Balanced Allocations: Caching and Packing, Twinning and Thinning

TL;DR

This paper introduces a unified framework for analyzing various biased load balancing processes, demonstrating they all maintain a small maximum load gap of O(log n) even under heavy loads.

Contribution

The paper develops a general analytical framework that covers multiple biased allocation processes, establishing their stability and small load gaps.

Findings

All processes achieve an O(log n) load gap.

Framework applies to heavy load scenarios with m ≥ n.

Analysis uses potential functions for stabilization proof.

Abstract

We consider the sequential allocation of $m$ balls (jobs) into $n$ bins (servers) by allowing each ball to choose from some bins sampled uniformly at random. The goal is to maintain a small gap between the maximum load and the average load. In this paper, we present a general framework that allows us to analyze various allocation processes that slightly prefer allocating into underloaded, as opposed to overloaded bins. Our analysis covers several natural instances of processes, including: The Caching process (a.k.a. memory protocol) as studied by Mitzenmacher, Prabhakar and Shah (2002): At each round we only take one bin sample, but we also have access to a cache in which the most recently used bin is stored. We place the ball into the least loaded of the two. The Packing process: At each round we only take one bin sample. If the load is below some threshold (e.g., the average load), then we place as many balls until the threshold is reached; otherwise, we place only one ball. The Twinning process: At each round, we only take one bin sample. If the load is below some threshold, then we place two balls; otherwise, we place only one ball. The Thinning process as recently studied by Feldheim and Gurel-Gurevich (2021): At each round, we first take one bin sample. If its load is below some threshold, we place one ball; otherwise, we place one ball into a $\textit{second}$ bin sample. As we demonstrate, our general framework implies for all these processes a gap of $\mathcal{O}(\log n)$ between the maximum load and average load, even when an arbitrary number of balls $m \geq n$ are allocated (heavily loaded case). Our analysis is inspired by a previous work of Peres, Talwar and Wieder (2010) for the $(1+\beta)$-process, however here we rely on the interplay between different potential functions to prove stabilization.