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

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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.

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Acknowledgments

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

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Research Bldg. No. 7, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan

    Sushmita Gupta

  2. School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada

    Shahin Kamali & Alejandro López-Ortiz

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  1. Sushmita Gupta
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  2. Shahin Kamali
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  3. Alejandro López-Ortiz
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Correspondence to Shahin Kamali.

Additional information

Preliminary version of this paper appeared in the Proceedings of the International Colloquium on Structural Information and Communication Complexity, SIROCCO’13 [24].

*We use log x to denote log2(x).

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Gupta, S., Kamali, S. & López-Ortiz, A. On the Advice Complexity of the k-server Problem Under Sparse Metrics. Theory Comput Syst 59, 476–499 (2016). https://doi.org/10.1007/s00224-015-9649-x

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