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Consensus with Stochastic Broadcast

Authors Pierre Fraigniaud , Boaz Patt-Shamir , Sergio Rajsbaum

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LIPIcs.DISC.2026.18.pdf
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ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Distributed computing models
  • Theory of computation → Random network models
  • Theory of computation → Distributed algorithms
Keywords
  • stochastic communication
  • distributed agreement
  • distributed consensus
  • Kripke graph
  • topology of distributed algorithms

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Abstract

We study binary consensus in the stochastic broadcast model, which assumes n ≥ 2 processes communicating synchronously by message broadcasts. At each round, every process broadcasts a message to all the other processes. Each broadcast succeeds independently with some probability p, where p ∈ [0,1] is a parameter of the model. If a broadcast succeeds, all processes receive the message, and if it fails, no process receives the message. The sender does not know whether its broadcast was successful or not. In this model, consensus is not solvable; the objective is to design, for a given number of rounds r, consensus algorithms that terminate in r rounds, minimizing the probability of error (disagreement). This problem has been studied in depth for 2 processes [DISC 2025]. We extend the study to n > 2.
We identify two key algorithms in the 1-round setting. In Algorithm pref1, each process decides 1 unless it knows of no input 1. In Algorithm courteous, each process decides the more frequent value in its view, and, in case of a tie, the process decides the negation of its own input value. It turns out that for small values of p courteous is better, while for larger values, pref1 is better.
Our main results are lower bounds. For n = 3 processes and one round, we show that courteous is optimal for p ∈ [0,2/3], and pref1 is optimal for p ∈ [2/3,1]. To obtain this result, we develop a specialized theory that extends the combinatorial topology approach to a probabilistic setting.
For an arbitrary (but fixed) number n > 3 of processes, we show that among all 1-round consensus algorithms, courteous is optimal for p ∈ [0,1/n], and that pref1 is asymptotically optimal for p → 1. 
For multi-round algorithms, we show that (unlike the case of n = 2 processes) repeating r times an optimal 1-round algorithm is not optimal for n ≥ 3 processes. Nevertheless, we present a 2-round algorithm, called sweep, that guarantees the best error probability for the number of transmissions. Specifically, for every number n of processes, repeating sweep r times entails at most rn broadcasts, and results in error probability q^{rn}, where q = 1-p is the probability that a broadcast fails.
We believe that our results provide another step towards the goal of a general topological theory of randomized distributed computing.

Cite As Get BibTex

Pierre Fraigniaud, Boaz Patt-Shamir, and Sergio Rajsbaum. Consensus with Stochastic Broadcast. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.18

Author Details

Pierre Fraigniaud
  • Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Université Paris Cité, France
Boaz Patt-Shamir
  • Tel Aviv University, Israel
Sergio Rajsbaum
  • Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Université Paris Cité, France

Funding

  • Fraigniaud, Pierre: Additional support from ANR Projects ENEDISC (ANR-24-CE48-7768-01) and PREDICTIONS (ANR-23-CE48-0010), and Émergence Project METALG from IDEX Paris Cité.
  • Patt-Shamir, Boaz: Work supported in part by Israel Science Foundation grant 1948/21 and Fondation des Sciences Mathématiques de Paris (FSMP).
  • Rajsbaum, Sergio: Support from project PAPIIT 37-IN109926

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