# On the complexity of pattern matching for highly compressed two-dimensional texts

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## Abstract

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 1-dimensional 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 pattern-matching 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 2d-pattern *P* and of a hierarchical description of a 2d-text *T*, 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 *Σ* _{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.

## Keywords

Polynomial Time Truth Assignment Algorithm Check Hierarchical Graph Pattern Check## Preview

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