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On the size of transducers for bidirectional decoding of prefix codes

Published online by Cambridge University Press:  23 February 2012

Laura Giambruno
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy. lgiambr@math.unipa.it; sabrina@math.unipa.it
Sabrina Mantaci
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy. lgiambr@math.unipa.it; sabrina@math.unipa.it
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Abstract

In a previous paper [L. Giambruno and S. Mantaci, Theoret. Comput. Sci. 411 (2010) 1785–1792] a bideterministic transducer is defined for the bidirectional deciphering of words by the method introduced by Girod [IEEE Commun. Lett. 3 (1999) 245–247]. Such a method is defined using prefix codes. Moreover a coding method, inspired by the Girod’s one, is introduced, and a transducer that allows both right-to-left and left-to-right decoding by this method is defined. It is proved also that this transducer is minimal. Here we consider the number of states of such a transducer, related to some features of the considered prefix code X. We find some bounds of such a number of states in relation with different notions of “size” of X. In particular, we give an exact formula for the number of states of transducers associated to maximal prefix codes. We moreover consider two special cases of codes: maximal uniform codes and a class of codes, that we name string-codes. We show that they represent, for maximal codes, the extreme cases with regard to the number of states in terms of different sizes. Moreover we prove that prefix codes corresponding to isomorphic trees have transducers that are isomorphic as unlabeled graphs.

Type
Research Article
Copyright
© EDP Sciences 2012

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References

M.-P. Béal, J. Berstel, B.H. Marcus, D. Perrin, C. Reutenauer and P.H. Siegel, Variable-length codes and finite automata, in Selected Topics in Information and Coding Theory, edited by I. Woungang. World Scientific. To appear.
J. Berstel and D. Perrin, Theory of Codes. Academic Press (1985).
J. Berstel, D. Perrin and C. Reutenauer, Codes and Automata. Cambridge University Press (2010).
Fraenkel, A.S. and Klein, S.T., Bidirectional Huffman coding.Comput. J. 33 (1990) 296307. Google Scholar
Giambruno, L. and Mantaci, S., Transducers for the bidirectional decoding of prefix codes, Theoret. Comput. Sci. 411 (2010) 17851792. Google Scholar
Girod, B., Bidirectionally decodable streams of prefix code words. IEEE Commun. Lett. 3 (1999) 245247. Google Scholar
M. Lothaire, Applied combinatorics on words, Encyclopedia of Mathematics and its Applications 104(2005).
D. Salomon, Variable-Length Codes for Data Compression. Springer-Verlag (2007).

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