Tight bounds for parallel randomized load balancing
Christoph Lenzen, Roger P. Wattenhofer · 2011
We explore the fundamental limits of distributed balls-into-bins algorithms, i.e., algorithms where balls act in parallel, as separate agents. This problem was introduced by Adler et al., who showed that non-adaptive and symmetric algorithms cannot reliably perform better than a maximum bin load of Theta(log log n / log log log n) within the same number of rounds. We present an adaptive symmetric algorithm that achieves a bin load of two in log* n+O(1) communication rounds using O(n) messages in total. Moreover, larger bin loads can be traded in for smaller time complexities. We prove a matching lower bound of (1-o(1))log* n on the time complexity of symmetric algorithms that guarantee small bin loads at an asymptotically optimal message complexity of O(n). The essential preconditions of the proof are (i) a limit of O(n) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls are not 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.