How asymmetry helps load balancing

Abstract

This paper deals with balls and bins processes related to randomized load balancing, dynamic resource allocation and hashing. Suppose n balls have to be assigned to n bins, where each ball has to be placed without knowledge about the distribution of previously placed balls. The goal is to achieve an allocation that is as even as possible so that no bin gets much more balls than the average. A well known and good solution for this problem is to choose d possible locations for each ball at random, to look into each of these bins, and to place the ball into the least full among these bins. This class of algorithms has been investigated intensively in the past but almost all previous analyses assume that the d locations for each ball are chosen uniform and independently at random from the set of all bins. We investigate whether a non-uniform and possibly dependent choice of the d locations for a ball can improve the load balancing. Three types of selections are distinguished: 1) uniform and independent 2) non-uniform and independent 3) non-uniform and dependent. Our first result shows that choosing the locations in a non-uniform way (type 2) results in a better load balancing than choosing the locations uniformly (type 1). Surprising, this smooth load balancing is obtained by an algorithm called "Always-Go-Left" which creates an asymmetric assignment of the balls to the bins. Our second result is a lower bound on the smallest-possible maximum load that can be achieved by any allocation algorithm of type 1, 2, or 3.