Tight bounds for parallel randomized load balancing

Abstract

Given a distributed system of $$n$$n balls and $$n$$n bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires $$\varTheta (\log \log n)$$?(loglogn) rounds, and any such algorithm running for $$r\in {\mathcal {O}}(1)$$r?O(1) rounds incurs a bin load of $$\varOmega ((\log n/\log \log n)^{1/r})$$Ω((logn/loglogn)1/r). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in $$\log ^* n+{\mathcal {O}}(1)$$log?n+O(1) communication rounds using $${\mathcal {O}}(n)$$O(n) messages in total. Our main result, however, is a matching lower bound of $$(1-o(1))\log ^* n$$(1-o(1))log?n on the time complexity of symmetric algorithms that guarantee small bin loads. The essential preconditions of the proof are (i) a limit of $${\mathcal {O}}(n)$$O(n) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls need not be globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time.