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Day 2 : Find K-th Row

PROBLEM STATEMENT

Given a square binary matrix ‘mat[n][n]’, find ‘K’ such that all elements in the Kth row are ‘0’ and all elements in the Kth column are ‘1’. The value of mat[k][k] can be anything (either ‘0’ or ‘1’). If no such k exists, return ‘-1’.

For example

Consider the following matrix :
0 1 1 
0 1 0 
1 1 0 

You can see that row 1 (0-based) contains all 0’s except mat[1][1] and column 1 contains all 1’s. Hence the answer for the above case is 1.

Input format :

The first line contains a single integer ‘T’ denoting the number of test cases to be run. Then the test cases follow.

The first line of each test case contains an integer N representing the number in the problem statement.

Then N lines follow, each of which contains N space-separated integers denoting the elements of the matrix.

Output Format :

 For each test case print an integer denoting the value of K if such K exists else print -1.

 Output for each test case will be printed in a separate line.

Note:

You are not required to print anything; it has already been taken care of. Just implement the function.

Constraints:

1 <= T <= 5
1 <= N <= 1000
0 <= Aij <= 1
Time Limit: 1sec

Approach 1

Approach:

For each row, we assume this to be the answer. For each number except mat[k][k] we check if it is 1 and for every number in the kth column we check if it is 1. If both the conditions match, we return this row. If no rows match we return -1.

Algorithm:

Start K from 0 to n - 1.
Check if all numbers in kth row except mat[k][k] are 0.
Check if all numbers in kth column except mat[k][k] are 1.
If the current K satisfies the above conditions return it.
Return -1 if no such k is found.

Try Problem

Approach 2

There can be at most one ‘K’ that can be qualified to be an answer. If we traverse the given matrix from a corner (preferably from top right and bottom left), we can quickly discard the complete row or complete column based on the below rules.

a) If mat[i][j] is 0 and i != j, then column j cannot be the solution.

b) If mat[i][j] is 1 and i != j, then row i cannot be the solution.

Algorithm:

Start from top right corner, i.e., i = 0, j = n-1. Initialize result as -1.
Do the following until we find the result or reach outside the matrix.
1. If mat[i][j] is 0, then check all elements on left of j in current row.
  1. If all elements on the left of j are also 0, then set the result as i.
  2. If all left side elements of i'th row are not 0, then this row cannot be a solution, increment i.
If mat[i][j] is 1, then check all elements below i in current column.
1. If all elements below i are 1, then set the result as j.
2. If all elements of j'th column are not 1, then this column cannot be a solution decrement j.
If the result is -1, return it.
Else check the validity of the result by checking all row and column elements of the result.

Try Problem