The Dynamic And-Or Quorum System

Abstract

We investigate issues related to the probe complexity of the And-Or quorum system and its implementation in a dynamic environment. Our contribution is twofold: We first analyze the algorithmic probe complexity of the And-Or quorum system, and present two optimal algorithms. The first is a non-adaptive algorithm with $O(\sqrt{n}log n)$ probe complexity, which matches a known lower bound. The second is an adaptive algorithm with a probe complexity that is linear in the cardinality of a quorum set ($O(\sqrt{n})$), and requires at most O(loglogn) rounds. To the best of our knowledge, all other adaptive algorithms with same parameters (load and probe complexity) require $\theta(\sqrt{n})$ rounds.

Our second contribution is presenting the ‘dynamic And-Or’ quorum system – an adaptation of the above quorum system to a dynamic environment, where processors join and leave the network. It is based on a dynamic overlay network that emulates the De-Bruijn network and maintains the good properties of the quorum system(e.g.,load and availability). The algorithms suggested for the maintenance of these dynamic data structures are strongly coupled with the dynamic overlay network. This fact enables the use of gossip protocols which saves in message complexity and keeps the protocols simple and local. All these qualities make the ‘dynamic And-Or’ an excellent candidate for an implementation of dynamic quorums.