Parallel balanced allocations: The heavily loaded case

We study parallel algorithms for the classical balls-into-bins problem, in which m balls acting in parallel as separate agents are placed into n bins. Algorithms operate in synchronous rounds, in each of which balls and bins exchange messages once. The goal is to minimize the maximal load over all bins using a small number of rounds and few messages. While the case of m = n balls has been extensively studied, little is known about the heavily loaded case. In this work, we consider parallel algorithms for this somewhat neglected regime of m ≫ n. The naïve solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load m/n + Θ(√m/n · log n) (for m ≥ n log n) with high probability (w.h.p.). In contrast, for the sequential setting Berenbrink et al. [5] showed that letting each ball join the least loaded bin of two randomly selected bins reduces the maximal load to m/n + O(log log m) w.h.p. To date, no parallel variant of such a result is known. We present a simple parallel threshold algorithm that obtains a maximal load of m/n + O(1) w.h.p. within O(log log(m/n) + log^∗ n) rounds. The algorithm is symmetric (balls and bins all “look the same”), and balls send O(1) messages in expectation. The additive term of O(log^∗ n) in the complexity is known to be tight for such algorithms [10]. We also prove that our analysis is tight, i.e., algorithms of the type we provide must run for Ω(min{log log(m/n), n}) rounds w.h.p. Finally, we give a simple asymmetric algorithm (i.e., balls are aware of a common labeling of the bins) that achieves a maximal load of m/n + O(1) in a constant number of rounds w.h.p. Again, balls send only a single message per round, and bins receive (1 + o(1))m/n + O(log n) messages w.h.p. This goes to show that, similar to the case of m = n, asymmetry allows for highly efficient solutions.