Abstract
We study the following balls and bins stochastic game between a player and an adversary: there are B bins and a sequence of ball arrival and extraction events. In an arrival event a ball is stored in an empty bin chosen by the adversary and discarded if no bin is empty. In an extraction event, an algorithm selects a bin, clears it, and gains its content. We are interested in analyzing the gain of an algorithm which serves in the dark without any feedback at all, i.e., does not see the sequence, the content of the bins, and even the content of the cleared bins (i.e. an oblivious algorithm). We compare that gain to the gain of an optimal, open eyes, strategy that gets the same online sequence. We name this gain ratio the “loss of serving in the dark”.
The randomized algorithm that was previously analyzed is choosing a bin independently and uniformly at random, which resulted in a competitive ratio of about 1.69. We show that although no information is ever provided to the algorithm, using non-uniform probability distribution reduces the competitive ratio. Specifically, we design a 1.55-competitive algorithm and establish a lower bound of 1.5. We also prove a lower bound of 2 against any deterministic algorithm. This matches the performance of the round robin 2-competitive strategy. Finally, we present an application relating to a prompt mechanism for bounded capacity auctions.
Supported in part by the Israel Science Foundation and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, New York (2000)
Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced allocations. SIAM J. Comput 29(1), 180–200 (1999)
Azar, Y., Cohen, E., Fiat, A., Kaplan, H., Räcke, H.: Optimal oblivious routing in polynomial time. J. Comput. Syst. Sci 69(3), 383–394 (2004)
Azar, Y., Cohen, I.R., Gamzu, I.: The loss of serving in the dark. In: Proceedings 45th Annual ACM Symposium on Theory of Computing, pp. 951–960 (2013)
Azar, Y., Richter, Y,: The zero-one principle for switching networks. In: Proceedings 36th Annual ACM Symposium on Theory of Computing, pp. 64–71 (2004)
Chekuri, C., Goel, A., Khanna, S., Kumar, A.: Multi-processor scheduling to minimize flow time with epsilon resource augmentation. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13–16, 2004, pp. 363–372 (2004)
Cole, R., Dobzinski, S., Fleischer, L.K.: Prompt mechanisms for online auctions. In: Monien, B., Schroeder, U.-P. (eds.) SAGT 2008. LNCS, vol. 4997, pp. 170–181. Springer, Heidelberg (2008)
Dubhashi, P.D., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press (2009)
Johnson, N.L., Kotz, S.: Urn Models and Their Applications. John Wiley & Sons (1977)
Kesselman, A., Lotker, Z., Mansour, Y., Patt-Shamir, B., Schieber, B., Sviridenko, M.: Buffer overflow management in qos switches. SIAM J. Comput 33(3), 563–583 (2004)
Kolchin, V.F., Sevastyanov, B.A., Chistyakov, V.P.: Random Allocations. John Wiley & Sons (1978)
McDiarmid, C.: Concentration. In: Probabilistic Methods for Algorithmic Discrete Mathematics, Springer (1998)
Mitzenmacher, M., Richa, A.W., Sitaraman, R.: The power of two random choices: a survey of techniques and results. In: Handbook of Randomized Computing. Springer
Mitzenmacher, M., Upfal, E.: Probability and computing - randomized algorithms and probabilistic analysis. Cambridge University Press (2005)
Räcke, H.: Minimizing congestion in general networks. In: 43rd Symposium on Foundations of Computer Science, pp. 43–52. IEEE Computer Society (2002)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings 40th Annual ACM Symposium on Theory of Computing, pp. 255–264 (2008)
Sanders, P.: On the competitive analysis of randomized static load balancing. In: Proceedings of the first Workshop on Randomized Parallel Algorithms, RANDOM (1996)
Schulz, A.S., Skutella, M.: Scheduling unrelated machines by randomized rounding. SIAM J. Discrete Math. 15(4), 450–469 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Azar, Y., Cohen, I.R. (2015). Serving in the Dark should be done Non-Uniformly. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-47672-7_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-47671-0
Online ISBN: 978-3-662-47672-7
eBook Packages: Computer ScienceComputer Science (R0)