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 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.
| Original language | English (US) |
|---|---|
| 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) |
|---|---|
| Volume | 1264 |
| ISSN (Print) | 0302-9743 |
| ISSN (Electronic) | 1611-3349 |
Other
| Other | 8th Annual Symposium on Combinatorial Pattern Matching, CPM 1997 |
|---|---|
| Country | Denmark |
| City | Aarhus |
| Period | 6/30/97 → 7/2/97 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)
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On the complexity of pattern matching for highly compressed Two-Dimensional texts. / Berman, Piotr; Karpinski, Marek; Larmore, Lawrence L.; Plandowski, Wojclech; Rytter, Wojciech.
Combinatorial Pattern Matching - 8th Annual Symposium, CPM 1997, Proceedings. ed. / Alberto Apostolico; Jotun Hein; Alberto Apostolico. Springer Verlag, 1997. p. 40-51 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1264).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.
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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 -