Question

This is an interview question: given a boolean matrix find the size of the largest contiguous square sub-matrix, which contains only “true” elements.

I found this question on SO but did not understand the answer. The proposed algorithm moves a diagonal line from the left top to the right bottom and keeps track of the rectangles of “true” elements as the line goes.

My questions:

• How to keep track of the rectangles in the algorithm?
• Why do we move the diagonal line? What if we move a vertical/horizontal line or both?
• How to calculate the algorithm complexity?

Answer 1

I can’t understand the answer in the link you posted, and I don’t think the complexity given there is optimal for your problem. (It claim that there is an O(N^(3/2)*logN) algorithm where N=n*n is the number of elements in the original matrix.)

For your largest square sub-matrix problem, there is a DP algorithm whose complexity is linear with the number of elements:

Let the original matrix is A[n][n], we are trying to find a matrix B[n][n], where B[i][j] indicates the size of largest square sub-matrix whose bottom-right element is A[i][j]. So

for (i = 0 ; i < n ; ++i)
  for (j = 0 ; j < n ; ++j) 
    if (A[i][j] == 0) B[i][j] = 0;
    else {
      if (B[i-1][j] != B[i][j-1]) {
        B[i][j] = min(B[i-1][j], B[i][j-1]) + 1
      } else {
        if (A[i-B[i-1][j]][j-B[i-1][j]] == 1)
          B[i][j] = B[i-1][j] + 1;
        else
          B[i][j] = B[i-1][j];
      }
    }

And the largest B[i][j] is the answer.

p.s. I didn’t check the array range for simplification. You can just consider the out-of-range elements are zero.