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Theory of Computing Systems

, Volume 59, Issue 3, pp 476–499 | Cite as

On the Advice Complexity of the k-server Problem Under Sparse Metrics

  • Sushmita Gupta
  • Shahin Kamali
  • Alejandro López-Ortiz
Article

Abstract

We consider the k-Server problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs. These algorithms require advice of (almost) linear size. We show that for graphs of size N and treewidth α, there is an online algorithm that receives O (n(log α + log log N))* bits of advice and optimally serves any sequence of length n. We also prove that if a graph admits a system of μ collective tree (q, r)-spanners, then there is a (q + r)-competitive algorithm which requires O (n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, when provided with O (n log log N) bits of advice. On the other side, we prove that advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of length n even for the 2-server problem on a path metric of size N ≥ 3. Through another lower bound argument, we show that at least \(\frac {n}{2}(\log \alpha - 1.22)\) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 ≤ α < 2k.

Keywords

k-Server problem Advice complexity Competitive analysis 

Notes

Acknowledgments

We gratefully acknowledge the useful reviews by the anonymous reviewers on an earlier version of this manuscript.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sushmita Gupta
    • 1
  • Shahin Kamali
    • 2
  • Alejandro López-Ortiz
    • 2
  1. 1.Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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