Maximize the number of palindromic Strings

Given N binary strings b1, b2, b3…. bn. The task is to find the maximum number of binary strings that you can make palindromic by swapping any pair of characters any number of times. Characters can be either from the same string or from different strings

Examples:

Input: N=3
1110
100110
010101
Output: 2
Explanation:
b1 = 1110
b2 = 100110 - > 110010
b3 = 010101 ->  110010
Now swap last 0 in s2 and s3
with 1's in s1
Final string become
b1 = 1000
b2 = 110011
b3 = 110011

where b1 and b2 are a palindrome

Input: N=3
1
1000
111110
Output: 3

Approach:



The lengths of the strings don’t change. It is also important to observe that if we are given a binary string of odd length, then we can always swap the characters in such a way to convert that string to be palindromic. This is because if the length is odd then we will have either (even numbers of zeros and an odd number of ones) or(even no of ones and an odd number of zeros). So it can always be placed in such a way to make that string palindromic.

Now as our ans can be either N or N-1. We have to think about the cases when our ans will be N. So, we are given N binary strings. If there is at least 1 string of odd length, then our ans will surely be N.

    This can be explained as follows:

  • Grab all the 0’s and 1’s and remove them from their spots. Then we will have at least a pair of 0’s or 1’s and then place them into their free spots symmetrically (skipping the middles of odd length). So by now, all the strings of even length are filled and strings of odd length have a free spot in the middle which can be easily filled with remaining characters. So in this case, our ans will be N.
  • Now, Another case. If we have all of the N strings of even length individually and total no of 1’s and 0’s are even (i.e total count of 1’s are even and the total count of 0’s are even), then in this case also our ans will be N. This is because 1’s and 0’s can be placed symmetrically in all of the N strings to make them palindrome. Otherwise, our ans will be N-1.

Below is the implementation of the above approach:

C++

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// C++ program for the
// above approach
#include <bits/stdc++.h>
using namespace std;
  
int max_palindrome(string s[], int n)
{
    int flag = 0;
    for (int i = 0; i < n; i++) {
        // To check if there is
        // any string of odd length
        if (s[i].size() % 2 != 0) {
            flag = 1;
        }
    }
  
    // If there is at least
    // 1 string of odd
    // length.
    if (flag == 1) {
        return n;
    }
  
    int z = 0, o = 0;
  
    // If all the strings are
    // of even length.
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < s[i].size(); j++) {
            // Count of 0's in all
            // the strings
            if (s[i][j] == '0')
                z++;
            // Count of 1's in
            // all the strings
            else
                o++;
        }
    }
    // If z is even
    // and o is even
    // then ans will be N.
    if (o % 2 == 0 && z % 2 == 0) {
        return n;
    }
    // Otherwise ans will be N-1.
    else {
        return n - 1;
    }
}
  
// Driver code
int main()
{
    int n = 3;
    string s[n] = { "1110", "100110", "010101" };
    cout << max_palindrome(s, n);
  
    return 0;
}

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Java

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// Java program for the above approach 
class GFG
{
      
    static int max_palindrome(String []s, int n) 
    
        int flag = 0
        for (int i = 0; i < n; i++)
        
            // To check if there is 
            // any string of odd length 
            if (s[i].length() % 2 != 0
            
                flag = 1
            
        
      
        // If there is at least 
        // 1 string of odd 
        // length. 
        if (flag == 1)
        
            return n; 
        
      
        int z = 0;
        int o = 0
      
        // If all the strings are 
        // of even length. 
        for (int i = 0; i < n; i++)
        
            for (int j = 0; j < s[i].length(); j++)
            
                // Count of 0's in all 
                // the strings 
                if (s[i].charAt(j) == '0'
                    z += 1;
                      
                // Count of 1's in 
                // all the strings 
                else
                    o += 1
            
        
          
        // If z is even 
        // and o is even 
        // then ans will be N. 
        if (o % 2 == 0 && z % 2 == 0
        
            return n; 
        
          
        // Otherwise ans will be N-1. 
        else 
        
            return n - 1
        
    
      
    // Driver code 
    public static void main (String[] args) 
    
        int n = 3;
        String s[] = {"1110", "100110", "010101" }; 
        System.out.println(max_palindrome(s, n)); 
    }
}
  
// This code is contributed by AnkitRai01

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Python3

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# Python3 program for the above approach
def max_palindrome(s, n) : 
  
    flag = 0
    for i in range(n) : 
          
        # To check if there is 
        # any string of odd length 
        if (len(s[i]) % 2 != 0) :
            flag = 1
  
    # If there is at least 
    # 1 string of odd 
    # length. 
    if (flag == 1) :
        return n; 
  
    z = 0; o = 0
  
    # If all the strings are 
    # of even length. 
    for i in range(n) : 
        for j in range(len(s[i])) :
              
            # Count of 0's in all 
            # the strings 
            if (s[i][j] == '0') :
                z += 1
                  
            # Count of 1's in 
            # all the strings 
            else :
                o += 1
                  
    # If z is even 
    # and o is even 
    # then ans will be N. 
    if (o % 2 == 0 and z % 2 == 0) :
        return n; 
      
    # Otherwise ans will be N-1. 
    else :
        return n - 1
  
# Driver code 
if __name__ == "__main__"
  
    n = 3
    s = [ "1110", "100110", "010101" ]; 
      
    print(max_palindrome(s, n)); 
  
# This code is contributed by AnkitRai01

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C#

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// C# program for the above approach 
using System;
  
class GFG
{
      
    static int max_palindrome(string []s, int n) 
    
        int flag = 0; 
        for (int i = 0; i < n; i++)
        
            // To check if there is 
            // any string of odd length 
            if (s[i].Length % 2 != 0) 
            
                flag = 1; 
            
        
      
        // If there is at least 
        // 1 string of odd 
        // length. 
        if (flag == 1)
        
            return n; 
        
      
        int z = 0;
        int o = 0; 
      
        // If all the strings are 
        // of even length. 
        for (int i = 0; i < n; i++)
        
            for (int j = 0; j < s[i].Length; j++)
            
                // Count of 0's in all 
                // the strings 
                if (s[i][j] == '0'
                    z += 1;
                      
                // Count of 1's in 
                // all the strings 
                else
                    o += 1; 
            
        
          
        // If z is even 
        // and o is even 
        // then ans will be N. 
        if (o % 2 == 0 && z % 2 == 0) 
        
            return n; 
        
          
        // Otherwise ans will be N-1. 
        else
        
            return n - 1; 
        
    
      
    // Driver code 
    public static void Main () 
    
        int n = 3;
        string []s = {"1110", "100110", "010101" }; 
        Console.WriteLine(max_palindrome(s, n)); 
    }
}
  
// This code is contributed by AnkitRai01

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Output:

2


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