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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
#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
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