# Parallel Balanced Allocations: The Heavily Loaded Case

Christoph Lenzen, Merav Parter, Eylon Yogev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

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 naive solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load m/n 0(Vm/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 0(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 + 0(1) w.h.p. within 0(log log(m/n) + log* n) rounds. The algorithm is symmetric (balls and bins all "look the same"), and balls send 0(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 11(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 + 0(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 0 (log n) messages w.h.p. This goes to show that, similar to the case of m = n, asymmetry allows for highly efficient solutions.

Original language English SPAA '19: The 31st ACM Symposium on Parallelism in Algorithms and Architectures Association for Computing Machinery (ACM) 313-322 10 9781450361842 https://doi.org/10.1145/3323165.3323203 Published - 17 Jun 2019 31st ACM Symposium on Parallelism in Algorithms and Architecturess (SPAA) - Phoenix, AzerbaijanDuration: 22 Jun 2019 → 24 Jun 2019

### Publication series

Name Annual ACM Symposium on Parallelism in Algorithms and Architectures

### Conference

Conference 31st ACM Symposium on Parallelism in Algorithms and Architecturess (SPAA) Azerbaijan Phoenix 22/06/19 → 24/06/19

## All Science Journal Classification (ASJC) codes

• Software
• Theoretical Computer Science
• Hardware and Architecture

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