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Consensus vs Broadcast, with and Without Noise (Extended Abstract)

Authors Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, Luca Trevisan

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Andrea Clementi
  • Università Tor Vergata di Roma, Italy
Luciano Gualà
  • Università Tor Vergata di Roma, Italy
Emanuele Natale
  • Université Côte d’Azur, Sophia Antipolis, France
Francesco Pasquale
  • Università Tor Vergata di Roma, Italy
Giacomo Scornavacca
  • Università degli Studi di Sassari, Italy
Luca Trevisan
  • Università Bocconi, Milano, Italy

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Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, and Luca Trevisan. Consensus vs Broadcast, with and Without Noise (Extended Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 42:1-42:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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 Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise). We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Random network models
  • Distributed Computing
  • Consensus
  • Broadcast
  • Gossip Models
  • Noisy Communication Channels


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