Advertisement

Lempel-Ziv Dimension for Lempel-Ziv Compression

  • Maria Lopez-Valdes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

This paper describes the Lempel-Ziv dimension (Hausdorff like dimension inspired in the LZ78 parsing), its fundamental properties and relation with Hausdorff dimension. It is shown that in the case of individual infinite sequences, the Lempel-Ziv dimension matches with the asymptotical Lempel-Ziv compression ratio. This fact is used to describe results on Lempel-Ziv compression in terms of dimension of complexity classes and vice versa.

Keywords

Compression Ratio Complexity Class Compression Algorithm Ergodic Measure Entropy Rate 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billingsley, P.: Probability and Measure. John Wiley & Sons, Inc., New York (1979)MATHGoogle Scholar
  2. 2.
    Bourke, C., Hitchcock, J.M., Vinodchandran, N.V.: Entropy rates and finite-state dimension. Theoretical Computer Science 349 (2004)Google Scholar
  3. 3.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. John Wiley & Sons, Inc., New York (1991)MATHCrossRefGoogle Scholar
  4. 4.
    Dai, J.J., Lathrop, J.I., Lutz, J.H., Mayordomo, E.: Finite-state dimension. Theoretical Computer Science 310, 1–33 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hojo, K., Ryabko, B., Suzuki, J.: Performance of data compression in terms of hausdorff dimension. TIEICE: IEICE Transactions on Communications/Electronics/ Information and Systems 310, 1761–1764 (2001)Google Scholar
  6. 6.
    Pierce II, L.A., Shields, P.C.: Sequences incompressible by SLZ (LZW) yet fully compressible by ULZ. In: Numbers, Information and Complexity, pp. 385–390. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  7. 7.
    Lathrop, J.I., Strauss, M.J.: A universal upper bound on the performance of the Lempel-Ziv algorithm on maliciously-constructed data. In: Carpentieri, B. (ed.) Compression and Complexity of Sequences 1997, pp. 123–135. IEEE Computer Society Press, Los Alamitos (1998)CrossRefGoogle Scholar
  8. 8.
    López-Valdés, M., Mayordomo, E.: Dimension is compression. In: Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, pp. 676–685. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lutz, J.H.: The dimensions of individual strings and sequences. Information and Computation 187, 49–79 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Scheinwald, D.: On the lempel-ziv proof and related topics. Proceedings of the IEEE 82, 866–871 (1994)CrossRefGoogle Scholar
  12. 12.
    Ziv, J., Lempel, A.: Compression of individual sequences via variable rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Zvonkin, A.K., Levin, L.A.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25, 83–124 (1970)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Maria Lopez-Valdes
    • 1
  1. 1.Departamento de Informática e Ing. de SistemasUniversidad de ZaragozaZaragozaSpain

Personalised recommendations