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Byzantine gathering in polynomial time

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Abstract

Gathering is a key task in distributed and mobile systems, which becomes significantly harder if some agents are subject to Byzantine faults, known as being the worst ones. We propose here to study the task of Byzantine gathering in an arbitrary graph: despite the presence of Byzantine agents, the goal is to ensure that all the other (good) agents, executing the same algorithm, eventually meet at the same node and stop. Initially, each agent gets as input a different label and some global knowledge that is common to all agents. The agents move in synchronous rounds and communicate with each other only when located at the same node. There are f Byzantine agents. These agents act in an unpredictable way, e.g., they may convey arbitrary informations or forge any label. In the literature, the gathering algorithms working in such a context all have an exponential time complexity in the number n of nodes and the labels of the good agents. In this paper, we design a deterministic algorithm to solve Byzantine gathering in time polynomial in n and the logarithm \(\ell \) of the smallest label of a good agent, provided the agents are a strong team i.e., a team where the number of good agents is at least some quadratic polynomial in f. Our algorithm requires global knowledge that can be coded in \(O(\log \log \log n)\) bits: we prove this size is of optimal order of magnitude to obtain a polynomial time complexity in n and \(\ell \) with strong teams.

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Notes

  1. As the reader will notice, our ultimate goal is to get a running time that is polynomial in N and \(|l_{min}|\), and thus polynomial in n and \(|l_{min}|\) as N is a polynomial upper bound on n. If \({\mathcal {G}}{\mathcal {K}}\) was equal to \(\lceil \log \log \log n\rceil \), then we could only “deduce” a quasi-polynomial upper bound on n (in particular, we should then replace \(2^{(2^{{\mathcal {G}}{\mathcal {K}}})}\) with \(2^{(2^{(2^{{\mathcal {G}}{\mathcal {K}}})})}\) at line 2 of Algorithm 1 and at line 1 of Algorithm 2). As a result, we would get a worse running time that would be sub-exponential in n and \(|l_{min}|\), but that would already be an improvement compared to the exponential complexity of the algorithms from [7, 18].

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Funding

This work was performed within Project ESTATE (Ref. ANR-16-CE25-0009-03). It is supported by French state funds managed by the ANR (Agence Nationale de la Recherche).

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Authors and Affiliations

  1. CNRS, LIP6, Sorbonne Université, 75005, Paris, France

    Sébastien Bouchard

  2. Laboratoire MIS, Université de Picardie Jules Verne, Amiens, France

    Yoann Dieudonné

  3. Ecole internationale des sciences du traitement de l’information EISTI, Cergy, France

    Anissa Lamani

Authors
  1. Sébastien Bouchard
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  2. Yoann Dieudonné
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  3. Anissa Lamani
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Correspondence to Yoann Dieudonné.

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A preliminary version of this paper appeared in the Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP 2018)

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Bouchard, S., Dieudonné, Y. & Lamani, A. Byzantine gathering in polynomial time. Distrib. Comput. 35, 235–263 (2022). https://doi.org/10.1007/s00446-022-00419-9

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