Tight Bounds for Asymptotic and Approximate Consensus

Abstract

Agreeing on a common value among a set of agents is a fundamental problem in distributed computing, which occurs in several variants: In contrast to exact consensus, approximate variants are studied in systems where exact agreement is not possible or required, e.g., in human-made distributed control systems and in the analysis of natural distributed systems, such as bird flocking and opinion dynamics. We study the time complexity of two classical agreement problems: non-terminating asymptotic consensus and terminating approximate consensus. Asymptotic consensus, requires agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values; approximate consensus requires agents to eventually stop setting their outputs, which must then lie within a predefined distance of each other. We prove tight lower bounds on the contraction ratios of asymptotic consensus algorithms subject to oblivious message adversaries, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented by Charron-Bost et al. (ICALP’16) for certain systems, including the strongest oblivious message adversary in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to the lower-bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.

A Speedup Theorem for Asynchronous Computation with Applications to Consensus and Approximate Agreement

“…Aspnes and Herlihy [1] proved a lower bound of ⌈log 3 1/𝜖⌉ rounds and an upper bound of ⌈log 2 1/𝜖⌉ rounds in the wait-free SWMR model, a gap that was later closed by Hoest and Shavit [28], who proved a tight bound on the number of rounds needed to solve approximate agreement: ⌈log 3 1/𝜖⌉ for 𝑛 = 2, and ⌈log 2 1/𝜖⌉ for 𝑛 ≥ 3. A similar result was recently proved in the context of dynamic networks by Függer, Nowak and Schwarz [20], where the reader can find a thorough discussion about the literature studying approximate agreement, and about the importance of this problem w.r.t. applications.…”

“…applications. Hoest and Shavit [28] prove their lower bounds using a global analysis of the protocol complex, while Függer, Nowak and Schwarz [20] extend the notion of valency used for consensus [18,33] for proving their results. We establish these lower bounds in a novel way, by computing the closure of the approximate agreement task.…”

Non-negotiating Distributed Computing

“…The multidimensional approximate agreement version, where processes start with values in R d , has also received recently much attention, e.g. [4,19,16]. It was considered in the context of message-passing Byzantine failures systems [29] as well as in shared memory systems with crash failures [26].…”

“…It would also be interesting to compare our results to dynamic graph message passing [16] or shared-memory iterated models [31], where processes operate in rounds, and hence the round number is a counter available for free. For instance, it has been shown [9] that the waitfree dynamic graph model is strong enough to implement any two process task, without an exponential slowdown, even using only beeps.…”

Two-Agent Approximate Agreement from an Epistemic Logic Perspective

“…arguments e.g. [18,21], here we will see a different explanation: in less than k rounds, they cannot acquire sufficient knowledge about their inputs. Also, while the two algorithms seem rather different, the agents learn exactly the same information about their inputs in both.…”