Balanced Allocations in Batches: Simplified and Generalized

We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the Two-Choice process, for each of $m$ sequentially arriving balls, two randomly chosen bins are sampled and the ball is placed in the least loaded bin. It is well-known that the maximum load is $m/n+\log_2 \log n + O(1)$ w.h.p. Berenbrink, Czumaj, Englert, Friedetzky and Nagel (2012) introduced a parallel version of this process, where $m$ balls arrive in consecutive batches of size $b=n$ each. Balls within the same batch are allocated in parallel, using the load information of the bins at the beginning of the batch. They proved that the gap of this process is $O(\log n)$ with high probability. In this work, we present a new analysis of this setting, which is based on exponential potential functions. This allows us to both simplify and generalize the analysis of [BCE12] in different ways: $\quad 1.$ Our analysis covers a broad class of processes. This includes not only Two-Choice, but also processes with fewer bin samples like $(1+\beta)$, processes which can only receive one bit of information from each bin sample and graphical allocation, where bins correspond to vertices in a graph. $\quad 2.$ Balls may be of different weights, as long as their weights are independent samples from a distribution satisfying a technical condition on its moment generating function. $\quad 3.$ For arbitrary batch sizes $b \geq n$, we prove a gap of $O(b/n \cdot \log n)$. For any $b \in [n , n^3]$, we improve this to $O(b/n + \log n)$ and show that it is tight for a family of processes. This implies the unexpected result that for e.g. $(1+\beta)$ with constant $\beta \in (0, 1]$, the gap is $\Theta(\log n)$ for all $b \in [n,n \log n]$. We also conduct experiments which support our theoretical results, and even hint at a superiority of less powerful processes like $(1+\beta)$ for large batch sizes.