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 ith row and jth 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.