Optimal Ball Recycling

TL;DR

This paper introduces the ball recycling game, a model for load balancing and memory-access heuristics, analyzing strategies for maximizing recycling rate with insights into optimal methods depending on distribution.

Contribution

It studies the ball recycling game, revealing that Random Ball is optimal in general, while Fullest Bin is near-optimal only under uniform distribution.

Findings

Random Ball strategy is constant-optimal for general distributions.

Fullest Bin strategy can be pessimal for general distributions.

Fullest Bin is near-optimal when the distribution is uniform.

Abstract

Balls-and-bins games have been a wildly successful tool for modeling load balancing problems. In this paper, we study a new scenario, which we call the ball recycling game, defined as follows: Throw m balls into n bins i.i.d. according to a given probability distribution p. Then, at each time step, pick a non-empty bin and recycle its balls: take the balls from the selected bin and re-throw them according to p. This balls-and-bins game closely models memory-access heuristics in databases. The goal is to have a bin-picking method that maximizes the recycling rate, defined to be the expected number of balls recycled per step in the stationary distribution. We study two natural strategies for ball recycling: Fullest Bin, which greedily picks the bin with the maximum number of balls, and Random Ball, which picks a ball at random and recycles its bin. We show that for general p, Random Ball is constant-optimal, whereas Fullest Bin can be pessimal. However, when p = u, the uniform distribution, Fullest Bin is optimal to within an additive constant.