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- W4226478312 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 $t=1,2,ldots$, 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: $quad bullet$ For any $n leq m leq mathrm{poly}(n)$, we prove a lower bound of $Omega(m/n cdot log n)$ on the maximum load. For the special case $m=n$, this matches the upper bound of $O(log n)$, as shown in [BCNPP19]. It also provides a positive answer to the conjecture in [BCNPP19] that for $m=n$ the maximum load is $omega(log n/ log log n)$ at least once in a polynomially large time interval. For $min [omega(n),nlog n]$, our new lower bound disproves the conjecture in [BCNPP19] that the maximum load remains $O(log n)$. $quad bullet$ For any $nleq mleqmathrm{poly}(n)$, we prove an upper bound of $O(m/ncdotlog n)$ on the maximum load for all steps of a polynomially large time interval. This matches our lower bound up to multiplicative constants. $quad bullet$ For any $mgeq n$, our analysis also implies an $O(m^2/n)$ waiting time to reach a configuration with a $O(m/ncdotlog m)$ maximum load, even for worst-case initial distributions. $quad bullet$ For any $m geq n$, we show that every ball visits every bin in $O(mlog m)$ rounds. For $m = n$, this improves the previous upper bound of $O(n log^2 n)$ in [BCNPP19]. We also prove that the upper bound is tight up to multiplicative constants for any $n leq m leq mathrm{poly}(n)$." @default.
- W4226478312 created "2022-05-05" @default.
- W4226478312 creator A5017390373 @default.
- W4226478312 creator A5028940316 @default.
- W4226478312 date "2022-03-23" @default.
- W4226478312 modified "2023-09-25" @default.
- W4226478312 title "Tight Bounds for Repeated Balls-into-Bins" @default.
- W4226478312 doi "https://doi.org/10.48550/arxiv.2203.12400" @default.
- W4226478312 hasPublicationYear "2022" @default.
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