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

Given two integers n and x, where x <= n. Find the maximum number of one’s in a n * n binary matrix can have such that every sub-matrix of size x * x 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

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 \left \lfloor (\frac{n}{x})^2 \right \rfloor number of zeroes, in the minimum arrangement. The total number of cells available is n^2 in a NxN matrix.

max(ones) = n^2 - \left \lfloor (\frac{n}{x})^2 \right \rfloor.

Below is the implementation of the above approach:



C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to get Maximum Number of
// ones in a matrix with given constraints
#include <bits/stdc++.h>
  
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;
}

chevron_right






                        filter_none









Java














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// 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 



















                        chevron_right







                        filter_none









Python3














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















# 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 



















                        chevron_right







                        filter_none









C#














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// 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 



















                        chevron_right







                        filter_none









PHP














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















<?php 


// PHP program to get Maximum  


// Number of ones in a matrix 


// with given constraints 


  


// Function that returns  


// the maximum number of ones  


function 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; 


echo getMaxOnes($n, $x); 


  


// This code is contributed 


// by akt_mit 


?> 



















                        chevron_right







                        filter_none











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.





    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
AayushChaturvedi
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