Set Matrix Zeroes

Problem

Given a m x n matrix, if an element is 0, set its entire row and column to 0. Do it in place.

Example

Given a matrix

[
  [1,2],
  [0,3]
],

return

[
  [0,2],
  [0,0]
]

Challenge

Did you use extra space?

A straight forward solution using O(mn) space is probably a bad idea.

A simple improvement uses O(m + n) space, but still not the best solution.

Could you devise a constant space solution?

Code - Java

public class Solution {
    /**
     * @param matrix: A list of lists of integers
     * @return: Void
     */
    public void setZeroes(int[][] matrix) {
        if (matrix == null || matrix.length == 0) {
            return;
        }

        int n = matrix.length;
        int m = matrix[0].length;

        boolean firstColZero = false;
        boolean firstRowZero = false;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (matrix[i][j] == 0) {
                    if (i == 0) {
                        firstRowZero = true;
                    }
                    if (j == 0) {
                        firstColZero = true;
                    }

                    matrix[i][0] = 0;
                    matrix[0][j] = 0;
                }
            }
        }

        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                if (matrix[i][0] == 0 || matrix[0][j] == 0) {
                    matrix[i][j] = 0;
                }
            }
        }

        if (firstColZero == true) {
            for (int i = 0; i < n; i++) {
                matrix[i][0] = 0;
            }
        }
        if (firstRowZero == true) {
            for (int j = 0; j < m; j++) {
                matrix[0][j] = 0;
            }
        }
    }
}