Advertisement

On the Advice Complexity of the k-Server Problem

  • Hans-Joachim Böckenhauer
  • Dennis Komm
  • Rastislav Královič
  • Richard Královič
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

Competitive analysis is the established tool for measuring the output quality of algorithms that work in an online environment. Recently, the model of advice complexity has been introduced as an alternative measurement which allows for a more fine-grained analysis of the hardness of online problems. In this model, one tries to measure the amount of information an online algorithm is lacking about the future parts of the input. This concept was investigated for a number of well-known online problems including the k-server problem.

In this paper, we first extend the analysis of the k-server problem by giving both a lower bound on the advice needed to obtain an optimal solution, and upper bounds on algorithms for the general k-server problem on metric graphs and the special case of dealing with the Euclidean plane. In the general case, we improve the previously known results by an exponential factor, in the Euclidean case we design an algorithm which achieves a constant competitive ratio for a very small (i.e., constant) number of advice bits per request.

Furthermore, we investigate the relation between advice complexity and randomized online computations by showing how lower bounds on the advice complexity can be used for proving lower bounds for the competitive ratio of randomized online algorithms.

Keywords

Competitive Ratio Online Algorithm Euclidean Plane Competitive Analysis Online Problem 
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.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the k-server problem. Technical Report 703, ETH Zürich (2010)Google Scholar
  2. 2.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the advice complexity of online problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 331–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  4. 4.
    Chaitin, G.J.: On the length of programs for computing finite binary sequences. Journal of the ACM 13(4), 547–569 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dobrev, S., Královič, R., Pardubská, D.: How much information about the future is needed? In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 247–258. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 427–438. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Hromkovič, J.: Design and Analysis of Randomized Algorithms: Introduction to Design Paradigms. Texts in Theoretical Computer Science. An EATCS Series. Springer, New York (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hromkovič, J., Královič, R., Královič, R.: Information complexity of online problems. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 24–36. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Kolmogorov, A.N.: Three approaches to the definition of the concept “quantity of information”. Problemy Peredachi Informatsii 1, 3–11 (1965)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Koutsoupias, E.: The k-server problem. Computer Science Review 3(2), 105–118 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Shannon, C.E.: A mathematical theory of communication. Mobile Computing and Communications Review 5(1), 3–55 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yao, A.C.-C.: Probabilistic computations: Toward a unified measure of complexity (extended abstract). In: FOCS, pp. 222–227. IEEE, Los Alamitos (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Dennis Komm
    • 1
  • Rastislav Královič
    • 2
  • Richard Královič
    • 1
  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland
  2. 2.Department of Computer ScienceComenius UniversityBratislavaSlovakia

Personalised recommendations