{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-stable-majority-population-protocol-using","title":"A stable majority population protocol using logarithmic time and states","arxiv_id":"2012.15800","date":"2020-12-31","proceeding":null,"authors":["David Doty","Mahsa Eftekhari","Eric Severson"],"abstract":"We study population protocols, a model of distributed computing appropriate for modeling well-mixed chemical reaction networks and other physical systems where agents exchange information in pairwise interactions, but have no control over their schedule of interaction partners. The well-studied *majority* problem is that of determining in an initial population of $n$ agents, each with one of two opinions $A$ or $B$, whether there are more $A$, more $B$, or a tie. A *stable* protocol solves this problem with probability 1 by eventually entering a configuration in which all agents agree on a correct consensus decision of $A$, $B$, or $T$, from which the consensus cannot change. We describe a protocol that solves this problem using $O(\\log n)$ states ($\\log \\log n + O(1)$ bits of memory) and optimal expected time $O(\\log n)$. The number of states $O(\\log n)$ is known to be optimal for the class of stable protocols that are \"output dominant\" and \"monotone\". These are two natural constraints satisfied by our protocol, making it state-optimal for that class. We use, and develop novel analysis of, a key technique called a \"fixed resolution clock\" due to Gasieniec, Stachowiak, and Uznanski, who showed a majority protocol using $O(\\log n)$ time and states that has a positive probability of error. Our protocol is *nonuniform*: the transition function has the value $\\left \\lceil {\\log n} \\right \\rceil$ encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to $\\Theta(\\log n \\log \\log n)$.","url_abs":"https://arxiv.org/abs/2012.15800v2","url_pdf":"https://arxiv.org/pdf/2012.15800v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-stable-majority-population-protocol-using","repo_url":"https://github.com/eftekhari-mhs/population-protocols/tree/master/Exact_Majority","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}