### 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 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 T_{1} 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.

Original language | English (US) |
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Title of host publication | Combinatorial Pattern Matching - 8th Annual Symposium, CPM 1997, Proceedings |

Editors | Alberto Apostolico, Jotun Hein, Alberto Apostolico |

Publisher | Springer Verlag |

Pages | 40-51 |

Number of pages | 12 |

ISBN (Print) | 9783540632207 |

State | Published - Jan 1 1997 |

Event | 8th Annual Symposium on Combinatorial Pattern Matching, CPM 1997 - Aarhus, Denmark Duration: Jun 30 1997 → Jul 2 1997 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1264 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 8th Annual Symposium on Combinatorial Pattern Matching, CPM 1997 |
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Country | Denmark |

City | Aarhus |

Period | 6/30/97 → 7/2/97 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Combinatorial Pattern Matching - 8th Annual Symposium, CPM 1997, Proceedings*(pp. 40-51). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1264). Springer Verlag.

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*Combinatorial Pattern Matching - 8th Annual Symposium, CPM 1997, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1264, Springer Verlag, pp. 40-51, 8th Annual Symposium on Combinatorial Pattern Matching, CPM 1997, Aarhus, Denmark, 6/30/97.

**On the complexity of pattern matching for highly compressed Two-Dimensional texts.** / Berman, Piotr; Karpinski, Marek; Larmore, Lawrence L.; Plandowski, Wojclech; Rytter, Wojciech.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - On the complexity of pattern matching for highly compressed Two-Dimensional texts

AU - Berman, Piotr

AU - Karpinski, Marek

AU - Larmore, Lawrence L.

AU - Plandowski, Wojclech

AU - Rytter, Wojciech

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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 ∑2p-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.

AB - 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 ∑2p-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.

UR - http://www.scopus.com/inward/record.url?scp=84948995379&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948995379&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783540632207

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 40

EP - 51

BT - Combinatorial Pattern Matching - 8th Annual Symposium, CPM 1997, Proceedings

A2 - Apostolico, Alberto

A2 - Hein, Jotun

A2 - Apostolico, Alberto

PB - Springer Verlag

ER -