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MinMax algorithms for stabilizing consensus

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Abstract

In the stabilizing consensus problem each agent of a networked system has an input value and is repeatedly writing an output value; it is required that eventually all the output values stabilize to the same value which, moreover, must be one of the input values. We study this problem for a synchronous model with identical and anonymous agents that are connected by a time-varying topology and may join the system at any time (asynchronous start). Our main result is a generic MinMax algorithm that solves the stabilizing consensus problem in this model when, in each sufficiently long but bounded period of time, there is an agent, called a root, that can send messages, possibly indirectly, to all other agents. We stress that the bound on the time required for achieving this rootedness property is unknown to the agents. Such topologies are highly dynamic (in particular, roots may change arbitrarily over time) and may have very weak connectivity properties (an agent may be never a root). Our distributed MinMax algorithms thus require neither central control nor any global information and are also quite efficient in terms of message size and storage requirements.

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Notes

  1. To simplify notations, there are self-loops at all nodes of \(\mathbb {G}^{{\mathrm {a}}}(t)\), including those corresponding to the passive agents at round t.

  2. This condition holds if \(\mathbb {G}\) is permanently strongly connected, or if \(\mathbb {G}(t) =G\) for some rooted digraph G and for every positive integer t.

  3. Observe that the network model consisting of all dynamic graphs with non-empty kernel is not closed.

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Acknowledgements

We would like to thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. CNRS, École polytechnique, 91128, Palaiseau, France

    Bernadette Charron-Bost

  2. Department of Computer Science, Technion, 32000, Haifa, Israel

    Shlomo Moran

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  1. Bernadette Charron-Bost
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  2. Shlomo Moran
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Correspondence to Shlomo Moran.

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Charron-Bost, B., Moran, S. MinMax algorithms for stabilizing consensus. Distrib. Comput. 34, 195–206 (2021). https://doi.org/10.1007/s00446-021-00392-9

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