An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model

El-Hayek A, Elsässer R, Schmid S. 2025. An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model. Proceedings of the ACM Symposium on Principles of Distributed Computing. PODC: Symposium on Principles of Distributed Computing.

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Conference Paper | Published | English
Author
El-Hayek, AntoineISTA ; Elsässer, Robert; Schmid, Stefan

Corresponding author has ISTA affiliation

Abstract
We revisit the majority problem in the population protocol communication model, as first studied by Angluin et al. (Distributed Computing 2008). We consider a more general version of this problem known as plurality consensus, which has already been studied intensively in the literature. In this problem, each node in a system of n nodes, has initially one of k different opinions, and they need to agree on the (relative) majority opinion. In particular, we consider the important and intensively studied model of Undecided State Dynamics. Our main contribution is an almost tight lower bound on the stabilization time: we prove that there exists an initial configuration, even with bias \Delta = \omega(\sqrt{n\log n}), where stabilization requires \Omega(kn\log \frac {\sqrt n} {k \log n}) interactions, or equivalently, \Omega(k\log \frac {\sqrt n} {k \log n}) parallel time for any k = o\left(\frac {\sqrt n}{\log n}\right). This bound is tight for any k \le n^{\frac 1 2 - \epsilon}, where \epsilon >0 can be any small constant, as Amir et al.~(PODC'23) gave a O(k\log n) parallel time upper bound for k = O\left(\frac {\sqrt n} {\log ^2 n}\right).
Publishing Year
Date Published
2025-06-13
Proceedings Title
Proceedings of the ACM Symposium on Principles of Distributed Computing
Publisher
Association for Computing Machinery
Acknowledgement
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MoDynStruct, No. 101019564) and the Austrian Science Fund (FWF) grant DOI 10.55776/I5862,grant DOI 10.55776/I5982, and grant DOI 10.55776/P33775 with additional funding from the netidee SCIENCE Stiftung, 2020–2024 and the German Research Foundation (DFG), grant 470029389 (FlexNets).
Conference
PODC: Symposium on Principles of Distributed Computing
Conference Location
Huatulco, Mexico
Conference Date
2025-06-16 – 2025-06-20
IST-REx-ID

Cite this

El-Hayek A, Elsässer R, Schmid S. An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model. In: Proceedings of the ACM Symposium on Principles of Distributed Computing. Association for Computing Machinery; 2025. doi:10.1145/3732772.3733505
El-Hayek, A., Elsässer, R., & Schmid, S. (2025). An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model. In Proceedings of the ACM Symposium on Principles of Distributed Computing. Huatulco, Mexico: Association for Computing Machinery. https://doi.org/10.1145/3732772.3733505
El-Hayek, Antoine, Robert Elsässer, and Stefan Schmid. “An Almost Tight Lower Bound for Plurality Consensus with Undecided State Dynamics in the Population Protocol Model.” In Proceedings of the ACM Symposium on Principles of Distributed Computing. Association for Computing Machinery, 2025. https://doi.org/10.1145/3732772.3733505.
A. El-Hayek, R. Elsässer, and S. Schmid, “An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model,” in Proceedings of the ACM Symposium on Principles of Distributed Computing, Huatulco, Mexico, 2025.
El-Hayek A, Elsässer R, Schmid S. 2025. An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model. Proceedings of the ACM Symposium on Principles of Distributed Computing. PODC: Symposium on Principles of Distributed Computing.
El-Hayek, Antoine, et al. “An Almost Tight Lower Bound for Plurality Consensus with Undecided State Dynamics in the Population Protocol Model.” Proceedings of the ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2025, doi:10.1145/3732772.3733505.
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2025-08-04
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