Parallel Randomized Load Balancing: A Lower Bound for a More General Model

We extend the lower bound of Adler et. al [1] and Berenbrink [3] for parallel randomized load balancing algorithms. The setting in these asynchronous and distributed algorithms is of n balls and n bins. The algorithms begin by each ball choosing d bins independently and uniformly at random. The balls and bins communicate to determine the assignment of each ball to a bin. The goal is to minimize the maximum load, i.e., the number of balls that are assigned to the same bin. In [1, 3], a lower bound of Ω( r √ log n/ log log n) is proved if the communication is limited to r rounds. Three assumptions appear in the proofs in [1, 3]: the topological assumption, random choices of confused balls, and symmetry. We extend the proof of the lower bound so that it holds without these three assumptions. This lower bound applies to every parallel randomized load balancing algorithm we are aware of [1, 3, 8, 5].