Complexity of Matching Sets of Two-Dimensional Patterns by Two-Dimensional On-Line Tessellation Automaton

Abstract

We study the two-dimensional pattern matching implemented using the deterministic two-dimensional on-line tessellation automaton. This restricted two-dimensional cellular automaton is able to simulate the Baker–Bird algorithm, which was proposed as the first algorithm for the two-dimensional pattern matching. We explore capabilities of this automaton to carry out the matching task against an arbitrary set of equal-sized patterns. To measure amount of resources needed to accomplish it, we introduce the pattern complexity of a picture language. We show that this complexity ranges from a constant to exponential one. All of these are illustrated by giving examples of two-dimensional on-line tessellation automata matching sets of patterns, describing general techniques of how to construct them and proving lower bounds on the pattern complexity of some picture languages as well as operations over them.