Count of binary strings of length N with even set bit count and at most K consecutive 1s

Given two integers N and K, the task is to find the number of binary strings of length N having an even number of 1’s out of which less than K are consecutive.
Examples: 

Input: N = 4, K = 2 
Output:
Explanation: 
The possible binary strings are 0000, 0101, 1001, 1010. They all have even number of 1’s with less than 2 of them occuring consecutively.

Input: N = 3, K = 2 
Output:
Explanation: 
The possible binary strings are 000, 101. All other strings that is 001, 010, 011, 100, 110, 111 does not meet the criteria.

Approach: 
This problem can be solved by Dynamic Programming.
Let us consider a 3D table dp[][][] to store the solution of each subproblem, such that, dp[n][i][s] denotes the number of binary strings of length n having i consecutive 1’s and sum of 1’s = s. As it is only required to check whether the total number of 1’s is even or not we store s % 2. So, dp[n][i][s] can be calculated as follows: 

  1. If we place 0 at the nth position, the number of 1’s remain unchanged. Hence, dp[n][i][s] = dp[n – 1][0][s].
  2. If we place 1 at the nth position, dp[n][i][s] = dp[n – 1][i + 1][(s + 1) % 2] .
  3. From the above two points the recurrence relation formed is given by: 



Below is the implementation of the above approach:

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// C++ program for the above approach
#include <bits/stdc++.h> 
using namespace std;
  
// Table to store solution
// of each subproblem
int dp[100001][20][2];
  
// Function to calculate
// the possible binary
// strings
int possibleBinaries(int pos,
                     int ones,
                     int sum,
                     int k)
{
    // If number of ones
    // is equal to K
    if (ones == k)
        return 0;
  
    // pos: current position
    // Base Case: When n
    // length is traversed
    if (pos == 0)
  
        // sum: count of 1's
        // Return the count
        // of 1's obtained
        return (sum == 0) ? 1 : 0;
  
    // If the subproblem has already
    // been solved
    if (dp[pos][ones][sum] != -1)
        // Return the answer
        return dp[pos][ones][sum];
  
    // Recursive call when current
    // position is filled with 1
    int ret = possibleBinaries(pos - 1,
                               ones + 1,
                               (sum + 1) % 2,
                               k)
              // Recursive call when current
              // position is filled with 0
              + possibleBinaries(pos - 1, 0,
                                 sum, k);
  
    // Store the solution
    // to this subproblem
    dp[pos][ones][sum] = ret;
  
    return dp[pos][ones][sum];
}
  
// Driver Code
int main()
{
    int N = 3;
    int K = 2;
  
    // Initialising the
    // table with -1
    memset(dp, -1, sizeof dp);
  
    cout << possibleBinaries(N, 0, 0, K);
}
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// Java program for the above approach
class GFG{
  
// Table to store solution
// of each subproblem
static int [][][]dp = new int[100001][20][2];
  
// Function to calculate
// the possible binary
// Strings
static int possibleBinaries(int pos, int ones,
                            int sum, int k)
{
      
    // If number of ones
    // is equal to K
    if (ones == k)
        return 0;
  
    // pos: current position
    // Base Case: When n
    // length is traversed
    if (pos == 0)
  
        // sum: count of 1's
        // Return the count
        // of 1's obtained
        return (sum == 0) ? 1 : 0;
  
    // If the subproblem has already
    // been solved
    if (dp[pos][ones][sum] != -1)
      
        // Return the answer
        return dp[pos][ones][sum];
  
    // Recursive call when current
    // position is filled with 1
    int ret = possibleBinaries(pos - 1,
                              ones + 1,
                              (sum + 1) % 2, k) + 
                                
              // Recursive call when current
              // position is filled with 0
              possibleBinaries(pos - 1, 0,
                               sum, k);
                                 
    // Store the solution
    // to this subproblem
    dp[pos][ones][sum] = ret;
  
    return dp[pos][ones][sum];
}
  
// Driver Code
public static void main(String[] args)
{
    int N = 3;
    int K = 2;
  
    // Initialising the
    // table with -1
    for(int i = 0; i < 100001; i++)
    {
        for(int j = 0; j < 20; j++)
        {
            for(int l = 0; l < 2; l++)
                dp[i][j][l] = -1;
        }
    }
  
    System.out.print(possibleBinaries(N, 0, 0, K));
}
}
  
// This code is contributed by Rohit_ranjan
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# Python3 program for the above approach 
import numpy as np 
  
# Table to store solution 
# of each subproblem 
dp = np.ones(((100002, 21, 3)))
dp = -1 * dp
  
# Function to calculate 
# the possible binary 
# strings 
def possibleBinaries(pos, ones, sum, k):
      
    # If number of ones 
    # is equal to K 
    if (ones == k):
        return 0
  
    # pos: current position 
    # Base Case: When n 
    # length is traversed 
    if (pos == 0):
  
        # sum: count of 1's 
        # Return the count 
        # of 1's obtained 
        return 1 if (sum == 0) else 0
  
    # If the subproblem has already 
    # been solved 
    if (dp[pos][ones][sum] != -1):
          
        # Return the answer 
        return dp[pos][ones][sum
  
    # Recursive call when current 
    # position is filled with 1 
    ret = (possibleBinaries(pos - 1,
                           ones + 1
                           (sum + 1) % 2, k) + 
      
           # Recursive call when current 
           # position is filled with 0 
           possibleBinaries(pos - 1, 0, sum, k)) 
  
    # Store the solution 
    # to this subproblem 
    dp[pos][ones][sum] = ret 
  
    return dp[pos][ones][sum
  
# Driver Code 
N = 3
K = 2
              
print(int(possibleBinaries(N, 0, 0, K)))
  
# This code is contributed by sanjoy_62
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// C# program for the above approach
using System;
  
class GFG{
  
// Table to store solution
// of each subproblem
static int [,,]dp = new int[100001, 20, 2];
  
// Function to calculate the
// possible binary Strings
static int possibleBinaries(int pos, int ones,
                            int sum, int k)
{
      
    // If number of ones
    // is equal to K
    if (ones == k)
        return 0;
  
    // pos: current position
    // Base Case: When n
    // length is traversed
    if (pos == 0)
  
        // sum: count of 1's
        // Return the count
        // of 1's obtained
        return (sum == 0) ? 1 : 0;
  
    // If the subproblem has already
    // been solved
    if (dp[pos, ones, sum] != -1)
      
        // Return the answer
        return dp[pos, ones, sum];
  
    // Recursive call when current
    // position is filled with 1
    int ret = possibleBinaries(pos - 1,
                                ones + 1,
                                (sum + 1) % 2, k) + 
              // Recursive call when current
              // position is filled with 0
              possibleBinaries(pos - 1, 0,
                               sum, k);
                                 
    // Store the solution
    // to this subproblem
    dp[pos, ones, sum] = ret;
  
    return dp[pos, ones, sum];
}
  
// Driver Code
public static void Main(String[] args)
{
    int N = 3;
    int K = 2;
  
    // Initialising the
    // table with -1
    for(int i = 0; i < 100001; i++)
    {
        for(int j = 0; j < 20; j++)
        {
            for(int l = 0; l < 2; l++)
                dp[i, j, l] = -1;
        }
    }
    Console.Write(possibleBinaries(N, 0, 0, K));
}
}
  
// This code is contributed by Amit Katiyar
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Output: 
2

Time Complexity: O(2*N*K)
 




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