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
N1 - Funding Information:
1Part of this work was done while four of the authors were visiting the Department of Computer Science, University of Bonn. An extended abstract of this work was presented at CPM’97 of [5]. 2Supported in part by NSF Grant CCR-9700053. 3Research was partially supported by the DFG Grant KA 673/4-1, ESPRIT BR Grants 7079, 21726, and EC-US 030, and the Max-Planck Research Prize. 4Research partially supported by National Science Foundation Grants CCR-9503441, CCR-9821009. 5To whom correspondence should be addressed.
PY - 1997
Y1 - 1997
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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U2 - 10.1007/3-540-63220-4_48
DO - 10.1007/3-540-63220-4_48
M3 - Conference contribution
AN - SCOPUS:84948995379
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 - Apostolico, Alberto
A2 - Hein, Jotun
PB - Springer Verlag
T2 - 8th Annual Symposium on Combinatorial Pattern Matching, CPM 1997
Y2 - 30 June 1997 through 2 July 1997
ER -