Maximize given function by selecting equal length substrings from given Binary Strings

Given two binary strings s1 and s2. The task is to choose substring from s1 and s2 say sub1 and sub2 of equal length such that it maximizes the function:

fun(s1, s2) = len(sub1) / (2^{xor(sub1, sub2)})

Examples:

Input: s1= “1101”, s2= “1110”
Output: 3
Explanation: Below are the substrings chosen from s1 and s2
Substring chosen from s1 -> “110”
Substring chosen from s2 -> “110”
Therefore, fun(s1, s2) = 3/ (2^{xor(110, 110)}) = 3, which is maximum possible. 

Input: s1= “1111”, s2= “1000”
Output: 1

Approach: In order to maximize the given function large substrings needed to be chosen with minimum XOR. To minimize the denominator, choose substrings in a way such that XOR of sub1 and sub2 is always 0 so that the denominator term will always be 1 (2⁰). So for that, find the longest common substring from the two strings s1 and s2, and print its length that would be the required answer. 

Below is the implementation of above approach:

C++

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

int dp[1000][1000];

// Function to find longest common substring.
int lcs(string s, string k, int n, int m)
{
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= m; j++) {
            if (i == 0 or j == 0) {
                dp[i][j] = 0;
            }
            else if (s[i - 1] == k[j - 1]) {
                dp[i][j] = 1 + dp[i - 1][j - 1];
            }
            else {
                dp[i][j] = max(dp[i - 1][j],
                               dp[i][j - 1]);
            }
        }
    }

    // Return the result
    return dp[n][m];
}

// Driver Code
int main()
{
    string s1 = "1110";
    string s2 = "1101";

    cout << lcs(s1, s2,
                s1.size(), s2.size());

    return 0;
}

Java

// Java program for above approach
class GFG{

  static int dp[][] = new int[1000][1000];

  // Function to find longest common substring.
  static int lcs(String s, String k, int n, int m)
  {
      for (int i = 0; i <= n; i++) {
          for (int j = 0; j <= m; j++) {
              if (i == 0 || j == 0) {
                  dp[i][j] = 0;
              }
              else if (s.charAt(i - 1) == k.charAt(j - 1)) {
                  dp[i][j] = 1 + dp[i - 1][j - 1];
              }
              else {
                  dp[i][j] = Math.max(dp[i - 1][j],
                                 dp[i][j - 1]);
              }
          }
      }

      // Return the result
      return dp[n][m];
  }

  // Driver Code
  public static void main(String [] args)
  {
      String s1 = "1110";
      String s2 = "1101";

      System.out.print(lcs(s1, s2,
                  s1.length(), s2.length()));


  }
}

// This code is contributed by AR_Gaurav

Python3

# Python3 program for above approach

import numpy as np;

dp = np.zeros((1000,1000));

# Function to find longest common substring.
def lcs( s,  k,  n,  m) :

    for i in range(n + 1) :
        for j in range(m + 1) :
            if (i == 0 or j == 0) :
                dp[i][j] = 0;

            elif (s[i - 1] == k[j - 1]) :
                dp[i][j] = 1 + dp[i - 1][j - 1];

            else :
                dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);

    # Return the result
    return dp[n][m];

# Driver Code
if __name__ == "__main__" :

    s1 = "1110";
    s2 = "1101";

    print(lcs(s1, s2,len(s1), len(s2)));

   # This code is contributed by AnkThon

C#

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

  static int [,]dp = new int[1000,1000];

  // Function to find longest common substring.
  static int lcs(string s, string k, int n, int m)
  {
      for (int i = 0; i <= n; i++) {
          for (int j = 0; j <= m; j++) {
              if (i == 0 || j == 0) {
                  dp[i, j] = 0;
              }
              else if (s[i - 1] == k[j - 1]) {
                  dp[i, j] = 1 + dp[i - 1, j - 1];
              }
              else {
                  dp[i, j] = Math.Max(dp[i - 1, j],
                                 dp[i, j - 1]);
              }
          }
      }

      // Return the result
      return dp[n, m];
  }

  // Driver Code
  public static void Main(string [] args)
  {
      string s1 = "1110";
      string s2 = "1101";

      Console.Write(lcs(s1, s2, s1.Length, s2.Length));
  }
}

// This code is contributed by AnkThon

Javascript

<script>
// JavaScript program for above approach
var dp = new Array(1000);
for (var i = 0; i < 1000; i++) {
  dp[i] = new Array(1000);
}

// Function to find longest common substring.
function lcs( s,  k,  n,  m)
{
    for (var i = 0; i <= n; i++) {
        for (var j = 0; j <= m; j++) {
            if (i == 0 || j == 0) {
                dp[i][j] = 0;
            }
            else if (s[i - 1] == k[j - 1]) {
                dp[i][j] = 1 + dp[i - 1][j - 1];
            }
            else {
                dp[i][j] = Math.max(dp[i - 1][j],
                               dp[i][j - 1]);
            }
        }
    }

    // Return the result
    return dp[n][m];
}

// Driver Code
var s1 = "1110";
var s2 = "1101";

document.write(lcs(s1, s2, s1.length, s2.length))

// This code is contributed by AnkThon
</script>

Output

3

Time Complexity: O(N*M), where N is the size of s1 and M is the size of s2.

Auxiliary Space: O(N*M), where N is the size of s1 and M is the size of s2.