An Improved Drift Theorem for Balanced Allocations

TL;DR

This paper presents an improved drift theorem for balanced allocations, providing tighter bounds and broader applicability to various allocation processes, which enhances understanding of load balancing in distributed systems.

Contribution

The authors improve the drift inequality for balanced allocations, making it asymptotically tighter, more general, and applicable to processes with variable allocation strategies.

Findings

Enhanced drift inequality with weaker preconditions

Broader applicability to multiple and variable ball allocations

Potential for further research in load balancing analysis

Abstract

In the balanced allocations framework, there are $m$ jobs (balls) to be allocated to $n$ servers (bins). The goal is to minimize the gap, the difference between the maximum and the average load. Peres, Talwar and Wieder (RSA 2015) used the hyperbolic cosine potential function to analyze a large family of allocation processes including the $(1+\beta)$-process and graphical balanced allocations. The key ingredient was to prove that the potential drops in every step, i.e., a drift inequality. In this work we improve the drift inequality so that (i) it is asymptotically tighter, (ii) it assumes weaker preconditions, (iii) it applies not only to processes allocating to more than one bin in a single step and (iv) to processes allocating a varying number of balls depending on the sampled bin. Our applications include the processes of (RSA 2015), but also several new processes, and we believe that our techniques may lead to further results in future work.