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Fast Graphical Population Protocols

Authors Dan Alistarh, Rati Gelashvili, Joel Rybicki



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Author Details

Dan Alistarh
  • IST Austria, Klosterneuburg, Austria
Rati Gelashvili
  • Novi Research, Menlo Park, CA, USA
Joel Rybicki
  • IST Austria, Klosterneuburg, Austria

Acknowledgements

We grateful to Giorgi Nadiradze for pointing out a generalisation of the phase clock construction to non-regular graphs. We also thank anonymous reviewers for their useful comments on earlier versions of this manuscript.

Cite AsGet BibTex

Dan Alistarh, Rati Gelashvili, and Joel Rybicki. Fast Graphical Population Protocols. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 14:1-14:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.OPODIS.2021.14

Abstract

Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other’s states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node. In this work, we consider the more general setting where G is an arbitrary regular graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ^{-2} ⋅ n polylog n pairwise interactions, with high probability, using at most φ^{-2} ⋅ polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • population protocols
  • leader election
  • exact majority
  • graphs

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