Tight Bounds for Repeated Balls-Into-Bins

Authors Dimitrios Los , Thomas Sauerwald



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Dimitrios Los
  • Department of Computer Science & Technology, University of Cambridge, UK
Thomas Sauerwald
  • Department of Computer Science & Technology, University of Cambridge, UK

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Dimitrios Los and Thomas Sauerwald. Tight Bounds for Repeated Balls-Into-Bins. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 45:1-45:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.45

Abstract

We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta (2019). This process starts with m balls arbitrarily distributed across n bins. At each round t = 1,2,…, one ball is selected from each non-empty bin, and then placed it into a bin chosen independently and uniformly at random. We prove the following results: - For any n ⩽ m ⩽ poly(n), we prove a lower bound of Ω(m/n ⋅ log n) on the maximum load. For the special case m = n, this matches the upper bound of 𝒪(log n), as shown in [Luca Becchetti et al., 2019]. It also provides a positive answer to the conjecture in [Luca Becchetti et al., 2019] that for m = n the maximum load is ω(log n/ log log n) at least once in a polynomially large time interval. For m ∈ [ω(n), n log n], our new lower bound disproves the conjecture in [Luca Becchetti et al., 2019] that the maximum load remains 𝒪(log n). - For any n ⩽ m ⩽ poly(n), we prove an upper bound of 𝒪(m/n ⋅ log n) on the maximum load for all steps of a polynomially large time interval. This matches our lower bound up to multiplicative constants. - For any m ⩾ n, our analysis also implies an 𝒪(m²/n) waiting time to reach a configuration with a 𝒪(m/n ⋅ log m) maximum load, even for worst-case initial distributions. - For m ⩾ n, we show that every ball visits every bin in 𝒪(m log m) rounds. For m = n, this improves the previous upper bound of 𝒪(n log² n) in [Luca Becchetti et al., 2019]. We also prove that the upper bound is tight up to multiplicative constants for any n ⩽ m ⩽ poly(n).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Repeated balls-into-bins
  • self-stabilizing systems
  • balanced allocations
  • potential functions
  • random walks

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