On the complexity of pattern matching for highly compressed Two-Dimensional texts

We consider the complexity of problems related to 2-dimensional texts (2d-texts) described succinctly. In a succinct description, larger rectangular sub-texts are defined in terms of smaller parts in a way similar to that of Lempel-Ziv compression for Idimensional texts, or in shortly described strings as in [9], or in hierarchical graphs described by context-free graph grammars. A given 2d-text T with many internal repetitions can have a hierarchical description (denoted Compress(T)) which is up to exponentially smaller and which can be the only part of the input for a patternmatching algorithm which gives information about T. Such a hierarchical description is given in terms of a straight-line program, see [9] or, equivalently, a 2-dimensional grammar. We consider compressed pattern-matching, where the input consists of a 2dpattern P and of a hierarchical description of a 2d-text T₁ and fully compressed pattern-matching, where the input consists of hierarchical descriptions of both the pattern P and the text T. For 1-dimensional strings there exist polynomial-time deterministic algorithms for these problems, for similar types of succinct text descriptions [2, 6, 8, 9]. We show that the complexity dramatically increases in a 2-dimensional setting. For example, compressed 2d-matching is NP-complete, fully compressed 2d-matching is ∑₂^p-complete, and testing a given occurrence of a two dimensional compressed pattern is co-NP-complete. On the other hand, we give efficient algorithms for the related problems of randomized equality testing and testing for a given occurrence of an uncompressed pattern. We also show the surprising fact that the compressed size of a subrectangle of a compressed 2d-text can grow exponentially, unlike the one dimensional case.