Multidimensional Balanced Allocation for Multiple Choice & (1 + Beta) Processes

TL;DR

This paper analyzes multidimensional balanced allocation processes, providing tight bounds on the gap for various models including (1+beta) processes, with significant improvements over prior bounds and implications for load balancing in parallel and weighted settings.

Contribution

It introduces the first tight bounds on the gap for multidimensional (1+beta) processes and extends analysis to weighted, parallel, and non-uniform cases, improving prior results.

Findings

Upper bound on gap for symmetric d choice process is O(lnln(n))

Expected gap for m>>n is bounded by O(lnln(n))

Weighted and parallel models also achieve O(log(n)) or better bounds

Abstract

Allocation of balls into bins is a well studied abstraction for load balancing problems.The literature hosts numerous results for sequential(single dimensional) allocation case when m balls are thrown into n bins. In this paper we study the symmetric multiple choice process for both unweighted and weighted balls as well as for both multidimensional and scalar models.Additionally,we present the results on bounds on gap for (1+beta) choice process with multidimensional balls and bins. We show that for the symmetric d choice process and with m=O(n), the upper bound on the gap is O(lnln(n)) w.h.p.This upper bound on the gap is within D=f factor of the lower bound. This is the first such tight result.For the general case of m>>n the expected gap is bounded by O(lnln(n)).For variable f and non-uniform distribution of the populated dimensions,we obtain the upper bound on the expected gap as O(log(n)). Further,for the multiple round parallel balls and bins,we show that the gap is also bounded by O(loglog(n)) for m=O(n).The same bound holds for the expected gap when m>>n. Our analysis also has strong implications in the sequential scalar case.For the weighted balls and bins and general case m>>n,we show that the upper bound on the expected gap is O(log(n)) which improves upon the best prior bound of n^c.Moreover,we show that for the (1 + beta) choice process and m=O(n) the upper bound(assuming uniform distribution of f populated dimensions over D total dimensions) on the gap is O(log(n)/beta),which is within D=f factor of the lower bound.For fixed f with non-uniform distribution and for random f with Binomial distribution the expected gap remains O(log(n)/beta) independent of the total number of balls thrown. This is the first such tight result for (1 +beta) paradigm with multidimensional balls and bins.