research-article

Metrical Task Systems on Trees via Mirror Descent and Unfair Gluing

Authors: Sébastien Bubeck,

Michael B. Cohen,

James R. Lee,

Yin Tat LeeAuthors Info & Claims

SIAM Journal on Computing, Volume 50, Issue 3

Pages 909 - 923

https://doi.org/10.1137/19M1237879

Published: 01 January 2021 Publication History

Abstract

We consider metrical task systems on tree metrics and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitive randomized algorithm for every $n$-point metric space.

References

[1]

Y. Bartal, Probabilistic approximations of metric spaces and its algorithmic applications, in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, FOCS '96, Burlington, VT, 1996, pp. 184--193.

Google Scholar

[2]

Y. Bartal, A. Blum, C. Burch, and A. Tomkins, A polylog(n)-competitive algorithm for metrical task systems, in Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, STOC '97, ACM, New York, 1997, pp. 711--719.

Google Scholar

[3]

Y. Bartal, B. Bollobás, and M. Mendel, Ramsey-type theorems for metric spaces with applications to online problems, J. Comput. System Sci., 72 (2006), pp. 890--921.

Google Scholar

[4]

N. Bansal, N. Buchbinder, A. Madry, and J. Naor, A polylogarithmic-competitive algorithm for the $k$-server problem, J. ACM, 62 (2015), 40.

Google Scholar

[5]

N. Bansal, N. Buchbinder, and J. Naor, Metrical task systems and the k-server problem on hsts, in Proceedings of the 37th International Colloquium Conference on Automata, Languages and Programming, ICALP'10, Springer-Verlag, Berlin, Heidelberg, 2010, pp. 287--298.

Google Scholar

[6]

N. Bansal, N. Buchbinder, and J. Naor, A primal-dual randomized algorithm for weighted paging, J. ACM, 59 (2012), 19.

Google Scholar

[7]

S. Bubeck, M. B. Cohen, Y. T. Lee, J. R. Lee, and A. Madry, k-server via multiscale entropic regularization, in Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, 2018, pp. 3--16.

Google Scholar

[8]

A. Blum, H. Karloff, Y. Rabani, and M. Saks, A decomposition theorem for task systems and bounds for randomized server problems, SIAM J. Comput., 30 (2000), pp. 1624--1661, https://doi.org/10.1137/S0097539799351882.

Google Scholar

[9]

Y. Bartal, N. Linial, M. Mendel, and A. Naor, On metric Ramsey-type phenomena, Ann. of Math. (2), 162 (2005), pp. 643--709.

Google Scholar

[10]

A. Borodin, N. Linial, and M. E. Saks, An optimal on-line algorithm for metrical task system, J. ACM, 39 (1992), pp. 745--763.

Google Scholar

[11]

A. Fiat and M. Mendel, Better algorithms for unfair metrical task systems and applications, SIAM J. Comput., 32 (2003), pp. 1403--1422, https://doi.org/10.1137/S0097539700376159.

Google Scholar

[12]

J. Fakcharoenphol, S. Rao, and K. Talwar, A tight bound on approximating arbitrary metrics by tree metrics, J. Comput. System Sci., 69 (2004), pp. 485--497.

Google Scholar

[13]

G. Leoni, A First Course in Sobolev Spaces, AMS, Providence, RI, 2009.

Google Scholar

[14]

S. Seiden, Unfair problems and randomized algorithms for metrical task systems, Inform. and Comput., 148 (1999), pp. 219--240.

Google Scholar

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SIAM Journal on Computing Volume 50, Issue 3

DOI:10.1137/smjcat.50.3

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This work is licensed under a Creative Commons Attribution 4.0 International License

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2021

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Wang KSong ZTheocharous GMahadevan SWilliams BChen YNeville J(2023)Smoothed online combinatorial optimization using imperfect predictionsProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i10.26430(12130-12137)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i10.26430
Lechowicz AChristianson NZuo JBashir NHajiesmaili MWierman AShenoy P(2023)The Online Pause and Resume Problem: Optimal Algorithms and An Application to Carbon-Aware Load ShiftingProceedings of the ACM on Measurement and Analysis of Computing Systems10.1145/36267767:3(1-32)Online publication date: 7-Dec-2023
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Räcke HSchmid SZabrodin RAgrawal KShun J(2023)Polylog-Competitive Algorithms for Dynamic Balanced Graph Partitioning for Ring DemandsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591097(403-413)Online publication date: 17-Jun-2023
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