Abstract
Given a two-dimensional array of symbols and a picture language over a finite alphabet, we investigate how to find rectangular subarrays that belong to the picture language. Two-dimensional pattern matching problems can be formulated by interpreting subarrays as matches and picture languages as patterns. We formulate four particular problems – finding maximum, minimum, any or all match(es) – and describe algorithms solving the problems for basic classes of picture languages, which include local picture languages and picture languages accepted by various types of deterministic two-dimensional finite automata such as four-way, three-way and two-way automata, on-line tessellation automata, and returning finite automata. We also prove that the pattern matching problems cannot be solved for the class of local picture languages in time linear in the input area unless the well known problem of triangle finding in a graph is solvable in quadratic time. This shows a fundamental difference between the complexity of one-dimensional and two-dimensional pattern matching.
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