Two-Dimensional Pattern Matching Against Basic Picture Languages

Given a two-dimensional array of symbols and a picture language over a finite alphabet, we study the problem of finding rectangular subarrays of the array that belong to the picture language. We formulate four particular problems – finding maximum, minimum, any or all match(es) – and describe algorithms solving them for basic classes of picture languages, including local picture languages and picture languages accepted by deterministic on-line tessellation automata or deterministic four-way finite automata. We also prove that the matching problems cannot be solved for the class of local picture languages in linear time unless the problem of triangle finding is solvable in quadratic time. This shows there is a fundamental difference in the pattern matching complexity regarding the one-dimensional and two-dimensional setting.