Competitive Algorithms for Generalized k-Server in Uniform Metrics
Article No.: 8, Pages 1 - 15
Abstract
The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server si lies in its own metric space Mi. A request is a k-tuple r = (r1,r2,…,rk, which is served by moving some server si to the point ri ∈ Mi, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines.
Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio k · 2k and O(k3 log k), respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of 2k-1. We also give a 22O(k)-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.
References
[1]
Dimitris Achlioptas, Marek Chrobak, and John Noga. 2000. Competitive analysis of randomized paging algorithms. Theoretical Computer Science 234, 1–2 (2000), 203–218. DOI:DOI:
[2]
John Augustine and Nick Gravin. 2010. On the continuous CNN problem. In Proceeding of the ISAAC. 254–265.
[3]
Nikhil Ayyadevara and Ashish Chiplunkar. 2021. The randomized competitive ratio of weighted k-server is at least exponential. In Proceedings of the European Symposium on Algorithms, ESA (LIPIcs), Petra Mutzel, Rasmus Pagh, and Grzegorz Herman (Eds.), Vol. 204. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 9:1–9:11. DOI:DOI:
[4]
Nikhil Bansal, Niv Buchbinder, Aleksander Madry, and Joseph Naor. 2015. A polylogarithmic-competitive algorithm for the k-server Problem. Journal of the ACM 62, 5 (2015), 40.
[5]
Nikhil Bansal, Marek Eliáš, and Grigorios Koumoutsos. 2017. Weighted k-server bounds via combinatorial dichotomies. In Symposium on Foundations of Computer Science, FOCS. 493–504. DOI:DOI:
[6]
Nikhil Bansal, Marek Eliáš, Grigorios Koumoutsos, and Jesper Nederlof. 2018. Competitive algorithms for generalized k-server in Uniform Metrics. In Proceedings of the Symposium on Discrete Algorithms, SODA. 992–1001. DOI:DOI:
[7]
Marcin Bienkowski, Łukasz Jeż, and Paweł Schmidt. 2019. Slaying hydrae: Improved bounds for generalized k-server in uniform metrics. In Proceedings of the Symposium on Algorithms and Computation, ISAAC 2019. 14:1–14:14.
[8]
Allan Borodin and Ran El-Yaniv. 1998. Online Computation and Competitive Analysis. Cambridge University Press.
[9]
Allan Borodin, Nathan Linial, and Michael E. Saks. 1992. An optimal on-line algorithm for metrical task system. Journal of the ACM 39, 4 (1992), 745–763. DOI:DOI:
[10]
Sébastien Bubeck, Michael B. Cohen, Yin Tat Lee, James R. Lee, and Aleksander Madry. 2018. k-server via multiscale entropic regularization. In Proceedings of the Symposium on Theory of Computing, STOC. 3–16. DOI:DOI:
[11]
Ashish Chiplunkar. Oct 2016. (Oct 2016). Personal Communication.
[12]
Ashish Chiplunkar and Sundar Vishwanathan. 2020. Randomized memoryless algorithms for the weighted and the generalized k-server Problems. ACM Transactions on Algorithms 16, 1 (2020), 14:1–14:28. DOI:DOI:
[13]
Marek Chrobak. 2003. SIGACT news online algorithms column 1. SIGACT News 34, 4 (2003), 68–77. DOI:DOI:
[14]
Marek Chrobak, Howard J. Karloff, Thomas H. Payne, and Sundar Vishwanathan. 1991. New results on server problems. SIAM Journal on Discrete Mathematics 4, 2 (1991), 172–181.
[15]
Marek Chrobak and Lawrence L. Larmore. 1991. An optimal on-line algorithm for k-servers on trees. SIAM Journal on Computing 20, 1 (1991), 144–148. DOI:DOI:
[16]
Z. Dvir. 2009. On the size of kakeya sets in finite fields. Journal of the American Mathematical Society 22 (2009), 1093–1097.
[17]
J. S. Ellenberg and D. Gijswijt. 2017. On large subsets of \(F_q^n\) with no three-term arithmetic progression. Ann. Math. 185, 1 (2017), 339–343.
[18]
Amos Fiat, Richard M. Karp, Michael Luby, Lyle A. McGeoch, Daniel Dominic Sleator, and Neal E. Young. 1991. Competitive paging algorithms. Journal of Algorithms 12, 4 (1991), 685–699. DOI:
[19]
Amos Fiat and Moty Ricklin. 1994. Competitive algorithms for the weighted server problem. Theoretical Computer Science 130, 1 (1994), 85–99. DOI:DOI:
[20]
L. Guth. 2016. Polynomial Methods in Combinatorics. American Mathematical Society.
[21]
Kazuo Iwama and Kouki Yonezawa. 2001. Axis-bound CNN problem. IEICE TRANS (2001), 1–8.
[22]
Kazuo Iwama and Kouki Yonezawa. 2004. The orthogonal CNN problem. Information Processing Letters 90, 3 (2004), 115–120.
[23]
Bart M. P. Jansen and Jesper Nederlof. 2019. Computing the chromatic number using graph decompositions via matrix rank. Theoretical Computer Science 795, 26 (2019), 520–539. DOI:DOI:
[24]
Stasys Jukna. 2011. Extremal Combinatorics - With Applications in Computer Science. Springer.
[25]
Elias Koutsoupias and Christos H. Papadimitriou. 1995. On the k-server conjecture. Journal of the ACM 42, 5 (1995), 971–983.
[26]
Elias Koutsoupias and Christos H. Papadimitriou. 1996. The 2-evader problem. Information Processing Letters 57, 5 (1996), 249–252. DOI:DOI:
[27]
Elias Koutsoupias and David Scot Taylor. 2004. The CNN problem and other k-server variants. Theoretical Computer Science 324, 2–3 (2004), 347–359. DOI:DOI:
[28]
Mark S. Manasse, Lyle A. McGeoch, and Daniel D. Sleator. 1990. Competitive algorithms for server problems. Journal of the ACM 11, 2 (1990), 208–230.
[29]
Jiří Matoušek. 2010. In Proceedings of the 33rd Miniatures: Mathematical and Algorithmic Applications of Linear Algebra. American Mathematical Society.
[30]
Lyle A. McGeoch and Daniel Dominic Sleator. 1991. A strongly competitive randomized paging algorithm. Algorithmica 6, 6 (1991), 816–825. DOI:DOI:
[31]
René Sitters. 2014. The generalized work function algorithm is competitive for the generalized 2-server problem. SIAM Journal on Computing 43, 1 (2014), 96–125. DOI:DOI:
[32]
René Sitters, Leen Stougie, and Willem de Paepe. 2003. A competitive algorithm for the general 2-server problem. In ICALP. 624–636.
[33]
René A. Sitters and Leen Stougie. 2006. The generalized two-server problem. Journal of the ACM 53, 3 (2006), 437–458. DOI:DOI:
[34]
Daniel Dominic Sleator and Robert Endre Tarjan. 1985. Amortized efficiency of list update and paging rules. Communications of the ACM 28, 2 (1985), 202–208. DOI:DOI:
Index Terms
- Competitive Algorithms for Generalized k-Server in Uniform Metrics
Recommendations
The (h,k)-Server Problem on Bounded Depth Trees
Special Issue on Soda'17 and Regular PapersWe study the k-server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform ...
A Polylogarithmic-Competitive Algorithm for the k-Server Problem
FOCS '11: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer ScienceWe give the first polylogarithmic-competitive randomized algorithm for the k-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of O~(log^3 n log^2 k) for any metric space on n points. This ...
A Polylogarithmic-Competitive Algorithm for the k-Server Problem
We give the first polylogarithmic-competitive randomized online algorithm for the k-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log2 k) for any metric space on n points. Our ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
January 2023
254 pages
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 20 February 2023
Online AM: 13 December 2022
Accepted: 19 September 2022
Received: 15 May 2020
Published in TALG Volume 19, Issue 1
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- NWO Vici
- NWO Vidi
- ERC consolidator
- FNRS
- NWO Veni
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 193Total Downloads
- Downloads (Last 12 months)67
- Downloads (Last 6 weeks)4
Reflects downloads up to 16 Nov 2024
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderFull Text
View this article in Full Text.
Full TextHTML Format
View this article in HTML Format.
HTML Format