Tight Bounds for Repeated Balls-into-Bins
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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 step t=1,2,…, we select one ball from each non-empty bin, and then place 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 [BCNPP19]. It also provides a positive answer to the conjecture in [BCNPP19] that for m=n the maximum load is ω(log n/ loglog n) in a polynomially large window. For m ∈ [ω(n), n log n], our new lower bound disproves the conjecture in [BCNPP19] 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 a polynomially large window. This matches our lower bound up to multiplicative constants. ∙ For any m ≥ n, our analysis also implies an O( m^2 / n) waiting time to a configuration with 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) steps. 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 ≤ m ≤poly(n).
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