Abstract
We consider efficiency of NC-algorithms for pattern-searching in highly compressed one- and two-dimensional texts. “Highly compressed” means that the text can be exponentially large with respect to its compressed version, and “fast” means “in polylogarithmic time”. Given an uncompressed pattern P and a compressed version of a text T, the compressed matching problem is to test if P occurs in T. Two types of closely related compressed representations of 1-dimensional texts are considered: the Lempel-Ziv encodings (LZ, in short) and restricted LZ encodings (RLZ, in short). For highly compressed texts there is a small difference between them, in extreme situations both of them compress text exponentially, e.g. Fibonacci words of size N have compressed versions of size O(logN) for LZ and Restricted LZ encodings. Despite similarities we prove that LZ-compressed matching is P-complete while RLZ-compressed matching is rather trivially in NC. We show how to improve a naive straightforward NC algorithm and obtain almost optimal parallel RLZ-compressed matching applying tree-contraction techniques to directed acyclic graphs with polynomial tree-size. As a corollary we obtain an almost optimal parallel algorithm for LZW-compressed matching which is simpler than the (more general) algorithm in [11]. Highly compressed 2-dimensional texts are also considered.
Supported in part by NUF-NAL (The Nuffield Foundation Award to Newly Appointed Lecturers) award.
Supported partially by the grant KBN 8T11C03915.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Amir, G. Benson, and M. Farach, Let sleeping files lie: Pattern matching in Z-compressed files, Proc. of 5th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1994.
A. Apostolico, C. Iliopoulos, G.M. Landau, B. Schieber, U. Vishkin, Parallel construction of a suffix tree with applications, Algorithmica 3 (1988) 347–365
P. Berman, M. Karpiński, L. Larmore, W. Plandowski, and W. Rytter, The complexity of pattern matching of highly compressed two-dimensional texts, Proc. of 8th Annual Symposium on Combinatorial Pattern Matching, CPM’97, July 1997.
M. Crochemore, W. Rytter, Text algorithms, Oxford University Press, 1994.
K. Culik and J. Karhumaki, Finite automata computing real functions, SIAM J. Comp (1994).
K. Culik and J. Kari, Image compression using weighted finite automata, Computer and Graphics 17, 305–313 (1993).
K. Culik and J. Kari, Fractal image compression: theory and applications, (ed. Y. Fisher), Springer Verlag 243–258 (1995).
S. De Agostino, P-complete problems in data compression, TCS 127, 181–186, 1994
M. Farach and M. Thorup, String Matching in Lempel-Ziv Compressed Strings, Proc. 27th ACM Symposium on Theory of Computing, pp. 703–713, 1994.
P. Ferragina, Personal communication
L. Gąsieniec, W. Rytter, Almost optimal fully LZW-compressed pattern matching, IEEE Data Compression Conference 1999
L. Gąsieniec, M. Karpiński, W. Plandowski, and W. Rytter, Efficient algorithms for Lempel-Ziv encoding, Proc. 5th Scandinavian Workshop on Algorithms Theory, pp. 392–403, LNCS 1097, July 1996.
A. Gibbons, W. Rytter. Efficient parallel algorithms, Cambridge Univ. Press 1988
M. Gu, M. Farach, R. Beigel, An efficient algorithm for dynamic text indexing, SODA’94, 697–704
J. Karhumaki, W. Plandowski, W. Rytter, Pattern matching for images generated by finite automata, FCT’97, in LNCS Springer Verlag 1997
J. Kari, P. Franti, Arithmetic coding of weighted finite automata, RAIRO Theor. Inform. Appl. 28 343–360 (1994).
H.J. Karloff, W.L. Ruzzo, The iterated mod problem, Inf. & Comp. 80(3) 193–204, 1989
M. Karpiński, W. Rytter, and A. Shinohara, A pattern matching for strings with short description, Proc. 6th Combinatorial Pattern Matching, June 1995.
A. Lempel and J. Ziv On the complexity of finite sequences, IEEE Transactions on Information Theory, pp. 22:75–81, 1976.
W. Plandowski and W. Rytter, Application of Lempel-Ziv encodings to the solution of words equations, Proc. of 25th International Colloquium on Automata, Languages, and Programming, to appear.
T.A. Welch, A technique for high performance data compression, IEEE transactions on Computers, pp. 17:8–19, 1984.
J. Ziv and A. Lempel, A universal algorithm for sequential data compression, IEEE Transactions on Information Theory, pp. IT-23(3):337–343, 1977.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gąsieniec, L., Gibbons, A., Rytter, W. (1999). Efficiency of Fast Parallel Pattern Searching in Highly Compressed Texts. In: Kutyłowski, M., Pacholski, L., Wierzbicki, T. (eds) Mathematical Foundations of Computer Science 1999. MFCS 1999. Lecture Notes in Computer Science, vol 1672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48340-3_5
Download citation
DOI: https://doi.org/10.1007/3-540-48340-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66408-6
Online ISBN: 978-3-540-48340-3
eBook Packages: Springer Book Archive