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The Lempel-Ziv Complexity of Fixed Points of Morphisms

  • Sorin Constantinescu
  • Lucian Ilie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

The Lempel–Ziv complexity is a fundamental measure of complexity for words, closely connected with the famous LZ77, LZ78 compression algorithms. We investigate this complexity measure for one of the most important families of infinite words in combinatorics, namely the fixed points of morphisms. We give a complete characterisation of the complexity classes which are Θ(1), Θ(logn), and Θ(n \(^{\rm 1/{\it k}}\)), k ∈ ℕ, k ≥2, depending on the periodicity of the word and the growth function of the morphism. The relation with the well-known classification of Ehrenfeucht, Lee, Rozenberg, and Pansiot for factor complexity classes is also investigated. The two measures complete each other, giving an improved picture for the complexity of these infinite words.

Keywords

Spike Train Complexity Measure Growth Function Factor Complexity Growth Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Amigo, J.M., Szczepanski, J., Wajnryb, E., Sanchez-Vives, M.V.: Estimating the entropy rate of spike trains via Lempel-Ziv complexity. Neural Computation 16(4), 717–736 (2004)MATHCrossRefGoogle Scholar
  2. 2.
    de Bruijn, N.G.: A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49, 758–764 (1946)MathSciNetGoogle Scholar
  3. 3.
    Chaitin, G.: On the length of programs for computing finite binary sequences. J. Assoc. Comput. Mach. 13, 547–569 (1966)MATHMathSciNetGoogle Scholar
  4. 4.
    Chen, X., Kwong, S., Li, M.: A compression algorithm for DNA sequences. IEEE Engineering in Medicine and Biology Magazine 20(4), 61–66 (2001)CrossRefGoogle Scholar
  5. 5.
    Choffrut, C., Karhumäki, J.: Combinatorics on words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, pp. 329–438. Springer, Heidelberg (1997)Google Scholar
  6. 6.
    Crochemore, M.: Linear searching for a square in a word. In: Apostolico, A., Galil, Z. (eds.) NATO Advanced Research Workshop on Combinatorial Algorithms on Words, 1984, pp. 66–72. Springer, Berlin (1985)Google Scholar
  7. 7.
    Ehrenfeucht, A., Lee, K.P., Rozenberg, G.: Subword complexities of various classes of deterministic developmental languages without interaction. Theoret. Comput. Sci. 1, 59–75 (1975)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ehrenfeucht, A., Rozenberg, G.: On the subword complexities of square-free D0L-languages. Theoret. Comput. Sci. 16, 25–32 (1981)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ehrenfeucht, A., Rozenberg, G.: On the subword complexities of D0L-languages with a constant distribution. Theoret. Comput. Sci. 13, 108–113 (1981)MATHMathSciNetGoogle Scholar
  10. 10.
    Ehrenfeucht, A., Rozenberg, G.: On the subword complexities of homomorphic images of languages. RAIRO Informatique Théorique 16, 303–316 (1982)MATHMathSciNetGoogle Scholar
  11. 11.
    Ehrenfeucht, A., Rozenberg, G.: On the subword complexities of locally catenative D0L-languages. Information Processing Letters 16, 7–9 (1982)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Ehrenfeucht, A., Rozenberg, G.: On the subword complexities of m-free D0L-languages. Information Processing Letters 17, 121–124 (1983)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Farach, M., Noordewier, M.O., Savari, S.A., Shepp, L.A., Wyner, A.D., Ziv, J.: On the entropy of DNA: algorithms and measurements based on memory and rapid convergence. In: Proc. of SODA 1995, pp. 48–57 (1995)Google Scholar
  14. 14.
    Gusev, V.D., Kulichkov, V.A., Chupakhina, O.M.: The Lempel-Ziv complexity and local structure analysis of genomes. Biosystems 30(1-3), 183–200 (1993)CrossRefGoogle Scholar
  15. 15.
    Ilie, L., Yu, S., Zhang, K.: Word complexity and repetitions in words. Internat. J. Found. Comput. Sci. 15(1), 41–55 (2004)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Inform. Transmission 1, 1–7 (1965)Google Scholar
  17. 17.
    Kolpakov, R., Kucherov, G.: Finding maximal repetitions in a word in linear time. In: Proc. of the 40th Annual Symposium on Foundations of Computer Science, pp. 596–604. IEEE Computer Soc., Los Alamitos (1999)Google Scholar
  18. 18.
    Lempel, A., Ziv, J.: On the Complexity of Finite Sequences. IEEE Trans. Inform. Theory 92(1), 75–81 (1976)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Lothaire, M.: Combinatorics on Words. Addison-Wesley, Reading (1983); Reprinted with corrections, Cambridge Univ. Press, Cambridge (1997)Google Scholar
  20. 20.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  21. 21.
    Lothaire, M.: Applied Combinatorics on Words. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  22. 22.
    Main, M.G.: Detecting leftmost maximal periodicities. Discrete Appl. Math. 25(1-2), 145–153 (1989)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pansiot, J.-J.: Bornes inférieures sur la complexité des facteurs des mots infinis engendrés par morphismes itérés. In: Fontet, M., Mehlhorn, K. (eds.) STACS 1984. LNCS, vol. 166, pp. 230–240. Springer, Heidelberg (1984)Google Scholar
  24. 24.
    Pansiot, J.-J.: Complexité des facteurs des mots infinis engendrés par morphismes itérés. In: Paredaens, J. (ed.) ICALP 1984. LNCS, vol. 172, pp. 380–389. Springer, Heidelberg (1984)Google Scholar
  25. 25.
    Rozenberg, G.: On subwords of formal languages. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 328–333. Springer, Heidelberg (1981)Google Scholar
  26. 26.
    Rozenberg, G., Salomaa, A.: The Mathematical Theory of L Systems. Academic Press, London (1980)MATHGoogle Scholar
  27. 27.
    Rytter, W.: Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theoret. Comput. Sci. 302(1-3), 211–222 (2003)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Salomaa, A., Soittola, M.: Automata-theoretic aspects of formal power series. Springer, New York (1978)MATHGoogle Scholar
  29. 29.
    Szczepanski, J., Amigo, M., Wajnryb, E., Sanchez-Vives, M.V.: Application of Lempel-Ziv complexity to the analysis of neural discharges. Network: Computation in Neural Systems 14(2), 335–350 (2003)CrossRefGoogle Scholar
  30. 30.
    Szczepanski, J., Amigo, J.M., Wajnryb, E., Sanchez-Vives, M.V.: Characterizing spike trains with Lempel-Ziv complexity. Neurocomputing 58-60, 79–84 (2004)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Inform. Theory 23(3), 337–343 (1977)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ziv, J., Lempel, A.: Compression of individual sequences via variable-rate coding. IEEE Trans. Inform. Theory 24(5), 530–536 (1978)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sorin Constantinescu
    • 1
  • Lucian Ilie
    • 1
  1. 1.Department of Computer ScienceUniversity of Western OntarioLondonCanada

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