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Stochastic analysis of average-based distributed algorithms

Part of: Special processes Markov processes

Published online by Cambridge University Press:  23 June 2021

Yves Mocquard ,
Frédérique Robin ,
Bruno Séricola  and
Emmanuelle Anceaume
Yves Mocquard*
Affiliation:
Inria, Univ. Rennes, CNRS, IRISA
Frédérique Robin*
Affiliation:
Inria, Univ. Rennes, CNRS, IRISA
Bruno Séricola*
Affiliation:
Inria, Univ. Rennes, CNRS, IRISA
Emmanuelle Anceaume*
Affiliation:
CNRS, Univ. Rennes, Inria, IRISA
*
*Postal address: Inria, Campus de Beaulieu, 35042 Rennes Cedex, France.
*Postal address: Inria, Campus de Beaulieu, 35042 Rennes Cedex, France.
*Postal address: Inria, Campus de Beaulieu, 35042 Rennes Cedex, France.
***Postal address: IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France.
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Abstract

We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

Keywords

Averaging stochastic processinteracting particle systemsMarkov chaincouplingdistributed algorithms

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 394 - 410
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aldous, D. and Lanoue, D. (2012). A lecture on the averaging process. Prob. Surv. 9, 90–102.CrossRefGoogle Scholar
Aldous, D., Lanoue, D. and Salez, J. (2015). The compulsive gambler process. Electron. J. Prob. 20, 35.10.1214/EJP.v20-3582CrossRefGoogle Scholar
Alistarh, D., Gelashvili, R. and Vojnovíc, M. (2015). Fast and exact majority in population protocols. In Proceedings of the 34th Annual ACM Symposium on Principles of Distributed Computing (PODC 2015), pp. 47–56. ACM.Google Scholar
Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M. and Peralta, R. (2006). Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18, 235–253.10.1007/s00446-005-0138-3CrossRefGoogle Scholar
Broom, M. and Cannings, C. (2013). A dynamic network population model with strategic link formation governed by individual preferences. J. Theoret. Biol. 335, 160–168.10.1016/j.jtbi.2013.06.024CrossRefGoogle Scholar
Cordasco, G. and Gargano, L. (2017). Space-optimal proportion consensus with population protocols. In Stabilization, Safety, and Security of Distributed Systems (SSS 2017) (Lecture Notes Comput. Sci. 10616), pp. 384–398. Springer, Cham.Google Scholar
Haslegrave, J. and Puljiz, M. (2017). Reaching consensus on a connected graph. J. Appl. Prob. 54, 181–198.10.1017/jpr.2016.94CrossRefGoogle Scholar
Lanoue, D. (2015). The iPod model. Electron. J. Prob. 20, 64.10.1214/EJP.v20-3559CrossRefGoogle Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Mocquard, Y., Anceaume, E., Aspnes, J., Busnel, Y. and Sericola, B. (2015). Counting with population protocols. In Proceedings of the 14th IEEE International Symposium on Network Computing and Applications (NCA 2015), pp. 35–42. IEEE.10.1109/NCA.2015.35CrossRefGoogle Scholar
Mocquard, Y., Anceaume, E. and Sericola, B. (2016). Optimal proportion computation with population protocols. In Proceedings of the 15th IEEE Symposium on Network Computing and Applications (NCA 2016), pp. 216–223. IEEE.Google Scholar
Mocquard, Y., Sericola, B. and Anceaume, E. (2020). Probabilistic analysis of rumor spreading time. INFORMS J. Computing 32, 172–181.10.1287/ijoc.2018.0845CrossRefGoogle Scholar
Sauerwald, T. and Sun, H. (2012). Tight bounds for randomized load balancing on arbitrary network topologies. In Proceedings of the 53rd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2012), pp. 341–350. IEEE.Google Scholar