Check if end of given Binary string can be reached by choosing jump value in between given range

Given two positive integers L and R and a binary string S of size N, the task is to check if the end of the string is reached from the index 0 by a series of jumps of indices, say i such that S[i] is 0 jumps are allowed in the range [L, R]. If it is possible to reach, then print Yes. Otherwise, print No.

Examples:

Input: S = "011010", L = 2, R = 3
Output: Yes
Explanation:
Following are the series of moves having characters at that indices as 0:
S[0](= 0) -> S[3](= 0) -> S[5](= 0) and S[5] is the end of the string S.
Therefore, print Yes.

Input: S = "01101110", L = 2, R = 3
Output: No

Approach: The above problem can be solved with the help of Dynamic Programming, the idea is to maintain 1D array, say dp[] where dp[i] will store the possibility of reaching the i^th position and update each index accordingly. Below are the steps:

• Initialize an array, dp[] such that dp[i] stores whether any index i can be reached from index 0 or not. Update the value of dp[0] as 1 as it is the current standing index.
• Initialize a variable, pre as 0 that stores the number of indices from which the current index is reachable.
• Iterate over the range [1, N) and update the value of pre variable as follows:
  • If the value of (i >= minJump), then increment the value of pre by dp[i - minJump].
  • If the value of (i > maxJump), then decrement the value of pre by dp[i - maxJump - 1].
• If the value of pre is positive, then there is at least 1 index from which the current index is reachable. Therefore, update the value of dp[i] = 1, if the value of S[i] = 0.
• After completing the above steps, if the value of dp[N - 1] is positive, then print Yes. Otherwise, print No.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to check if it is possible
// to reach the end of the binary string
// using the given jumps
bool canReach(string s, int L, int R)
{
    // Stores the DP states
    vector<int> dp(s.length());

    // Initial state
    dp[0] = 1;

    // Stores count of indices from which
    // it is possible to reach index i
    int pre = 0;

    // Traverse the given string
    for (int i = 1; i < s.length(); i++) {

        // Update the values of pre
        // accordingly
        if (i >= L) {
            pre += dp[i - L];
        }

        // If the jump size is out of
        // the range [L, R]
        if (i > R) {
            pre -= dp[i - R - 1];
        }

        dp[i] = (pre > 0) and (s[i] == '0');
    }

    // Return answer
    return dp[s.length() - 1];
}

// Driver Code
int main()
{
    string S = "01101110";
    int L = 2, R = 3;

    cout << (canReach(S, L, R) ? "Yes" : "No");

    return 0;
}

// Java program for the above approach

public class GFG {
    
    // Function to check if it is possible
    // to reach the end of the binary string
    // using the given jumps
    static int canReach(String s, int L, int R)
    {
      
        // Stores the DP states
        int dp[] = new int[s.length()];
        
        // Initial state
        dp[0] = 1;
    
        // Stores count of indices from which
        // it is possible to reach index i
        int pre = 0;
    
        // Traverse the given string
        for (int i = 1; i < s.length(); i++) {
    
            // Update the values of pre
            // accordingly
            if (i >= L) {
                pre += dp[i - L];
            }
    
            // If the jump size is out of
            // the range [L, R]
            if (i > R) {
                pre -= dp[i - R - 1];
            }
            
            if (pre > 0 && s.charAt(i) == '0')
                 dp[i] = 1;
            else
                dp[i] = 0;
        }
    
        // Return answer
        return dp[s.length() - 1];
    }
    
    // Driver Code
    public static void main (String[] args)
    {
        String S = "01101110";
        int L = 2, R = 3;
    
        if (canReach(S, L, R) == 1)
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}

// This code is contributed by AnkThon

# python program for the above approach

# Function to check if it is possible
# to reach the end of the binary string
# using the given jumps


def canReach(s, L, R):

    # Stores the DP states
    dp = [0 for _ in range(len(s))]

    # Initial state
    dp[0] = 1

    # Stores count of indices from which
    # it is possible to reach index i
    pre = 0

    # Traverse the given string
    for i in range(1, len(s)):

        # Update the values of pre
        # accordingly
        if (i >= L):
            pre += dp[i - L]

        # If the jump size is out of
        # the range [L, R]
        if (i > R):
            pre -= dp[i - R - 1]

        dp[i] = (pre > 0) and (s[i] == '0')

    # Return answer
    return dp[len(s) - 1]


# Driver Code
if __name__ == "__main__":

    S = "01101110"
    L = 2
    R = 3

    if canReach(S, L, R):
        print("Yes")
    else:
        print("No")

    # This code is contributed by rakeshsahni

// C#  program for the above approach
using System;
public class GFG
{

    // Function to check if it is possible
    // to reach the end of the binary string
    // using the given jumps
    static int canReach(String s, int L, int R)
    {

        // Stores the DP states
        int[] dp = new int[s.Length];

        // Initial state
        dp[0] = 1;

        // Stores count of indices from which
        // it is possible to reach index i
        int pre = 0;

        // Traverse the given string
        for (int i = 1; i < s.Length; i++)
        {

            // Update the values of pre
            // accordingly
            if (i >= L)
            {
                pre += dp[i - L];
            }

            // If the jump size is out of
            // the range [L, R]
            if (i > R)
            {
                pre -= dp[i - R - 1];
            }

            if (pre > 0 && s[i] == '0')
                dp[i] = 1;
            else
                dp[i] = 0;
        }

        // Return answer
        return dp[s.Length - 1];
    }

    // Driver Code
    public static void Main()
    {
        String S = "01101110";
        int L = 2, R = 3;

        if (canReach(S, L, R) == 1)
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}

// This code is contributed by Saurabh

<script>
    // JavaScript program for the above approach

    // Function to check if it is possible
    // to reach the end of the binary string
    // using the given jumps
    const canReach = (s, L, R) => {
        // Stores the DP states
        let dp = new Array(s.length).fill(1);

        // Stores count of indices from which
        // it is possible to reach index i
        let pre = 0;

        // Traverse the given string
        for (let i = 1; i < s.length; i++) {

            // Update the values of pre
            // accordingly
            if (i >= L) {
                pre += dp[i - L];
            }

            // If the jump size is out of
            // the range [L, R]
            if (i > R) {
                pre -= dp[i - R - 1];
            }

            dp[i] = (pre > 0) && (s[i] == '0');
        }

        // Return answer
        return dp[s.length - 1];
    }

    // Driver Code
    let S = "01101110";
    let L = 2, R = 3;

    if (canReach(S, L, R)) document.write("Yes");
    else document.write("No");

// This code is contributed by rakeshsahni

</script>

No

Time Complexity: O(N)
Auxiliary Space: O(N)