Tight Bounds for Repeated Balls-into-Bins

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 $m\in [\omega(n),n\log n]$, our new lower bound disproves the conjecture in [BCNPP19] that the maximum load remains $O(\log n)$. $\quad \bullet$ For any $n\leq m\leq\mathrm{poly}(n)$, we prove an upper bound of $O(m/n\cdot\log n)$ 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 $m\geq n$, our analysis also implies an $O(m^2/n)$ waiting time to reach a configuration with a $O(m/n\cdot\log m)$ maximum load, even for worst-case initial distributions. $\quad \bullet$ For any $m \geq n$, we show that every ball visits every bin in $O(m\log m)$ 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 $n \leq m \leq \mathrm{poly}(n)$.