Brief Announcement: Tight Bounds for Repeated Balls-into-Bins

Abstract

We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta [3]. 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 · logn) on the maximum load. For the special case m = n, this matches the upper bound of O (logn), as shown in [3]. It also provides a positive answer to the conjecture in [3] that for m = n the maximum load is ω(log n /log log n) in a polynomially large window. For m ∈ [ω (n), n logn], our new lower bound also disproves the conjecture in [3] that the maximum load remains O (logn). For any n ≤ m ≤ poly(n), we prove an upper bound of O (m/n · logn) on the maximum load for a polynomially large window, which matches our lower bound. For any m ≥ n, our analysis also implies an O (m2 /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 log2 n) in [3] and for any n ≤ m ≤ poly(n) this is tight up to multiplicative constants. Full version of the paper at: https://arxiv.org/abs/2203.12400.