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 step $t=1,2,\ldots$, 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: $\quad \bullet$ 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 $\omega(\log n/ \log \log n)$ in a polynomially large window. 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 a polynomially large window. 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 a configuration with $O(m/n \cdot \log m)$ maximum load, even for worst-case initial distributions. $\quad \bullet$ For $m \geq n$, 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 $n \leq m \leq \mathrm{poly}(n)$.Comment: 37 pages, 5 figure