Abstract
The context of this paper is the two-choice paradigm which is deeply used in balanced online resource allocation, priority scheduling, load balancing and more recently in population protocols. The model governing the evolution of these systems consists in throwing balls one by one and independently of each others into n bins, which represent the number of agents in the system. At each discrete instant, a ball is placed in the least filled bin among two bins randomly chosen among the n ones. A natural question is the evaluation of the difference between the number of balls in the most loaded bin and the one in the least loaded. At time t, this difference is denoted by \(\text {Gap}(t)\). A lot of work has been devoted to the derivation of asymptotic approximations of this gap for large values of n. In this paper we go a step further by showing that for all \(t \ge 0\), \(n \ge 2\) and \(\sigma > 0\), the variable \(\text {Gap}(t)\) is less than \(a(1+\sigma )\ln (n) + b\) with probability greater than \(1-1/n^\sigma \), where the constants a and b, which are independent of t, \(\sigma \) and n, are optimized and given explicitly, which to the best of our knowledge has never been done before.
This work was partially funded by the French ANR project SocioPlug (ANR-13-INFR-0003), and by the DeSceNt project granted by the Labex CominLabs excellence laboratory (ANR-10-LABX-07-01).
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 subscriptionsReferences
Adler, M., Berenbrink, P., Schröder, K.: Analyzing an infinite parallel job allocation process. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 417–428. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-68530-8_35
Adler, M., Chakrabarti, S., Mitzenmacher, M., Rasmussen, L.: Parallel randomized load balancing. Random Struct. Algorithms 13(2), 159–188 (1998)
Alistarh, D., Aspnes, J., Gelashvili, R.: Space-optimal majority in population protocols. In: Czumaj, A. (ed.) Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2221–2239 (2018)
Alistarh, D., Kopinsky, J., Li, J., Nadiradze, G.: The power of choice in priority scheduling. In: Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC) (2017)
Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced allocations (extended abstract). In: Proceedings of the ACM Symposium on Theory of Computing (STOC) (1994)
Berenbrink, P., Czumaj, A., Friedetzky, T., Vvedenskaya, N.D.: Infinite parallel job allocation (extended abstract). In: Proceedings of the ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 99–108 (2000)
Berenbrink, P., Czumaj, A., Steger, A., Vöcking, B.: Balanced allocations: the heavily loaded case. SIAM J. Comput. 35(6), 1350–1385 (2006)
Berenbrink, P., Meyer auf der Heide, F., Schröder, K.: Allocating weighted jobs in parallel. Theory Comput. Syst. 32(3), 281–300 (1999)
Mitzenmacher, M.: Load balancing and density dependent jump Markov processes. In: Proceedings of International Conference on Foundations of Computer Science (1996)
Mitzenmacher, M., Richa, A.W., Sitaraman, R.: The power of two random choices: a survey of techniques and results. In: Handbook of Randomized Computing, pp. 255–312. Kluwer (2000)
Mocquard, Y., Sericola, B., Anceaume, E.: Balanced allocations and global clock in population protocols: An accurate analysis (Full version), Technical report (2018). https://hal.archives-ouvertes.fr/hal-01790973
Peres, Y., Talwar, K., Wieder, U.: The (1+\(\beta \))-choice process and weighted balls into bins. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) (2010)
Peres, Y., Talwar, K., Wieder, U.: Graphical balanced allocations and the (1 + \(\beta \))-choice process. Random Struct. Algorithms 47(4), 760–775 (2015)
Raab, M., Steger, A.: “Balls into Bins” — a simple and tight analysis. In: Luby, M., Rolim, J.D.P., Serna, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 159–170. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-49543-6_13
Talwar, K., Wieder, U.: Balanced allocations: a simple proof for the heavily loaded case. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 979–990. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_81
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Mocquard, Y., Sericola, B., Anceaume, E. (2018). Balanced Allocations and Global Clock in Population Protocols: An Accurate Analysis. In: Lotker, Z., Patt-Shamir, B. (eds) Structural Information and Communication Complexity. SIROCCO 2018. Lecture Notes in Computer Science(), vol 11085. Springer, Cham. https://doi.org/10.1007/978-3-030-01325-7_26
Download citation
DOI: https://doi.org/10.1007/978-3-030-01325-7_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01324-0
Online ISBN: 978-3-030-01325-7
eBook Packages: Computer ScienceComputer Science (R0)