Abstract
The power of weighted finite automata to describe very complex images was widely studied, see [5, 6, 7]. Finite automata can be also used as an effective tool for compression of two-dimensional images. There are some software packages using this type of compression, see [12, 6]. We consider the complexity of some pattern-matching problems for two-dimensional images which are highly compressed using finite deterministic and weighted automata as small descriptions of images. Our basic problems are compressed pattern-matching, where the pattern is given explicitely and the text is compressed, and fully compressed pattern-matching (when also the pattern is compressed). We consider also fully compressed pattern-checking: testing of a given occurrence of the compressed pattern in a given position. We prove: Compressed matching for deterministic automata is in P. Compressed matching for weighted automata is NP-complete. Fully compressed pattern-checking for deterministic automata is in P. Fully compressed matching for deterministic automata is NP-complete.
Then we consider a 2-dimensional version of Lempel-Ziv compression (2LZ-compression), which results by traversing a given 2d-array by the Hilbert's curve and encoding the obtained string using Lempel-Ziv encoding (similar compression was considered in [16].) We investigate a relationship between finite automata encodings and LZ-encodings of images and show how to transform a description of an image T in terms of a deterministic automata A into a LZ-encoding of T of size polynomial w.r.t. |A|. This implies that searching for a compressed 1-dimensional pattern in a 2LZ-compressed 2d-text is NP-hard. We show also that there is no polynomial size transformation of images given by LZ-encoding to images described by automata.
Supported by Academy of Finland under grant 14047.
On leave from Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa, Poland. Supported by the grant KBN 8T11C01208.
Supported by the grant KBN 8T11C01208.
Preview
Unable to display preview. Download preview PDF.
References
A. Amir, G. Benson and M. Farach, Let sleeping files lie: pattern-matching in Z-compressed files, in SODA'94 (1994)
A. Amir and G. Benson, Efficient two dimensional compressed matching, Proc. of the 2nd IEEE Data Compression Conference 279–288 (1992).
A. Amir, G. Benson and M. Farach, Optimal two-dimensional compressed matching, in ICALP'94 215–225 (1994).
M. Crochemore and W. Rytter, Text Algorithms, Oxford University Press, New York (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).
D. Derencourt, J. Karhumäki, M. Letteux and A. Terlutte, On continuous functions computed by real functions, RAIRO Theor. Inform. Appl. 28, 387–404 (1994).
S. Eilenberg, Automata, Languages and Machines, Vol.A, Academic Press, New York (1974).
K. Culik and J. Kari, Fractal image compression: theory and applications, (ed. Y. Fisher); Springer Verlag 243–258 (1995).
M. Farach and M. Thorup, String matching in Lempel-Ziv compressed strings, in STOC'95, pp. 703–712.
Hopcroft J.E., Ullman J.D, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley (1979).
J. Kari, P. Franti, Arithmetic coding of weighted finite automata, RAIRO Theor. Inform. Appl. 28 343–360 (1994).
L. Gąsieniec, M. Karpiński, W. Plandowski and W. Rytter; Efficient Algorithms for Compressed Strings, in SWAT'96 (1996).
M. Karpinski, W. Rytter and A. Shinohara, Pattern-matching for strings with short description, in CPM'95 (1995).
A. Lempel and J. Ziv, On the complexity of finite sequences, IEEE Trans. on Inf. Theory 22, 75–81 (1976).
A. Lempel and J. Ziv, Compression of two-dimensional images sequences, Combinatorial algorithms on words (ed. A. Apostolico, Z.Galil) Springer Verlag 141–156 (1985).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karhumäki, J., Plandowski, W., Rytter, W. (1997). Pattern-matching problems for 2-dimensional images described by finite automata. In: Chlebus, B.S., Czaja, L. (eds) Fundamentals of Computation Theory. FCT 1997. Lecture Notes in Computer Science, vol 1279. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036188
Download citation
DOI: https://doi.org/10.1007/BFb0036188
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63386-0
Online ISBN: 978-3-540-69529-5
eBook Packages: Springer Book Archive