Guys I have been working on this problem since morning (4 hours ago) and still couldn’t find a proper answer. Can anyone help me?
problem link: https://codeforces.com/contest/1761/problem/C#

• Problem Explanation:
  The problem “Set Construction” from Codeforces requires constructing sets based on a given binary matrix, with certain rules for inclusion and comparison. Let’s break it down:

Problem Explanation

1. Input:

   • You have a binary matrix b with dimensions n x n, where each element is either 0 or 1.

2. Output:

   • You need to construct n sets ( A_1, A_2, dots, A_n ) consisting of distinct integers from 1 to ( n ).

3. Conditions:

   • Each set must be non-empty and distinct.
   • The matrix b indicates the subset relationship between these sets:
     • If b[i][j] = 1, then the set ( A_i ) must be a proper subset of ( A_j ).
     • If b[i][j] = 0, then the set ( A_i ) must not be a subset of ( A_j ).

4. Proper Subset:

   • A set ( A_i ) is a proper subset of ( A_j ) if all elements of ( A_i ) are in ( A_j ), and ( A_i ) is strictly smaller (i.e., ( A_i neq A_j )).

Construction of Sets

To create the sets:

1. Start with each set containing its index: ( A_i = {i+1} ).
2. For each 1 in the matrix b[i][j], add the elements of ( A_i ) to ( A_j ).

Example

For instance, if the matrix b is:

0 1 0
0 0 1
0 0 0

• It indicates ( A_1 ) should be a proper subset of ( A_2 ), and ( A_2 ) should be a proper subset of ( A_3 ). Thus, you might have:
  • ( A_1 = {1} )
  • ( A_2 = {1, 2} )
  • ( A_3 = {1, 2, 3} )

This ensures the proper subset relationships as indicated by the matrix b.

Implementation Notes

The solution involves iterating over the matrix and ensuring that the sets are constructed according to the proper subset relationships. For each 1 in the matrix, the corresponding set relationships are established. If a zero is encountered, the corresponding sets must not have a subset relationship.

Example:

- Input
2
4
0001
1001
0001
0000
3
011
001

- Output
3 1 2 3
2 1 3
2 2 4
4 1 2 3 4
1 1
2 1 2
3 1 2 3

• My C# code is here:

int t = int.Parse(Console.ReadLine());

for (int i = 0; i < t; i++)
{
    int n = int.Parse(Console.ReadLine());
    int[,] b = new int[n, n]; // nxn size, 2d array def

    for (int j = 0; j < n; j++)
    {     
        for (int k = 0; k < n; k++)
        {
            b[j, k] = int.Parse(Console.ReadLine());
        }   
    }

    List<HashSet<int>> sets = new List<HashSet<int>>();

    for (int j = 0; j < n; j++)
    {
        HashSet<int> set = new HashSet<int>();

        for (int k = 0; k < n; k++)
        {
            set.Add(k+1);  
            if(b[j, k] == 1) {
                set.Remove(j+1);
            }
              
        }
        sets.Add(set);
    }


}

What should I do after that?

I have tried to solve the problem but I couldn’t find a valid solution for the problem. I am expecting **help **for a *proper *and *valid *solution.