Largest submatrix with equal no of 1's and 0's

Question

Given a matrix of size mxn containing 0\'s and 1\'s only. I need to find the largest sub-matrix which has equal number of 1\'s and 0\'s in it. Brute force appro

Answer 1

I assume a submatrice is formed using only contiguous rows\columns of the original matrix (ie by removing first\last row or columns).

This way, a matrix can be represented as

Mat = {origin(row,col), rowCount, columnCount}

If the original matrix is of dimension M x N, then

rowCount =  M - row
columnCount = N - col
Mat = {origin(row,col), M - row, N - col}.

The variable row and col has respectively M and N possible values, which means there are O(MxN) of such matrices.

Idea of Algorithm

1. pop matrix (m, n)from queue
2. Test . If success, output matrix
3. Construct all matrices (m, n-1) and (m-1, n) and put on queue
4. go back to 1.

Now there are two points:

• When you reduce dimensions there are only 4 possible matrices (2 by removing rows, 2 by removing columns)
• You construct a sub matrix by remove either the first and last row\column. You just need remove the count for the row\column you removed, which takes O(n) or O(m) time. this is the dynamic programming step.

Which mean the complexity is O(max(M,N)MN)