Maximum number of ones in a N*N matrix with given constraints in C++

Given the task is to find the maximum number of ones in a binary matrix possible with the following constraints.

Two integers N and X are given where X<=N. The size of the binary matrix should be N*N and every sub-matrix of size X*X should contain at least one zero.

Let’s now understand what we have to do using an example −

Input − N=4, X=2

Output − 12

Explanation − The resultant matrix will be −

1 1 1 1
1 0 0 1
1 0 0 1
1 1 1 1

Input − N=7, X=3

Output − 45

Approach used in the below program as follows

• To obtain the maximum number of ones, first we will have to find the minimum number of zeros required in the given matrix.

  By observing a common pattern in all the matrices, it can be seen that the number of zeroes required = (N / X)²

  So the maximum number of ones = Total elements in matrix – number of zeros\

• In function MaxOne() create a variable Z of type int and store in it the minimum number of zeros required which is equal to (N / X)²

• Then initialize another variable total = N*N of type int to store the total size of the matrix.

• Then finally initialize int ans = total – Z to store the final answer and return ans.

Example

 Live Demo

#include <bits/stdc++.h>
using namespace std;
int MaxOne(int N, int X){
   // Minimum number of zeroes that are needed
   int Z = (N / X);
   Z = Z * Z;
   /* Totol elements in matrix = square of the size of the matrices*/
   int total =N * N;
   //Final answer
   int ans = total - Z;
   return ans;
}
int main(){
   int N = 4;
   int X = 2;
   cout << MaxOne(N, X);
   return 0;
}

Output

If we run the above code we will get the following output −

12