java - Fast counting of 2D sub-matrices withing a large, dense 2D matrix?

Question

What's a good algorithm for counting submatrices within a larger, dense matrix? If I had a single line of data, I could use a suffix tree, but I'm not sure if generalizing a suffix tree into higher dimensions is exactly straightforward or the best approach here.

Thoughts?

My naive solution to index the first element of the dense matrix and eliminate full-matrix searching provided only a modest improvement over full-matrix scanning.

What's the best way to solve this problem?

Example:

Input:

Full matrix:
123
212
421

Search matrix:
12  
21  

Output:
2

This sub-matrix occurs twice in the full matrix, so the output is 2. The full matrix could be 1000x1000, however, with a search matrix as large as 100x100 (variable size), and I need to process a number of search matrices in a row. Ergo, a brute force of this problem is far too inefficient to meet my sub-second search time for several matrices.

Answer 1

Algorithms and Theory of Computation Handbook suggests what is an N^2 * log(Alphabet Size) solution. Given a sub-matrix to search for, first of all de-dupe its rows. Now note that if you search the large matrix row by row at most one of the de-duped rows can appear at any position. Use Aho-Corasick to search this in time N^2 * log(Alphabet Size) and write down at each cell in the large matrix either null or an identifier for the matching row of the sub-matrix. Now use Aho-Corasick again to search down the columns of this matrix of row matches and signal a match where all the rows are present below each other.