A Generalization of Multiple Choice Balls-into-Bins: Tight Bounds

TL;DR

This paper extends the multiple choice balls-into-bins model to a generalized $(k,d)$-choice process, deriving tight bounds on maximum load and optimizing parameters for minimal load and message cost, with applications in distributed systems.

Contribution

It provides tight bounds on maximum bin load for the $(k,d)$-choice process and identifies optimal parameter settings for practical efficiency.

Findings

Achieves constant maximum load with 2n messages.

Derives tight bounds for maximum load in the generalized model.

Analyzes heavily loaded scenarios with $d \, \geq \, 2k$.

Abstract

This paper investigates a general version of the multiple choice model called the $(k,d)$-choice process in which $n$ balls are assigned to $n$ bins. In the process, $k<d$ balls are placed into $k$ least loaded out of $d$ bins chosen independently and uniformly at random in each of $\frac{n}{k}$ rounds. The primary goal is to derive tight bounds on the maximum bin load for $(k,d)$-choice for any $1 \leq k < d \leq n$. Our results enable one to choose suitable parameters $k$ and $d$ for which the $(k,d)$-choice process achieves the optimal tradeoff between the maximum bin load and message cost: a constant maximum load and $2n$ messages. It is also shown that the maximum load for a heavily loaded case, in which $m>n$ balls are placed into $n$ bins, if $d \geq 2k$. Potential applications are also discussed such as distributed storage as well as parallel job scheduling in a cluster.