Probabilistic leader election on rings of known size

Abstract

We are interested in the average case complexity of leader election (and related problems) on asynchronous processor rings, whose size, n, is known to all constituent processors.

Duris and Galil [10] prove an Ω(n log n) lower bound for the average (over all assignments of identifiers) of the number of messages required by any deterministic leader election algorithm, when n is a power of 2 and the processor identifier space is sufficiently (exponentially) large. More recently Bodlaender

Both Duris and Galil's and Bodlaender's lower bounds extend naturally (but significantly) to the expected message complexity of Las Vegas algorithms (randomized algorithms that terminate, always correctly, with probability 1) [7, 11, 5]. These bounds, in turn, can be shown to apply to the evaluation of several natural functions on rings, including AND, OR and XOR [7].

We show that the lower bound technique introduced by Duris and Galil can be modified to provide a direct proof of an Ω(n log n) lower bound for the expected message complexity of Las Vegas leader election on anonymous (identifier free) rings, that is substantially simpler than the original. This simplicity not only serves to highlight the important structure of technique but also facilitates its extension to both arbitrary ring size and to Monte Carlo algorithms (randomized algorithms that err with probability at most ε). Specifically, we prove that the expected message complexity of any probabilistic algorithm that selects a leader with probability at least 1 − ε on an anonymous ring of known size n, is Ω (n min (log n, log log (1/ε))). A number of common function evaluation problems (including AND, OR, PARITY, and SUM) on rings of known size, are shown to inherit this complexity bound; furthermore these bounds are tight to within a constant factor.

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada, the British Columbia Advanced Systems Institute and the Killam Foundation.