Brief Announcement: Communication Patterns for Optimal Resilience

Abstract

In response to the impossibility of solving the consensus problem in asynchronous systems subject to failures, various relaxations of the consensus problem have been proposed, including approximate agreement, crusader agreement, gather, and reliable broadcast. Some are interesting in their own right while others are useful building blocks for solving other problems. We focus on message-passing systems of n processes, up to f of which can experience malicious (Byzantine) failures. These problems all require that n > 3f and they frequently have fairly simple and efficient algorithms when n > 5f. Challenges arise when considering resilience between 3f+1 and 5f. For instance, nearly twenty years elapsed between the discovery of an approximate agreement algorithm for n > 5f [Danny Dolev et al., 1986] and one for n > 3f [Abraham et al., 2004]. A stumbling block could be too much focus on looking for algorithms in a certain natural and intuitive form, which we call canonical (asynchronous) rounds. In such an algorithm, each process repeatedly sends a message containing its entire state tagged with a round number, then waits to receive n-f messages with that same round number, does some local computation and proceeds to the next round number. The n > 5f approximate agreement algorithm is in canonical round form but the n > 3f one is not.
For algorithms in canonical round form, an obvious way of measuring time is the number of canonical rounds until the algorithm completes. However, this approach does not apply to other algorithms, such as those in which processes wait to receive a certain number of messages that have other properties besides simply having a certain round number. Attempts to rewrite these latter algorithms in canonical round form can result in drastically increased round complexity. This blow-up in the round complexity is inherent, as we show in this paper that for a wide set of problems, there is no algorithm in canonical round form that has a finite upper bound on the number of rounds if n ≤ 5f. In contrast, the standard way of measuring time results in constant time complexity.
We first show the impossibility of a bounded number of canonical rounds for a generic problem that captures the key properties needed in the proof. The result then follows immediately for, most notably, crusader agreement and flavors of approximate agreement. We then show via reductions that the same result holds for reliable broadcast and gather, since there are crusader agreement algorithms that use reliable broadcast and gather with no round overhead.