Maximum size square sub-matrix with all 1s

Question: Given a matrix consisting only 0s and 1s, find the maximum size square sub-matrix with all 1s.

Example: Consider the below matrix.

0  1  1  0  1

1  1  0  1  0

0  1  1  1  0

1  1  1  1  0

1  1  1  1  1

0  0  0  0  0

The maximum square sub-matrix with all '1' bits is from (2,1) to (4,3)

1  1  1

1  1  1

1  1  1

Answer:
This is a classic Dynamic Programming problem. Lets calculate the maximum size square sub-matrix as we traverse the original matrix M[][]. We will use a auxiliary matrix S[][] of same size for memoization. S[i][j] represents size of the square sub-matrix with all 1s including M[i][j]. 'i' and 'j' will be the last row and column respectively in square sub-matrix.

How to calculate S[i][j]:
(1) First copy the first row and first column as it is from M[][] to S[][]
(2) And for the remaining entries as mentioned do the following:


We should note that if M[i][j] is '0' then S[i][j] will obviously be '0'. If M[i][j] is '1' then S[i][j] depends on earlier values.

If M[i][j] is '1' then it will contribute to the all 1s square sub-matrix ending at either M[i][j-1] or M[i-1][j] or M[i-1][j-1]. If we visualize the conditions then, we will see:

S[i][j] = min(S[i][j-1], S[i-1][j], S[i-1][j-1]) + 1

So to sum up:

If M[i][j] is 1 then
    S[i][j] = min(S[i][j-1], S[i-1][j], S[i-1][j-1]) + 1
Else /*If M[i][j] is 0*/
    S[i][j] = 0

(3) Find the maximum entry in S[][] and use it to construct the maximum size square sub-matrix.

Example - How did we arrive at above relationship?
Note if we include M[i][j] in earlier calculated sub-matrix then we are adding S[i][j] elements from i^th row and j^th columns. They all should be '1' if we wanna include M[i][j]. On visualizing with some examples, readers will analyze why, minimum of 3 neighbors is taken.

For the given M[][] in above example, constructed S[][] would be:

0  1  1  0  1

1  1  0  1  0

0  1  1  1  0

1  1  2  2  0

1  2  2  3  1

0  0  0  0  0

The value of maximum entry in above matrix is 3 and coordinates of the entry are (4, 3). Using the maximum value and its coordinates, we can find out the required sub-matrix.

#define ROW 10
#define COL 10

void FindMaxSubSquare(bool M[ROW][COL], int &max_i,

int &max_j, int &size)
{
    int i,j;
    int S[ROW][COL];

    /* Set first column of S[][]*/
    for(i = 0; i < ROW; i++)
        S[i][0] = M[i][0];

    /* Set first row of S[][]*/
    for(j = 0; j < COL; j++)
        S[0][j] = M[0][j];

    /* Construct other entries of S[][]*/
    for(i = 1; i < ROW; i++)
    {
        for(j = 1; j < COL; j++)
        {
            if(M[i][j] == 1)
                S[i][j] = min(S[i][j-1], S[i-1][j], S[i-1][j-1]) + 1;
            else
                S[i][j] = 0;
        }
    } 

    /* Find the maximum entry, and indexes of maximum entry in S[][] */
    size = S[0][0]; max_i = 0; max_j = 0;
    for(i = 0; i < ROW; i++)
    {
        for(j = 0; j < COL; j++)
        {
            if(size < S[i][j])
            {
                size = S[i][j];
                max_i = i;
                max_j = j;
            }
        }
    }     
    return
}

Complexity:
Time Complexity: O(m*n) where m is number of rows and n is number of columns in the given matrix.
Space Complexity: O(m*n) where m is number of rows and n is number of columns in the given matrix.