Balanced Allocation on Graphs

TL;DR

This paper investigates the maximum load in a graph-constrained two-choice balls and bins process, establishing bounds for almost regular and regular graphs, and introduces a method to achieve improved load balancing with limited random accesses.

Contribution

It extends the two-choice load balancing analysis to graph-constrained settings and proposes a new scheme using contiguous bin groups for better load distribution with fewer random choices.

Findings

Maximum load for almost regular graphs is log log n + O(1/psilon)

Maximum load for elta-regular graphs is log log n + O(rac{log n}{log (Delta/log^4 n)}

Achieves O(log log n / d) maximum load with only two random accesses using contiguous bin groups.

Abstract

In this paper, we study the two choice balls and bins process when balls are not allowed to choose any two random bins, but only bins that are connected by an edge in an underlying graph. We show that for $n$ balls and $n$ bins, if the graph is almost regular with degree $n^\epsilon$, where $\epsilon$ is not too small, the previous bounds on the maximum load continue to hold. Precisely, the maximum load is $\log \log n + O(1/\epsilon) + O(1)$. For general $\Delta$-regular graphs, we show that the maximum load is $\log\log n + O(\frac{\log n}{\log (\Delta/\log^4 n)}) + O(1)$ and also provide an almost matching lower bound of $\log \log n + \frac{\log n}{\log (\Delta \log n)}$. V{\"o}cking [Voc99] showed that the maximum bin size with $d$ choice load balancing can be further improved to $O(\log\log n /d)$ by breaking ties to the left. This requires $d$ random bin choices. We show that such bounds can be achieved by making only two random accesses and querying $d/2$ contiguous bins in each access. By grouping a sequence of $n$ bins into $2n/d$ groups, each of $d/2$ consecutive bins, if each ball chooses two groups at random and inserts the new ball into the least-loaded bin in the lesser loaded group, then the maximum load is $O(\log\log n/d)$ with high probability.