Find length of longest subsequence of one string which is substring of another string

Try it on GfG Practice

Given two strings X and Y. The task is to find the length of the longest subsequence of string X which is a substring in sequence Y.

Examples: 

Input : X = "ABCD", Y = "BACDBDCD"
Output : 3
Explanation: "ACD" is longest subsequence of X which is substring of Y.

Input : X = "A", Y = "A"
Output : 1

Perquisites: Longest common subsequence problem will help you understand this problem in a snap. 

Using Recursion - O(n*m) Time and O(n) Space

Let n be the length of X and m be the length of Y. We will make a recursive function as follows with 4 arguments and the return type is int as we will get the length of a maximum possible subsequence of X which is the Substring of Y. We will take judgment for further process based on the last char in strings by using n-1 and m-1.

For recursion we need 2 things, we will make 1st the base case and 2nd is calls on smaller input (for that we will see the choice diagram). 

Base Case:

By seeing the argument of function we can see only 2 arguments that will change while recursion calls, i.e. lengths of both strings. So for the base case think of the smallest input we can give. You'll see the smallest input is 0, i.e. empty lengths. hence when n == 0 or m == 0 is the base case. n and m can't be less than zero. Now we know the condition and we need to return the length of subsequence as per the question. if the length is 0, then means one of the string is empty, and there is no common subsequence possible, so we have to return 0 for the same.

Children Calls:

We can see how to make calls by seeing if the last char of both strings are the same or not. We can see how this is slightly different from the LCS (Longest Common Subsequence) question.

int maxSubsequenceSubstring(string &X,string &Y,int n,int m) {

// Base Case
if (n==0 || m==0) return 0;

// Calls on smaller inputs
// if the last char of both strings are equal

if(X[n-1] == Y[m-1]) {

return 1 + maxSubsequenceSubstring(X,Y,n-1,m-1);

}

// if the last char of both strings are not equal
else {

return maxSubsequenceSubstring(X,Y,n-1,m);

}

}

Now here is the main crux of the question, we can see we are calling for X[0..n] and Y[0..m], in our recursion function it will return the answer of maximum length of the subsequence of X, and Substring of Y (and the ending char of that substring of Y ends at length m). This is very important as we want to find all intermediary substrings also. hence we need to use a for loop where we will call the above function for all the lengths from 0 to m of Y, and return the maximum of the answer there. Here is the final code in C++ for the same.

Below is the implementation of the above approach:

#include<bits/stdc++.h>
using namespace std;

int maxSubsequenceSubstring(string &X,string &Y,int n,int m)
{
    // Base Case
    if (n==0 || m==0) return 0;
    
    // Calls on smaller inputs
    
    // if the last char of both strings are equal
    if(X[n-1] == Y[m-1])
    {
        return  1 + maxSubsequenceSubstring(X,Y,n-1,m-1);
    }
    
    // if the last char of both strings are not equal
    else
    {
        return maxSubsequenceSubstring(X,Y,n-1,m);    
    }
}

int main()
{
     string X = "abcd";
      string Y = "bacdbdcd";
      int n = X.size(),m = Y.size();
      int maximum_length = 0; //as minimum length can be 0 only.
      for(int i = 0;i<=m;i++) // traversing for every length of Y.
    {
        int temp_ans = maxSubsequenceSubstring(X,Y,n,i);
          if(temp_ans > maximum_length) maximum_length = temp_ans;
    }
      cout<<"Length for maximum possible Subsequence of string X which is Substring of Y -> "<<maximum_length;
      return 0;
}

import java.util.*;
class GFG {

    static int maxSubsequenceSubString(String X, String Y,
                                       int n, int m)
    {
      
        // Base Case
        if (n == 0 || m == 0)
            return 0;

        // Calls on smaller inputs

        // if the last char of both Strings are equal
        if (X.charAt(n - 1) == Y.charAt(m - 1)) {
            return 1
                + maxSubsequenceSubString(X, Y, n - 1,
                                          m - 1);
        }

        // if the last char of both Strings are not equal
        else {
            return maxSubsequenceSubString(X, Y, n - 1, m);
        }
    }

  // Driver code
    public static void main(String[] args)
    {
        String X = "abcd";
        String Y = "bacdbdcd";
        int n = X.length(), m = Y.length();
        int maximum_length
            = 0; // as minimum length can be 0 only.
        for (int i = 0; i <= m;
             i++) // traversing for every length of Y.
        {
            int temp_ans
                = maxSubsequenceSubString(X, Y, n, i);
            if (temp_ans > maximum_length)
                maximum_length = temp_ans;
        }
        System.out.print(
            "Length for maximum possible Subsequence of String X which is SubString of Y->"
            + maximum_length);
    }
}

def maxSubsequenceSubstring(X, Y, n, m):

    # Base Case
    if (n == 0 or m == 0):
        return 0
    
    # Calls on smaller inputs
    
    # if the last char of both strings are equal
    if(X[n - 1] == Y[m - 1]):
        return 1 + maxSubsequenceSubstring(X, Y, n - 1, m - 1)
    
    # if the last char of both strings are not equal
    else:
        return maxSubsequenceSubstring(X, Y, n - 1, m)

# driver code
X = "abcd"
Y = "bacdbdcd"
n, m = len(X), len(Y)
maximum_length = 0 #as minimum length can be 0 only.
for i in range(m + 1):# traversing for every length of Y.

    temp_ans = maxSubsequenceSubstring(X, Y, n, i)
    if(temp_ans > maximum_length):
         maximum_length = temp_ans

print(f"Length for maximum possible Subsequence of string X which is Substring of Y -> {maximum_length}")

using System;

public class GFG {

    static int maxSubsequenceSubString(String X, String Y,
                                       int n, int m)
    {

        // Base Case
        if (n == 0 || m == 0)
            return 0;

        // Calls on smaller inputs

        // if the last char of both Strings are equal
        if (X[n - 1] == Y[m - 1]) {
            return 1
                + maxSubsequenceSubString(X, Y, n - 1,
                                          m - 1);
        }

        // if the last char of both Strings are not equal
        else {
            return maxSubsequenceSubString(X, Y, n - 1, m);
        }
    }

    // Driver code
    public static void Main(String[] args)
    {
        String X = "abcd";
        String Y = "bacdbdcd";
        int n = X.Length, m = Y.Length;
        int maximum_length
            = 0; // as minimum length can be 0 only.
        for (int i = 0; i <= m;
             i++) // traversing for every length of Y.
        {
            int temp_ans
                = maxSubsequenceSubString(X, Y, n, i);
            if (temp_ans > maximum_length)
                maximum_length = temp_ans;
        }
        Console.Write(
            "Length for maximum possible Subsequence of String X which is SubString of Y->"
            + maximum_length);
    }
}

<script>

function maxSubsequenceSubstring(X,Y,n,m)
{
    // Base Case
    if (n==0 || m==0) return 0;
    
    // Calls on smaller inputs
    
    // if the last char of both strings are equal
    if(X[n-1] == Y[m-1])
    {
        return 1 + maxSubsequenceSubstring(X,Y,n-1,m-1);
    }
    
    // if the last char of both strings are not equal
    else
    {
        return maxSubsequenceSubstring(X,Y,n-1,m);
    }
}

// driver code

let X = "abcd";
let Y = "bacdbdcd";
let n = X.length,m = Y.length;
let maximum_length = 0; //as minimum length can be 0 only.
for(let i = 0;i<=m;i++) // traversing for every length of Y.
{
    let temp_ans = maxSubsequenceSubstring(X,Y,n,i);
    if(temp_ans > maximum_length) maximum_length = temp_ans;
}
document.write("Length for maximum possible Subsequence of string X which is Substring of Y -> "+ maximum_length);

</script>

Length for maximum possible Subsequence of string X which is Substring of Y -> 3

Time Complexity: O(n*m) (For every call in the recursion function we are decreasing n, hence we will reach the base case exactly after n calls, and we are using for loop for m times for the different lengths of string Y).
Space Complexity: O(n) (For recursion calls we are using stacks for each call).

Using Top-Down DP (Memoization) –                                                                                                                                                                        

From the above recursion solution, there are multiple calls and further, we are using recursion in for loop, there is a high probability we have already solved the answer for a call. hence to optimize our recursion solution we will use? (See we have only 2 arguments that vary throughout the calls, hence the dimension of the array is 2 and the size is (n+1) * (m+1) because we need to store answers for all possible calls from 0..n and 0..m). Hence it is a 2D array.

we can use vectors for the same or dynamic allocation of the array.

// initialise a vector like this

vector<vector<int>> dp(n+1,vector<int>(m+1,-1));

// or Dynamic allocation

int **dp = new int*[n+1];

for(int i = 0;i<=n;i++)

{

dp[i] = new int[m+1];

for(int j = 0;j<=m;j++)

{

dp[i][j] = -1;

}

}

By initializing the 2D vector we will use this array as the 5th argument in recursion and store our answer. also, we have filled it with -1, which means we have not solved this call hence use traditional recursion for it, if dp[n][m] at any call is not -1, means we have already solved the call, hence use the answer of dp[n][m].

// In recursion calls we will check for if we have solved the answer for the call or not

if(dp[n][m] != -1) return dp[n][m];

// Else we will store the result and return that back from this call

if(X[n-1] == Y[m-1])

return dp[n][m] = 1 + maxSubsequenceSubstring(X,Y,n-1,m-1,dp);

else

return dp[n][m] = maxSubsequenceSubstring(X,Y,n-1,m,dp);

Below is the implementation of the above approach:

#include<bits/stdc++.h>
using namespace std;

int maxSubsequenceSubstring(string &X,string &Y,int n,int m,vector<vector<int>>& dp)
{
    // Base Case
    if (n==0 || m==0) return 0;
    
      // check if we have already solved it?
      if(dp[n][m] != -1) return dp[n][m];
  
      
    // Calls on smaller inputs
    
    // if the last char of both strings are equal
    if(X[n-1] == Y[m-1])
    {
        return  dp[n][m] = 1 + maxSubsequenceSubstring(X,Y,n-1,m-1,dp);
    }
    
    // if the last char of both strings are not equal
    else
    {
        return dp[n][m] = maxSubsequenceSubstring(X,Y,n-1,m,dp);    
    }
}

int main()
{
     string X = "abcd";
      string Y = "bacdbdcd";
      int n = X.size(),m = Y.size();
      int maximum_length = 0; //as minimum length can be 0 only.
  
      vector<vector<int>> dp(n+1,vector<int>(m+1,-1));
      for(int i = 0;i<=m;i++) // traversing for every length of Y.
    {
        int temp_ans = maxSubsequenceSubstring(X,Y,n,i,dp);
          if(temp_ans > maximum_length) maximum_length = temp_ans;
    }
      cout<<"Length for maximum possible Subsequence of string X which is Substring of Y -> "<<maximum_length;
      return 0;
}

/*package whatever //do not write package name here */

import java.io.*;
import java.util.*;

class GFG {
    static int maxSubsequenceSubstring(String X,String Y,int n,int m,int dp[][])
    {
        // Base Case
        if (n==0 || m==0) return 0;
        
        // check if we have already solved it?
        if(dp[n][m] != -1) return dp[n][m];
    
        
        // Calls on smaller inputs
        
        // if the last char of both strings are equal
        if(X.charAt(n-1) == Y.charAt(m-1))
        {
            return dp[n][m] = 1 + maxSubsequenceSubstring(X,Y,n-1,m-1,dp);
        }
        
        // if the last char of both strings are not equal
        else
        {
            return dp[n][m] = maxSubsequenceSubstring(X,Y,n-1,m,dp);
        }
    }
    
// Driver Code
public static void main(String args[])
{
    String X = "abcd";
    String Y = "bacdbdcd";
    int n = X.length(),m = Y.length();
    int maximum_length = 0; //as minimum length can be 0 only.
    
    int dp[][] = new int[n+1][m+1];
    for(int i = 0; i < n + 1; i++){
        Arrays.fill(dp[i], -1);
    }
    for(int i = 0;i<=m;i++) // traversing for every length of Y.
    {
        int temp_ans = maxSubsequenceSubstring(X,Y,n,i,dp);
        if(temp_ans > maximum_length) maximum_length = temp_ans;
    }
    System.out.println("Length for maximum possible Subsequence of string X which is Substring of Y -> "+maximum_length);
}
}

# Python implementation of above approach
def maxSubsequenceSubstring(X, Y, n, m, dp):
  
    # Base Case
    if (n == 0 or m == 0):
        return 0
    
    # check if we have already solved it?
    if(dp[n][m] != -1):
        return dp[n][m]
  
    # Calls on smaller inputs
    
    # if the last char of both strings are equal
    if(X[n - 1] == Y[m - 1]):
        dp[n][m] = 1 + maxSubsequenceSubstring(X, Y, n - 1, m - 1, dp)
        return dp[n][m]

    # if the last char of both strings are not equal
    else:
        dp[n][m] = maxSubsequenceSubstring(X, Y, n - 1, m, dp)
        return dp[n][m]

# driver code
X = "abcd"
Y = "bacdbdcd"
n,m = len(X),len(Y)
maximum_length = 0 #as minimum length can be 0 only.
  
dp = [[-1 for i in range(m+1)]for j in range(n+1)]

for i in range(m+1): # traversing for every length of Y.

    temp_ans = maxSubsequenceSubstring(X, Y, n, i, dp)
    if(temp_ans > maximum_length):
        maximum_length = temp_ans

print("Length for maximum possible Subsequence of string X which is Substring of Y -> "+str(maximum_length))

// C# code for the above approach
using System;

class GFG
{
  static int maxSubsequenceSubstring(string X, string Y, int n, int m, int[,] dp)
  {
    // Base Case
    if (n == 0 || m == 0) return 0;

    // check if we have already solved it?
    if (dp[n, m] != -1) return dp[n, m];

    // Calls on smaller inputs

    // if the last char of both strings are equal
    if (X[n - 1] == Y[m - 1])
    {
      return dp[n, m] = 1 + maxSubsequenceSubstring(X, Y, n - 1, m - 1, dp);
    }

    // if the last char of both strings are not equal
    else
    {
      return dp[n, m] = maxSubsequenceSubstring(X, Y, n - 1, m, dp);
    }
  }

  // Driver Code
  public static void Main(string[] args)
  {
    string X = "abcd";
    string Y = "bacdbdcd";
    int n = X.Length, m = Y.Length;
    int maximum_length = 0; //as minimum length can be 0 only.

    int[,] dp = new int[n + 1, m + 1];
    for (int i = 0; i < n + 1; i++)
    {
      for (int j = 0; j < m + 1; j++)
      {
        dp[i, j] = -1;
      }
    }
    for (int i = 0; i <= m; i++) // traversing for every length of Y.
    {
      int temp_ans = maxSubsequenceSubstring(X, Y, n, i, dp);
      if (temp_ans > maximum_length) maximum_length = temp_ans;
    }
    Console.WriteLine("Length for maximum possible Subsequence of string X which is Substring of Y -> " + maximum_length);
  }
}

<script>

// JavaScript implementation of above approach


function maxSubsequenceSubstring(X,Y,n,m,dp)
{
    // Base Case
    if (n==0 || m==0) return 0;
    
      // check if we have already solved it?
      if(dp[n][m] != -1) return dp[n][m];
  
      
    // Calls on smaller inputs
    
    // if the last char of both strings are equal
    if(X[n-1] == Y[m-1])
    {
        return  dp[n][m] = 1 + maxSubsequenceSubstring(X,Y,n-1,m-1,dp);
    }
    
    // if the last char of both strings are not equal
    else
    {
        return dp[n][m] = maxSubsequenceSubstring(X,Y,n-1,m,dp);    
    }
}

// driver code

let X = "abcd";
let Y = "bacdbdcd";
let n = X.length,m = Y.length;
let maximum_length = 0; //as minimum length can be 0 only.
  
let dp = new Array(n+1);
for(let i=0;i<n+1;i++){
    dp[i] = new Array(m+1).fill(-1);
}
for(let i = 0;i<=m;i++) // traversing for every length of Y.
{
    let temp_ans = maxSubsequenceSubstring(X,Y,n,i,dp);
    if(temp_ans > maximum_length) maximum_length = temp_ans;
}
document.write("Length for maximum possible Subsequence of string X which is Substring of Y -> ",maximum_length);


</script>

Length for maximum possible Subsequence of string X which is Substring of Y -> 3

Time Complexity: O(n*m) (It will be definitely better than the recursion solution, the worst case is possible only possible when none of the char of string X is there in String Y.)
Space Complexity: O(n*m + n) (the Size of Dp array and stack call size of recursion)

Using Bottom-Up DP (Tabulation) – O(n*m) Time and O(n*m + m) Space

Let n be length of X and m be length of Y. Create a 2D array 'dp[][]' of m + 1 rows and n + 1 columns. Value dp[i][j] is maximum length of subsequence of X[0....j] which is substring of Y[0....i]. Now for each cell of dp[][] fill value as : 

for (i = 1 to m)

for (j = 1 to n)

if (x[j-1] == y[i - 1])

dp[i][j] = dp[i-1][j-1] + 1;

else

dp[i][j] = dp[i][j-1];

And finally, the length of the longest subsequence of x which is substring of y is max(dp[i][n]) where 1 <= i <= m.

Below is the implementation of the above approach:

// C++ code to implement the approach
#include <bits/stdc++.h>
using namespace std;

const int MAX = 1000;

// Return the maximum size of substring of
// X which is substring in Y.
int maxSubsequenceSubstring(string x,string y,int n,int m)
{
    vector<vector<int>>dp(MAX,vector<int>(MAX,0));

    // Calculating value for each element.
    for (int i = 1; i <= m; i++) {
        for (int j = 1; j <= n; j++) {

            // If alphabet of string X and Y are
            // equal make dp[i][j] = 1 + dp[i-1][j-1]
            if (x[j - 1] == y[i - 1])
                dp[i][j] = 1 + dp[i - 1][j - 1];

            // Else copy the previous value in the
            // row i.e dp[i-1][j-1]
            else
                dp[i][j] = dp[i][j - 1];
        }
    }

    // Finding the maximum length.
    int ans = 0;
    for (int i = 1; i <= m; i++)
        ans = max(ans, dp[i][n]);

    return ans;
}

// Driver code
int main()
{
    string x = "ABCD";
    string y = "BACDBDCD";
    int n = x.length(), m = y.length();
    cout<<maxSubsequenceSubstring(x, y, n, m);
}

// Java program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.

public class GFG 
{
    static final int MAX = 1000;
    
    // Return the maximum size of substring of
    // X which is substring in Y.
    static int maxSubsequenceSubstring(char x[], char y[],
                                int n, int m)
    {
        int dp[][] = new int[MAX][MAX];
     
        // Initialize the dp[][] to 0.
        for (int i = 0; i <= m; i++)
            for (int j = 0; j <= n; j++)
                dp[i][j] = 0;
     
        // Calculating value for each element.
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
     
                // If alphabet of string X and Y are
                // equal make dp[i][j] = 1 + dp[i-1][j-1]
                if (x[j - 1] == y[i - 1])
                    dp[i][j] = 1 + dp[i - 1][j - 1];
     
                // Else copy the previous value in the
                // row i.e dp[i-1][j-1]
                else
                    dp[i][j] = dp[i][j - 1];
            }
        }
     
        // Finding the maximum length.
        int ans = 0;
        for (int i = 1; i <= m; i++)
            ans = Math.max(ans, dp[i][n]);
     
        return ans;
    }
    
    // Driver Method
    public static void main(String[] args)
    {
        char x[] = "ABCD".toCharArray();
        char y[] = "BACDBDCD".toCharArray();
        int n = x.length, m = y.length;
        System.out.println(maxSubsequenceSubstring(x, y, n, m));
    }
}

# Python3 program to find maximum 
# length of subsequence of a string
# X such it is substring in another
# string Y. 

MAX = 1000

# Return the maximum size of 
# substring of X which is
# substring in Y.
def maxSubsequenceSubstring(x, y, n, m):
    dp = [[0 for i in range(MAX)]
             for i in range(MAX)]
             
    # Initialize the dp[][] to 0.

    # Calculating value for each element.
    for i in range(1, m + 1):
        for j in range(1, n + 1):
            
            # If alphabet of string 
            # X and Y are equal make 
            # dp[i][j] = 1 + dp[i-1][j-1]
            if(x[j - 1] == y[i - 1]):
                dp[i][j] = 1 + dp[i - 1][j - 1]

            # Else copy the previous value 
            # in the row i.e dp[i-1][j-1]
            else:
                dp[i][j] = dp[i][j - 1]
                
    # Finding the maximum length
    ans = 0
    for i in range(1, m + 1):
        ans = max(ans, dp[i][n])
    return ans

# Driver Code 
x = "ABCD"
y = "BACDBDCD"
n = len(x)
m = len(y)
print(maxSubsequenceSubstring(x, y, n, m))

// C# program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.
using System;

public class GFG 
{
    static int MAX = 1000;
    
    // Return the maximum size of substring of
    // X which is substring in Y.
    static int maxSubsequenceSubstring(string x, string y,
                                            int n, int m)
    {
        int[ ,]dp = new int[MAX, MAX];
    
        // Initialize the dp[][] to 0.
        for (int i = 0; i <= m; i++)
            for (int j = 0; j <= n; j++)
                dp[i, j] = 0;
    
        // Calculating value for each element.
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
    
                // If alphabet of string X and Y are
                // equal make dp[i][j] = 1 + dp[i-1][j-1]
                if (x[j - 1] == y[i - 1])
                    dp[i, j] = 1 + dp[i - 1, j - 1];
    
                // Else copy the previous value in the
                // row i.e dp[i-1][j-1]
                else
                    dp[i, j] = dp[i, j - 1];
            }
        }
    
        // Finding the maximum length.
        int ans = 0;
        
        for (int i = 1; i <= m; i++)
            ans = Math.Max(ans, dp[i,n]);
    
        return ans;
    }
    
    // Driver Method
    public static void Main()
    {
        string x = "ABCD";
        string y = "BACDBDCD";
        int n = x.Length, m = y.Length;
        
        Console.WriteLine(maxSubsequenceSubstring(x,
                                            y, n, m));
    }
}

<script>


// Javascript program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.
var MAX = 1000;

// Return the maximum size of substring of
// X which is substring in Y.
function maxSubsequenceSubstring(x, y, n, m)
{
    var dp = Array.from(Array(MAX), ()=> Array(MAX));

    // Initialize the dp[][] to 0.
    for (var i = 0; i <= m; i++)
        for (var j = 0; j <= n; j++)
            dp[i][j] = 0;

    // Calculating value for each element.
    for (var i = 1; i <= m; i++) {
        for (var j = 1; j <= n; j++) {

            // If alphabet of string X and Y are
            // equal make dp[i][j] = 1 + dp[i-1][j-1]
            if (x[j - 1] == y[i - 1])
                dp[i][j] = 1 + dp[i - 1][j - 1];

            // Else copy the previous value in the
            // row i.e dp[i-1][j-1]
            else
                dp[i][j] = dp[i][j - 1];
        }
    }

    // Finding the maximum length.
    var ans = 0;
    for (var i = 1; i <= m; i++)
        ans = Math.max(ans, dp[i][n]);

    return ans;
}

// Driver Program
var x = "ABCD";
var y = "BACDBDCD";
var n = x.length, m = y.length;
document.write( maxSubsequenceSubstring(x, y, n, m));

</script>

3

Time Complexity: O(n*m), due to time required to fill the Dp array
Space Complexity: O(n*m + n), due the Size of Dp array

Greedy Approach - O(n*m) Time and O(1) Space

The idea is to use two pointers: one traverses Y and the other traverses X. For each position in Y, the inner loop checks if characters in X can match consecutively, and the length of the match is tracked. The maximum length of such consecutive matches is returned as the result.

Below is the implementation of the above approach:

#include<bits/stdc++.h>
using namespace std;

// Function to find the maximum subsequence substring length
int maxSubsequenceSubstring(string X, string Y, int N, int M) {
  
    // Initialize a variable to keep track of the maximum length
    int maxi = 0;
    
    // Outer loop to traverse through each character of Y
    for (int i = 0; i < M; i++) {
        int t = i; // Initialize a pointer 't' to track position in Y
        
        // Inner loop to traverse through each character of X
        for (int j = 0; j < N; j++) {
            
            // If characters from X and Y match, move pointer 't' in Y
            if (Y[t] == X[j]) {
                t++;
            }
            
            // Update the maximum length found so far
            maxi = max(maxi, t - i);
        }
    }
    
    // Return the maximum length of the subsequence substring
    return maxi;
}

int main() {
  
    // Hardcoded input values
    int N = 4, M = 8;
    string X = "abcd";
    string Y = "bacdbdcd";
    
    // Call the function and store the result
    int result = maxSubsequenceSubstring(X, Y, N, M);
    
    // Print the result
    cout << "Maximum subsequence substring length: " << result << endl;
    
    return 0;
}

class GfG {
  
    // Function to find the maximum subsequence substring length
    public static int maxSubsequenceSubstring(String X, String Y, int N, int M) {
      
        // Initialize a variable to keep track of the maximum length
        int maxi = 0;

        // Outer loop to traverse through each character of Y
        for (int i = 0; i < M; i++) {
            int t = i; // Initialize a pointer 't' to track position in Y

            // Inner loop to traverse through each character of X
            for (int j = 0; j < N; j++) {
                // If characters from X and Y match, move pointer 't' in Y
                if (Y.charAt(t) == X.charAt(j)) {
                    t++;
                }

                // Update the maximum length found so far
                maxi = Math.max(maxi, t - i);
            }
        }

        // Return the maximum length of the subsequence substring
        return maxi;
    }

    // Driver code
    public static void main(String[] args) {
      
        // Hardcoded input values
        int N = 4, M = 8;
        String X = "abcd";
        String Y = "bacdbdcd";

        // Call the function and store the result
        int result = maxSubsequenceSubstring(X, Y, N, M);

        // Print the result
        System.out.println("Maximum subsequence substring length: " + result);
    }
}

def maxSubsequenceSubstring(X, Y, N, M):
    # Initialize a variable to keep track of the maximum length
    maxi = 0
    
    # Outer loop to traverse through each character of Y
    for i in range(M):
        t = i  # Initialize a pointer 't' to track position in Y

        # Inner loop to traverse through each character of X
        for j in range(N):
            # If characters from X and Y match, move pointer 't' in Y
            if Y[t] == X[j]:
                t += 1
            
            # Update the maximum length found so far
            maxi = max(maxi, t - i)
    
    # Return the maximum length of the subsequence substring
    return maxi


def main():
    # Hardcoded input values
    N = 4
    M = 8
    X = "abcd"
    Y = "bacdbdcd"
    
    # Call the function and store the result
    result = maxSubsequenceSubstring(X, Y, N, M)
    
    # Print the result
    print(f"Maximum subsequence substring length: {result}")


# Driver code
if __name__ == "__main__":
    main()

using System;

class GfG {
    // Function to find the maximum subsequence substring length
    public static int MaxSubsequenceSubstring(string X, string Y, int N, int M) {
        // Initialize a variable to keep track of the maximum length
        int maxi = 0;

        // Outer loop to traverse through each character of Y
        for (int i = 0; i < M; i++) {
            int t = i; // Initialize a pointer 't' to track position in Y

            // Inner loop to traverse through each character of X
            for (int j = 0; j < N; j++) {
                // If characters from X and Y match, move pointer 't' in Y
                if (Y[t] == X[j]) {
                    t++;
                }

                // Update the maximum length found so far
                maxi = Math.Max(maxi, t - i);
            }
        }

        // Return the maximum length of the subsequence substring
        return maxi;
    }

    // Driver code
    public static void Main(string[] args) {
        // Hardcoded input values
        int N = 4, M = 8;
        string X = "abcd";
        string Y = "bacdbdcd";

        // Call the function and store the result
        int result = MaxSubsequenceSubstring(X, Y, N, M);

        // Print the result
        Console.WriteLine("Maximum subsequence substring length: " + result);
    }
}

function maxSubsequenceSubstring(X, Y, N, M) {
    // Initialize a variable to keep track of the maximum length
    let maxi = 0;
    
    // Outer loop to traverse through each character of Y
    for (let i = 0; i < M; i++) {
        let t = i;  // Initialize a pointer 't' to track position in Y

        // Inner loop to traverse through each character of X
        for (let j = 0; j < N; j++) {
            // If characters from X and Y match, move pointer 't' in Y
            if (Y[t] === X[j]) {
                t++;
            }
            
            // Update the maximum length found so far
            maxi = Math.max(maxi, t - i);
        }
    }
    
    // Return the maximum length of the subsequence substring
    return maxi;
}

// Hardcoded input values
const N = 4;
const M = 8;
const X = "abcd";
const Y = "bacdbdcd";

// Call the function and store the result
const result = maxSubsequenceSubstring(X, Y, N, M);

// Print the result
console.log("Maximum subsequence substring length:", result);

Time Complexity: O(n * m), where n and m represent the lengths of the strings X and Y, respectively.
Space Complexity: O(1), as the solution uses only a constant amount of extra space.