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

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 the k least loaded out of d bins chosen independently and uniformly at random in each of n k rounds. The primary goal is to derive tight bounds on the maximum bin load for (k, d)-choice for any 1 ≤ k < d ≤ 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 O(n) messages. The maximum load for a heavily loaded case where m > n balls are placed into n bins is also presented for the case d ≥ 2k. Potential applications are discussed such as distributed storage as well as parallel job scheduling in a cluster.