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.
