Consensus Needs Broadcast in Noiseless Models but can be Exponentially Easier in the Presence of Noise

TL;DR

This paper explores the complexity of Consensus and Broadcast problems in distributed systems, showing they are equivalent in noiseless models but Consensus becomes exponentially easier than Broadcast in noisy models.

Contribution

It establishes a reduction from Broadcast to Consensus in noiseless models and provides tight bounds demonstrating an exponential gap in rounds needed between Consensus and Broadcast in noisy models.

Findings

Consensus and Broadcast are equivalent in noiseless models.

A logarithmic lower bound on rounds for Consensus in the GOSSIP model.

An exponential gap in rounds between Consensus and Broadcast in noisy models.

Abstract

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast? In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent. We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An $\Omega(\epsilon^{-2} n)$ lower bound on the number of rounds needed for Broadcast is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] in one such model (noisy uniform PULL, where $\epsilon$ is a parameter that measures the amount of noise). In such model, we prove a new $\Theta(\epsilon^{-2} n \log n)$ bound for Broadcast and a $\Theta(\epsilon^{-2} \log n)$ bound for binary Consensus, thus establishing an exponential gap between the number of rounds necessary for Consensus versus Broadcast.