Count of Binary Strings of length at most N with set bit count as multiple of K

Given two integers N and K, the task is to find the count of binary strings of at most N length that can be formed such that the count of consecutive 1‘s is always a multiple of K.

Example:

Input: N = 3, K = 2
Output: 6
Explanation: Binary strings of atmost N length containing consecutive 1’s as a multiple of K are as follows:
Length 1: “0”, contains 0 consecutive 1.
Length 2: “00”, “11”, contains 0 and 2 consecutive 1’s respectively.
Length 3: “000”, “011”, “110”, contains 0 and two different combinations of 2 consecutive 1’s respectively.
So, total number of strings that can be formed is 6.
Input: N = 5, K = 4
Output: 8

Approach: The given problem can be solved with the help of Dynamic Programming using memoization. Follow the below steps to solve the given problem:

Create a recursive function cntStrings(N, K), which returns the number of strings of N length having the consecutive 1’s as multiples of K. This can be done by assigning 1 to the next K consecutive indices from the current index and recursively calling for the remaining string or assigning 0 to the current index and recursively calling for the remaining string.
Create an array dp[] which stores the memorized values of the above recursive function.
Call the function cntStrings(i, K) for all possible values of i in the range [1, N] and store their sum in a variable cnt.
The value stored in cnt is the required answer.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
int dp[1001];
 
// Recursive function to find the count
// of valid binary strings of length n
int cntString(int n, int k)
{
    // Base Case
    if (n == 0) {
        return 1;
    }
 
    // If current value is already calculated
    if (dp[n] != -1) {
        return dp[n];
    }
 
    // Stores the current answer
    int ans = 0;
 
    // Case for element at next k indices as 1
    if (n >= k) {
        ans += cntString(n - k, k);
    }
 
    // Case for element at current index as 0
    ans += cntString(n - 1, k);
 
    // Return ans with storing it in dp[]
    return dp[n] = ans;
}
 
// Function to find the count of valid
// binary strings of atmost N length
int cntStringAll(int N, int K)
{
    // Initializing all elements with -1
    memset(dp, -1, sizeof(dp));
 
    // Stores the final answer
    int cnt = 0;
 
    // Iterate and calculate the total
    // possible binary strings of each
    // length in the range [1, N]
    for (int i = 1; i <= N; i++) {
        cnt += cntString(i, K);
    }
 
    // Return Answer
    return cnt;
}
 
// Driver Code
int main()
{
    int N = 5;
    int K = 4;
 
    cout << cntStringAll(N, K);
 
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
class GFG {
 
    static int dp[] = new int[1001];
 
    // Recursive function to find the count
    // of valid binary strings of length n
    static int cntString(int n, int k)
    {
        // Base Case
        if (n == 0) {
            return 1;
        }
 
        // If current value is already calculated
        if (dp[n] != -1) {
            return dp[n];
        }
 
        // Stores the current answer
        int ans = 0;
 
        // Case for element at next k indices as 1
        if (n >= k) {
            ans += cntString(n - k, k);
        }
 
        // Case for element at current index as 0
        ans += cntString(n - 1, k);
 
        // Return ans with storing it in dp[]
        return dp[n] = ans;
    }
 
    // Function to find the count of valid
    // binary strings of atmost N length
    static int cntStringAll(int N, int K)
    {
        // Initializing all elements with -1
        for (int i = 0; i < 1001; i++)
            dp[i] = -1;
 
        // Stores the final answer
        int cnt = 0;
 
        // Iterate and calculate the total
        // possible binary strings of each
        // length in the range [1, N]
        for (int i = 1; i <= N; i++) {
            cnt += cntString(i, K);
        }
 
        // Return Answer
        return cnt;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int N = 5;
        int K = 4;
 
        System.out.println(cntStringAll(N, K));
    }
}
 
// This code is contributed by dwivediyash

Python3

# python program for the above approach
dp = [-1 for _ in range(1001)]
 
# Recursive function to find the count
# of valid binary strings of length n
def cntString(n, k):
 
        # Base Case
    if (n == 0):
        return 1
 
        # If current value is already calculated
    if (dp[n] != -1):
        return dp[n]
 
        # Stores the current answer
    ans = 0
 
    # Case for element at next k indices as 1
    if (n >= k):
        ans += cntString(n - k, k)
 
        # Case for element at current index as 0
    ans += cntString(n - 1, k)
 
    # Return ans with storing it in dp[]
    dp[n] = ans
 
    return dp[n]
 
 
# Function to find the count of valid
# binary strings of atmost N length
def cntStringAll(N, K):
 
        # Stores the final answer
    cnt = 0
 
    # Iterate and calculate the total
    # possible binary strings of each
    # length in the range [1, N]
    for i in range(1, N + 1):
        cnt += cntString(i, K)
 
    # Return Answer
    return cnt
 
# Driver Code
if __name__ == "__main__":
 
    N = 5
    K = 4
 
    print(cntStringAll(N, K))
 
    # This code is contributed by rakeshsahni

C#

// C# program for the above approach
using System;
 
public class GFG
{
 
    static int []dp = new int[1001];
 
    // Recursive function to find the count
    // of valid binary strings of length n
    static int cntString(int n, int k)
    {
       
        // Base Case
        if (n == 0) {
            return 1;
        }
 
        // If current value is already calculated
        if (dp[n] != -1) {
            return dp[n];
        }
 
        // Stores the current answer
        int ans = 0;
 
        // Case for element at next k indices as 1
        if (n >= k) {
            ans += cntString(n - k, k);
        }
 
        // Case for element at current index as 0
        ans += cntString(n - 1, k);
 
        // Return ans with storing it in dp[]
        return dp[n] = ans;
    }
 
    // Function to find the count of valid
    // binary strings of atmost N length
    static int cntStringAll(int N, int K)
    {
       
        // Initializing all elements with -1
        for (int i = 0; i < 1001; i++)
            dp[i] = -1;
 
        // Stores the final answer
        int cnt = 0;
 
        // Iterate and calculate the total
        // possible binary strings of each
        // length in the range [1, N]
        for (int i = 1; i <= N; i++) {
            cnt += cntString(i, K);
        }
 
        // Return Answer
        return cnt;
    }
 
    // Driver Code
    public static void Main(String[] args)
    {
        int N = 5;
        int K = 4;
 
        Console.WriteLine(cntStringAll(N, K));
    }
}
 
// This code is contributed by AnkThon

Javascript

<script>
    // JavaScript program for the above approach
 
    let dp = new Array(1001).fill(-1);
 
    // Recursive function to find the count
    // of valid binary strings of length n
    const cntString = (n, k) => {
        // Base Case
        if (n == 0) {
            return 1;
        }
 
        // If current value is already calculated
        if (dp[n] != -1) {
            return dp[n];
        }
 
        // Stores the current answer
        let ans = 0;
 
        // Case for element at next k indices as 1
        if (n >= k) {
            ans += cntString(n - k, k);
        }
 
        // Case for element at current index as 0
        ans += cntString(n - 1, k);
 
        // Return ans with storing it in dp[]
        return dp[n] = ans;
    }
 
    // Function to find the count of valid
    // binary strings of atmost N length
    const cntStringAll = (N, K) => {
 
        // Stores the final answer
        let cnt = 0;
 
        // Iterate and calculate the total
        // possible binary strings of each
        // length in the range [1, N]
        for (let i = 1; i <= N; i++) {
            cnt += cntString(i, K);
        }
 
        // Return Answer
        return cnt;
    }
 
    // Driver Code
    let N = 5;
    let K = 4;
 
    document.write(cntStringAll(N, K));
 
    // This code is contributed by rakeshsahni
 
</script>