Mean-Biased Processes for Balanced Allocations

TL;DR

This paper introduces a new class of balanced allocation processes that bias towards underloaded bins, demonstrating that such processes keep the load gap logarithmic in the number of bins, with broad applicability and tight bounds.

Contribution

The paper defines and analyzes mean-biased allocation processes, including new and existing variants, proving their effectiveness in maintaining balanced loads with high probability.

Findings

Load gap is logarithmic in number of bins for biased processes.

Processes are effective for any number of balls allocated, including heavily loaded cases.

Results are tight for many processes, including Mean-Thinning and Twinning.

Abstract

We introduce a new class of balanced allocation processes which bias towards underloaded bins (those with load below the mean load) either by skewing the probability by which a bin is chosen for an allocation (probability bias), or alternatively, by adding more balls to an underloaded bin (weight bias). A prototypical process satisfying the probability bias condition is Mean-Thinning: At each round, we sample one bin and if it is underloaded, we allocate one ball; otherwise, we allocate one ball to a second bin sample. Versions of this process have been in use since at least 1986. An example of a process, introduced by us, which satisfies the weight bias condition is Twinning: At each round, we only sample one bin. If the bin is underloaded, then we allocate two balls; otherwise, we allocate only one ball. Our main result is that for any process with a probability or weight bias, with high probability the gap between maximum and minimum load is logarithmic in the number of bins. This result holds for any number of allocated balls (heavily loaded case), covers many natural processes that relax the Two-Choice process, and we also prove it is tight for many such processes, including Mean-Thinning and Twinning. Our analysis employs a delicate interplay between linear, quadratic and exponential potential functions. It also hinges on a phenomenon we call "mean quantile stabilization", which holds in greater generality than our framework and may be of independent interest.