Algorithmica

pp 1–31 | Cite as

Self-Stabilizing Balls and Bins in Batches

The Power of Leaky Bins
  • Petra Berenbrink
  • Tom Friedetzky
  • Peter Kling
  • Frederik Mallmann-Trenn
  • Lars Nagel
  • Chris Wastell
Article
  • 14 Downloads

Abstract

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modeled as static balls into bins processes, where m balls (tasks) are to be distributed among n bins (servers). In a seminal work, Azar et al. (SIAM J Comput 29(1):180–200, 1999.  https://doi.org/10.1137/S0097539795288490) proposed the sequential strategy \(\textsc {Greedy}[{d}]\) for \(n=m\). Each ball queries the load of d random bins and is allocated to a least loaded of them. Azar et al. (1999) showed that \(d=2\) yields an exponential improvement compared to \(d=1\). Berenbrink et al. (SIAM J Comput 35(6):1350–1385, 2006.  https://doi.org/10.1137/S009753970444435X) extended this to \(m\gg n\), showing that for \(d=2\) the maximal load difference is independent of m (in contrast to the \(d=1\) case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of \(\lambda n\) new balls arrive and are distributed (in parallel) to the bins. Subsequently, each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server’s current load but receive no information about parallel requests. We study the \(\textsc {Greedy}[{d}]\) distribution scheme in this setting and show a strong self-stabilizing property: for any arrival rate \(\lambda =\lambda (n)<1\), the system load is time-invariant. Moreover, for any (even super-exponential) round t, the maximum system load is (w.h.p.) Open image in new window for \(d=1\) and Open image in new window for \(d=2\). In particular, \(\textsc {Greedy}[{2}]\) has an exponentially smaller system load for high arrival rates.

Keywords

Balls-into-bins Self-stabilizing 2-Choice Positive recurrent Maximum load 

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Petra Berenbrink
    • 1
    • 2
  • Tom Friedetzky
    • 3
  • Peter Kling
    • 1
    • 2
  • Frederik Mallmann-Trenn
    • 2
    • 4
  • Lars Nagel
    • 5
  • Chris Wastell
    • 3
  1. 1.Universität HamburgHamburgGermany
  2. 2.Simon Fraser UniversityBurnabyCanada
  3. 3.Durham UniversityDurhamUK
  4. 4.École normale supérieureParisFrance
  5. 5.Loughborough UniversityLoughboroughUK

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