Skip to main content
SpringerLink
Log in

Analysis of the Harmonic Algorithm for Three Servers

  • Conference paper
  • First Online:
Book cover STACS 2003 (STACS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2607))

Included in the following conference series:

Abstract

Harmonic is a randomized algorithm for the k-server problem that, at each step, given a request point r, chooses the server to be moved to r with probability inversely proportional to the distance to r. For general k, it is known that the competitive ratio of Harmonic is at least 1/2k(k + 1), while the best upper bound on this ratio is exponential in k. It has been conjectured that Harmonic is 1/2k(k + 1)-competitive for all k. This conjecture has been proven in a number of special cases, including k = 2 and for the so-called lazy adversary.

In this paper we provide further evidence for this conjecture, by proving that Harmonic is 6-competitive for k = 3. Our approach is based on the random walk techniques and their relationship to the electrical network theory. We propose a new potential function Φ and reduce the proof of the validity of Φ to several inequalities involving hitting costs. Then we show that these inequalities hold for k = 3.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y. Bartal, M. Chrobak, J. Noga, and P. Raghavan. More on random walks, electrical networks, and the harmonic k-server algorithm. Information Processing Letters, 84:271–276, 2002.

    MATH  MathSciNet  Google Scholar 

  2. Y. Bartal and E. Grove. The harmonic k-server algorithm is competitive. Journal of the ACM, 47(1):1–15, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Berman, H. Karlo., and G. Tardos. A competitive algorithm for three servers. In Proc. 1st Symp. on Discrete Algorithms, pages 280–290, 1990.

    Google Scholar 

  4. A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998.

    Google Scholar 

  5. A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari. The electrical resistance of a graph captures its commute and cover times. Computational Complexity, 6:312–340, 1997.

    Article  MathSciNet  Google Scholar 

  6. M. Chrobak and L. L. Larmore. HARMONIC is three-competitive for two servers. Theoretical Computer Science, 98:339–346, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Chrobak and L. L. Larmore. Metrical task systems, the server problem, and the work function algorithm. In Online Algorithms: State of the Art, pages 74–94. Springer-Verlag, 1998.

    Google Scholar 

  8. M. Chrobak and J. Sgall. A simple analysis of the harmonic algorithm for two servers. Information Processing Letters, 75:75–77, 2000.

    Article  MathSciNet  Google Scholar 

  9. M. Chrobak and J. Sgall. Analysis of the harmonic algorithm for three servers. Technical Report ITI Series 2002-102, Charles University, Prague, 2002. http://iti.mff.cuni.cz/series/index.html

  10. D. Coppersmith, P. G. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40:421–453, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. G. Doyle and J. L. Snell. Random Walks and Electrical Networks. Mathematical Association of America, 1984.

    Google Scholar 

  12. E. Koutsoupias and C. Papadimitriou. On the k-server conjecture. Journal of the ACM, 42:971–983, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Manasse, L. A. McGeoch, and D. Sleator. Competitive algorithms for server problems. Journal of Algorithms, 11:208–230, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Percival. Solution of passive electrical networks by means of mathematical trees. J. Inst. Elect. Engrs., 100:143–150, 1953.

    MathSciNet  Google Scholar 

  15. P. Raghavan and M. Snir. Memory versus randomization in on-line algorithms. IBM Journal on Research and Development, 38, 1994.

    Google Scholar 

  16. S. Seshu and M. B. Reed. Linear Graphs and Electrical Networks. Addison-Wesley, 1961.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of California, 92521, Riverside, CA

    Marek Chrobak

  2. Mathematical Inst., AS CR, Žitná 25, CZ-11567, Praha 1, Czech Republic

    Jiří Sgall

Authors
  1. Marek Chrobak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Sgall
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takusstr. 9, 14195, Berlin, Germany

    Helmut Alt

  2. LIRMM, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chrobak, M., Sgall, J. (2003). Analysis of the Harmonic Algorithm for Three Servers. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-36494-3_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00623-7

  • Online ISBN: 978-3-540-36494-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation