Year: 2022
Type: conference-paper
Abstract: 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) with high probability.Berenbrink, Czumaj, Englert, Friedetzky and Nagel [7] 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 [7] in different ways:(1) Our analysis covers a broad class of processes.This includes not only Two-Choice, but also processes with fewer bin samples like the (1 + 𝛽)-process, processes which can only receive one bit of information from each bin sample and graphical allocation, where bins correspond to vertices in a graph.(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.(3) For any batch sizes 𝑏 ⩾ 𝑛, we prove a gap of O (𝑏/𝑛 • log 𝑛).For any 𝑏 ∈ [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. the (1 + 𝛽)-process with constant 𝛽 ∈ (0, 1], the gap is Θ(log 𝑛) for all 𝑏 ∈ [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 + 𝛽) for large batch sizes.
Authors: Dimitrios Los, Thomas Sauerwald
Institution: University of Cambridge
Language: English
Subfield: Computer Networks and Communications
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4289296199