curr);
    }
 
    // If K lies in 2nd part
    // having inverted value
    else
    {
      if (curr == 0)
      {
        curr = 1;
      }
      else
      {
        curr = 0;
      }
 
      // Recursive Call
      return findDig(N - 1,
                     K - middle, curr);
    }
  }
 
  // Function to find first K characters
  // in Nth term of Thue-Morse sequence
  public static void firstKTerms(int N, int K)
  {
     
    // Loop to iterate all K bits
    for (int i = 1; i <= K; ++i)
    {
 
      // Print value of ith bit
      Console.Write(findDig(N, i, 0));
    }
  }
 
  // Driver Code
  public static void Main()
  {
    int N = 4;
    int K = 7;
 
    firstKTerms(N, K);
  }
}
 
// This code is contributed by saurabh_jaiswal.


Javascript




<script>
       // JavaScript code for the above approach
 
       // Recursive function to find the
       // value of the kth bit in Nth term
       function findDig(N, K, curr)
       {
           // Base Case
           if (N == 0) {
               return curr;
           }
 
           // Stores the middle index
           let middle = Math.floor(Math.pow(2, N) / 2);
 
           // If K lies in 1st part
           if (K <= middle) {
 
               // Recursive Call
               return findDig(N - 1, K, curr);
           }
 
           // If K lies in 2nd part
           // having inverted value
           else {
               if (curr == 0) {
                   curr = 1;
               }
               else {
                   curr = 0;
               }
 
               // Recursive Call
               return findDig(N - 1,
                   K - middle, curr);
           }
       }
 
       // Function to find first K characters
       // in Nth term of Thue-Morse sequence
       function firstKTerms(N, K) {
           // Loop to iterate all K bits
           for (let i = 1; i <= K; ++i) {
 
               // Print value of ith bit
               document.write(findDig(N, i, 0));
           }
       }
 
       // Driver Code
       let N = 4;
       let K = 7;
 
       firstKTerms(N, K);
 
        // This code is contributed by Potta Lokesh
   </script>


 
 

Output

0110100

 

Time Complexity: O(K*log N)
Auxiliary Space: O(1) 

 


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