Efficient Stable Population Protocols for Parity and Beyond

[2512.20163] Efficient Stable Population Protocols for Parity and Beyond Skip to main content Search Submit Donate Log in Computer Science > Distributed, Parallel, and Cluster Computing arXiv:2512.20163 (cs) [Submitted on 23 Dec 2025 ( v1 ), last revised 25 Aug 2026 (this version, v5)] Title: Efficient Stable Population Protocols for Parity and Beyond Authors: Leszek Gąsieniec , Tytus Grodzicki , Tomasz Jurdziński , Jakub Kowalski , Grzegorz Stachowiak View PDF HTML (experimental) Abstract: For nearly two decades, population protocols have been extensively studied, yielding efficient solutions for central problems in distributed computing, including leader election, and majority computation, a predicate type in Presburger Arithmetic closely tied to population protocols. Surprisingly, no protocols have achieved both time- and space-efficiency for congruency predicates, such as parity computation, which are complementary in this arithmetic framework. This gap highlights a significant challenge in the field. To address this gap, we explore the parity problem, where agents are tasked with computing the parity of the given sub-population size. Then we extend the solution for parity to compute congruences modulo an arbitrary $m$. Previous research on efficient population protocols has focused on protocols that minimise both stabilisation time and state utilisation for specific problems. In contrast, this work slightly relaxes this expectation, permitting protocols to place less emphasis on full optimisation and more on universality, robustness, and probabilistic guarantees. This allows us to propose a novel computing paradigm that integrates population weights (or simply weights), a robust clocking mechanism, and efficient anomaly detection coupled with a switching mechanism (which ensures slow but always correct solutions). This paradigm facilitates universal design of efficient multistage stable population protocols. Specifically, the first efficient parity and congruence protocols introduced here use both $O(\log^3 n)$ states and achieve silent stabilisation in $O(\log^3 n)$ time. We conclude by discussing the impact of implicit conversion between unary and binary representations enabled by the weight system, with applications to other problems, including the computation and representation of (sub-)population sizes. Comments: To be presented at DISC 2026 Subjects: Distributed, Parallel, and Cluster Computing (cs.DC) MSC classes: 68W15, 68W20 Cite as: arXiv:2512.20163 [cs.DC] (or arXiv:2512.20163v5 [cs.DC] for this version) https://doi.org/10.48550/arXiv.2512.20163 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Grzegorz Stachowiak [ view email ] [v1] Tue, 23 Dec 2025 08:41:10 UTC (644 KB) [v2] Sat, 10 Jan 2026 08:10:30 UTC (645 KB) [v3] Sat, 14 Mar 2026 08:20:32 UTC (40 KB) [v4] Sun, 7 Jun 2026 07:10:48 UTC (43 KB) [v5] Tue, 25 Aug 2026 07:57:39 UTC (35 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.DC < prev | next > new | recent | 2025-12 Change to browse by: cs References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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