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.
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© 1997 Springer-Verlag Berlin Heidelberg
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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
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DOI: https://doi.org/10.1007/BFb0036188
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