Multiple-Choice Balanced Allocation in (Almost) Parallel

Abstract

We consider the problem of resource allocation in a parallel environment where new incoming resources are arriving online in groups or batches.

We study this scenario in an abstract framework of allocating balls into bins. We revisit the allocation algorithm $\mbox{\sc Greedy} [2]$ due to Azar, Broder, Karlin, and Upfal (SIAM J. Comput. 1999), in which, for sequentially arriving balls, each ball chooses two bins at random, and gets placed into one of those two bins with minimum load. The maximum load of any bin after the last ball is allocated by $\mbox{\sc Greedy} [2]$ is well understood, as is, indeed, the entire load distribution, for a wide range of settings. The main goal of our paper is to study balls and bins allocation processes in a parallel environment with the balls arriving in batches. In our model, m balls arrive in batches of size n each (with n being also equal to the number of bins), and the balls in each batch are to be distributed among the bins simultaneously. In this setting, we consider an algorithm that uses $\mbox{\sc Greedy} [2]$ for all balls within a given batch, the answers to those balls’ load queries are with respect to the bin loads at the end of the previous batch, and do not in any way depend on decisions made by other balls from the same batch.

Our main contribution is a tight analysis of the new process allocating balls in batches: we show that after the allocation of any number of batches, the gap between maximum and minimum load is O(logn) with high probability, and is therefore independent of the number of batches used.

Research supported by the Centre for Discrete Mathematics and its Applications (DIMAP), and by EPSRC awards EP/D063191/1, EP/G069034/1, EP/F043333/1 and EP/F043333/1.