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

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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 occurring 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: 
     

dp[n][i][s] = dp[n-1][0][s] + dp[n - 1][i + 1][(s + 1)mod 2]

Below is the implementation of the above approach:
 

C++




// 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);
}


Java




// Java program for the above approach
import java.io.*;
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


Python3




# 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


C#




// 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


Javascript




<script>
// Javascript program for the above approach
 
// Table to store solution
// of each subproblem
let dp = new Array(100001).fill(-1).map((t) => new Array(20).fill(-1).map((r) => new Array(2).fill(-1)));
 
// Function to calculate
// the possible binary
// strings
function 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 (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
    let 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
let N = 3;
let K = 2;
 
// Initialising the
// table with -1
document.write(possibleBinaries(N, 0, 0, K));
 
// This code is contributed by _saurabh_jaiswal
</script>


Output

2

Time Complexity: O(2*N*K), where N and K represents the given two integers.
 Auxiliary Space: O(100001*20*2), no any other extra space is required, so it is a constant.

Efficient approach : Using DP Tabulation method ( Iterative approach )

The approach to solve this problem is same but DP tabulation(bottom-up) method is better then Dp + memoization(top-down) because memoization method needs extra stack space of recursion calls.

Steps to solve this problem :

  • Create a 3D DP table to store the solution of the subproblems.
  • Initialize the DP  with base cases
  • Now Iterate over subproblems to get the value of current problem form previous computation of subproblems stored in DP
  • Return the final solution stored in dp[N][0][0].

Implementation :

C++




// C++ code for above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate
// the possible binary
// strings
int possibleBinaries(int N, int K) {
     
    // Table to store solution
    // of each subproblem
    int dp[N+1][K+1][2];
    memset(dp, 0, sizeof(dp));
 
    // base case
    for(int i=0; i<=K; i++) {
        dp[1][i][0] = 1;
        dp[1][i][1] = 1;
    }
 
    // iterate over subproblems to  get the current
    // value from previous computation
    for(int i=2; i<=N; i++) {
        for(int j=0; j<=K; j++) {
            for(int k=0; k<=1; k++) {
                if(j == K) {
                    dp[i][j][k] = 0;
                } else if(k == 0) {
                    dp[i][j][k] = dp[i-1][j+1][1];
                } else {
                    dp[i][j][k] = dp[i-1][j+1][0] + dp[i-1][j][1];
                }
            }
        }
    }
     
    // return final answer
    return dp[N][0][0];
}
 
// Driver Code
int main() {
    int N = 3;
    int K = 2;
     
    // Function call
    cout << possibleBinaries(N, K);
}
 
// this code is contributed by bhardwajji


Java




import java.util.Arrays;
 
public class PossibleBinaries {
    // Function to calculate
    // the possible binary
    // strings
    static int possibleBinaries(int N, int K) {
 
        // Table to store solution
        // of each subproblem
        int[][][] dp = new int[N+1][K+1][2];
        for(int i=0; i<=N; i++) {
            for(int j=0; j<=K; j++) {
                Arrays.fill(dp[i][j], 0);
            }
        }
 
        // base case
        for(int i=0; i<=K; i++) {
            dp[1][i][0] = 1;
            dp[1][i][1] = 1;
        }
 
        // iterate over subproblems to  get the current
        // value from previous computation
        for(int i=2; i<=N; i++) {
            for(int j=0; j<=K; j++) {
                for(int k=0; k<=1; k++) {
                    if(j == K) {
                        dp[i][j][k] = 0;
                    } else if(k == 0) {
                        dp[i][j][k] = dp[i-1][j+1][1];
                    } else {
                        dp[i][j][k] = dp[i-1][j+1][0] + dp[i-1][j][1];
                    }
                }
            }
        }
 
        // return final answer
        return dp[N][0][0];
    }
 
    // Driver Code
    public static void main(String[] args) {
        int N = 3;
        int K = 2;
 
        // Function call
        System.out.println(possibleBinaries(N, K));
    }
}


Python3




# Function to calculate
# the possible binary
# strings
def possibleBinaries(N, K):
 
    # Table to store solution
    # of each subproblem
    dp = [[[0 for _ in range(2)] for _ in range(K+1)] for _ in range(N+1)]
 
    # base case
    for i in range(K+1):
        dp[1][i][0] = 1
        dp[1][i][1] = 1
 
    # iterate over subproblems to get the current
    # value from previous computation
    for i in range(2, N+1):
        for j in range(K+1):
            for k in range(2):
                if j == K:
                    dp[i][j][k] = 0
                elif k == 0:
                    dp[i][j][k] = dp[i-1][j+1][1]
                else:
                    dp[i][j][k] = dp[i-1][j+1][0] + dp[i-1][j][1]
 
    # return final answer
    return dp[N][0][0]
 
# Driver Code
N = 3
K = 2
 
# Function call
print(possibleBinaries(N, K))


C#




using System;
 
public class Program {
    // Function to calculate
    // the possible binary
    // strings
    public static int PossibleBinaries(int N, int K) {
        // Table to store solution
        // of each subproblem
        int[,,] dp = new int[N+1, K+1, 2];
 
        // base case
        for(int i=0; i<=K; i++) {
            dp[1, i, 0] = 1;
            dp[1, i, 1] = 1;
        }
 
        // iterate over subproblems to  get the current
        // value from previous computation
        for(int i=2; i<=N; i++) {
            for(int j=0; j<=K; j++) {
                for(int k=0; k<=1; k++) {
                    if(j == K) {
                        dp[i, j, k] = 0;
                    } else if(k == 0) {
                        dp[i, j, k] = dp[i-1, j+1, 1];
                    } else {
                        dp[i, j, k] = dp[i-1, j+1, 0] + dp[i-1, j, 1];
                    }
                }
            }
        }
         
        // return final answer
        return dp[N, 0, 0];
    }
 
    // Driver Code
    public static void Main() {
        int N = 3;
        int K = 2;
         
        // Function call
        Console.WriteLine(PossibleBinaries(N, K));
    }
}


Javascript




// Javascript implementation of given problem
 
// Function to calculate
// the possible binary
// strings
function possibleBinaries(N, K) {
 
    // Table to store solution
    // of each subproblem
    var dp = new Array(N+1);
    for (var i = 0; i < dp.length; i++) {
        dp[i] = new Array(K+1);
        for (var j = 0; j < dp[i].length; j++) {
            dp[i][j] = new Array(2);
            for (var k = 0; k < dp[i][j].length; k++) {
                dp[i][j][k] = 0;
            }
        }
    }
 
    // base case
    for (var i = 0; i < K+1; i++) {
        dp[1][i][0] = 1;
        dp[1][i][1] = 1;
    }
 
    // iterate over subproblems to get the current
    // value from previous computation
    for (var i = 2; i < N+1; i++) {
        for (var j = 0; j < K+1; j++) {
            for (var k = 0; k < 2; k++) {
                if (j == K) {
                    dp[i][j][k] = 0;
                } else if (k == 0) {
                    dp[i][j][k] = dp[i-1][j+1][1];
                } else {
                    dp[i][j][k] = dp[i-1][j+1][0] + dp[i-1][j][1];
                }
            }
        }
    }
 
    // return final answer
    return dp[N][0][0];
}
 
// Driver Code
var N = 3;
var K = 2;
 
// Function call
console.log(possibleBinaries(N, K));
 
// This code is contributed by Tapesh(tapeshdua420)


Output

2

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


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Last Updated : 06 Jul, 2023
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