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

Given two integers and , where . Find the maximum number of one’s in a binary matrix can have such that every sub-matrix of size has atleast one cell as zero.

Examples:

Input:5 3
Output: Maximum number of ones = 24
The matrix will be:
1 1 1 1 1
1 1 1 1 1
1 1 0 1 1
1 1 1 1 1
1 1 1 1 1

Input:5 2
Output: Maximum number of ones = 21
The matrix will be:
1 1 1 1 1
1 0 1 0 1
1 1 1 1 1
1 0 1 0 1
1 1 1 1 1

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach The problem can be solved using a greedy approach. Place a zero at the right-bottom corner of the first square sub-matrix, i.e. the sub-matrix with coordinates (1, 1) and (x, x), and create the rest of the matrix symmetrically, we can get the minimum number of zeros, or, the maximum number of ones. Thus by observing, a common conclusion can be drawn that there are number of zeroes, in the minimum arrangement. The total number of cells available is in a NxN matrix.

.

Below is the implementation of the above approach:

 // C++ program to get Maximum Number of // ones in a matrix with given constraints #include    using namespace std;    // Function that returns the maximum  // number  of ones  int getMaxOnes(int n, int x) {     // Minimum number of zeroes     int zeroes = (n / x);     zeroes = zeroes * zeroes;        // Totol cells = square of the size of the matrices     int total = n * n;        // Intialising the answer     int ans =  total - zeroes;        return ans;  }    // Driver code int main() {     // Intitalising the variables     int n = 5;     int x = 2;               cout << getMaxOnes(n, x);        return 0; }

 // Java program to get Maximum // Number of ones in a matrix  // with given constraints import java.io.*;    class GFG {        // Function that returns  // the maximum number of ones  static int getMaxOnes(int n,                        int x) {     // Minimum number of zeroes     int zeroes = (n / x);     zeroes = zeroes * zeroes;        // Totol cells = square of     // the size of the matrices     int total = n * n;        // Intialising the answer     int ans = total - zeroes;        return ans;  }    // Driver code public static void main (String[] args)  {    // Intitalising the variables int n = 5; int x = 2; System.out.println(getMaxOnes(n, x)); } }    // This code is contributed // by akt_mit

 # Python3 program to get  # Maximum Number of ones  # in a matrix with given # constraints    # Function that returns  # the maximum number of ones  def getMaxOnes(n, x):            # Minimum number     # of zeroes     zeroes = (int)(n / x);     zeroes = zeroes * zeroes;        # Totol cells = square of     # the size of the matrices     total = n * n;        # Intialising      # the answer     ans = total - zeroes;        return ans;     # Driver code    # Intitalising the variables n = 5; x = 2; print(getMaxOnes(n, x));    # This code is contributed # by mits

 // C# program to get Maximum // Number of ones in a matrix  // with given constraints using System;    class GFG {        // Function that returns  // the maximum number of ones  static int getMaxOnes(int n,                        int x) {     // Minimum number of zeroes     int zeroes = (n / x);     zeroes = zeroes * zeroes;        // Totol cells = square of     // the size of the matrices     int total = n * n;        // Intialising the answer     int ans = total - zeroes;        return ans;  }    // Driver code static public void Main () {                // Intitalising the     // variables     int n = 5;     int x = 2;     Console.WriteLine(getMaxOnes(n, x)); } }    // This code is contributed // by ajit



Output:
21

Time Complexity: O(1)
Auxiliary Space: O(1)

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : jit_t, Mithun Kumar

Article Tags :
Practice Tags :