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> |
0110100
Time Complexity: O(K*log N)
Auxiliary Space: O(1)