[{"oa":1,"year":"2026","month":"07","page":"77-87","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of k decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime \r\nk\r\n=\r\nO\r\n(\r\nn\r\n/\r\n(\r\nlog\r\n\r\nn\r\n)\r\n2\r\n)\r\n (for the population protocol model) and k = O((n/log n)1/3) (for the gossip model), often under restrictive assumptions on the initial configuration.\r\nIn this paper, we obtain the first consensus-time guarantees for USD that hold for arbitrary 2 ≤ k ≤ n and for arbitrary initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within \r\nO\r\n~\r\n(\r\nmin\r\n{\r\nk\r\n,\r\nn\r\n}\r\n)\r\n synchronous rounds with probability 1 - p⊥ - n-c, where p⊥ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within \r\nO\r\n~\r\n(\r\nmin\r\n{\r\nk\r\nn\r\n,\r\nn\r\n3\r\n/\r\n2\r\n}\r\n)\r\n asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal."}],"publisher":"Association for Computing Machinery","title":"Undecided state dynamics with many opinions","keyword":["consensus dynamics","undecided state dynamics","gossip model","population protocol model"],"date_updated":"2026-07-21T07:54:01Z","author":[{"full_name":"Cooper, Colin","first_name":"Colin","last_name":"Cooper"},{"first_name":"Frederik","id":"68748c44-84d5-11f1-b4f6-ca083374e553","last_name":"Mallmann-Trenn","full_name":"Mallmann-Trenn, Frederik"},{"full_name":"Radzik, Tomasz","last_name":"Radzik","first_name":"Tomasz"},{"full_name":"Shimizu, Nobutaka","last_name":"Shimizu","first_name":"Nobutaka"},{"full_name":"Shiraga, Takeharu","first_name":"Takeharu","last_name":"Shiraga"}],"supplementarymaterial":"no","date_published":"2026-07-01T00:00:00Z","department":[{"_id":"MoHe"}],"external_id":{"arxiv":["2603.02636"]},"article_processing_charge":"No","researchdata_availability":"no","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"das_tickbox":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"access_level":"open_access","success":1,"date_created":"2026-07-21T07:51:58Z","creator":"dernst","file_name":"2026_ACMPODC_Cooper.pdf","file_size":709077,"date_updated":"2026-07-21T07:51:58Z","checksum":"9ada61feba1e93fd72867a5ba4ee8bb8","content_type":"application/pdf","relation":"main_file","file_id":"22380"}],"file_date_updated":"2026-07-21T07:51:58Z","quality_controlled":"1","citation":{"ista":"Cooper C, Mallmann-Trenn F, Radzik T, Shimizu N, Shiraga T. 2026. Undecided state dynamics with many opinions. Proceedings of the Annual ACM Symposium on Principles of Distributed Computing. PODC: Symposium on Principles of Distributed Computing, 77–87.","apa":"Cooper, C., Mallmann-Trenn, F., Radzik, T., Shimizu, N., & Shiraga, T. (2026). Undecided state dynamics with many opinions. In Proceedings of the Annual ACM Symposium on Principles of Distributed Computing (pp. 77–87). Egham, United Kingdom: Association for Computing Machinery. https://doi.org/10.1145/3796701.3815920","short":"C. Cooper, F. Mallmann-Trenn, T. Radzik, N. Shimizu, T. Shiraga, in:, Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2026, pp. 77–87.","ama":"Cooper C, Mallmann-Trenn F, Radzik T, Shimizu N, Shiraga T. Undecided state dynamics with many opinions. In: Proceedings of the Annual ACM Symposium on Principles of Distributed Computing. Association for Computing Machinery; 2026:77-87. doi:10.1145/3796701.3815920","mla":"Cooper, Colin, et al. “Undecided State Dynamics with Many Opinions.” Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2026, pp. 77–87, doi:10.1145/3796701.3815920.","ieee":"C. Cooper, F. Mallmann-Trenn, T. Radzik, N. Shimizu, and T. Shiraga, “Undecided state dynamics with many opinions,” in Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Egham, United Kingdom, 2026, pp. 77–87.","chicago":"Cooper, Colin, Frederik Mallmann-Trenn, Tomasz Radzik, Nobutaka Shimizu, and Takeharu Shiraga. “Undecided State Dynamics with Many Opinions.” In Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, 77–87. Association for Computing Machinery, 2026. https://doi.org/10.1145/3796701.3815920."},"license":"https://creativecommons.org/licenses/by/4.0/","doi":"10.1145/3796701.3815920","scopus_import":"1","ddc":["000"],"has_accepted_license":"1","acknowledgement":"Nobutaka Shimizu is supported by JSPS KAKENHI Grant Number\r\n23K16837. Takeharu Shiraga is supported by JSPS KAKENHI Grant\r\nNumber 23K16840, and JST CRONOS Grant Number JPMJCS24K2.\r\nColin Cooper is supported by a Mercator Fellowship from DFG\r\nProject 491453517 at the University of Hamburg. We thank the\r\nanonymous reviewers for their helpful comments and suggestions.","OA_type":"gold","OA_place":"publisher","day":"01","publication_status":"published","publication_identifier":{"isbn":["9798400725128"]},"_id":"22367","publication":"Proceedings of the Annual ACM Symposium on Principles of Distributed Computing","status":"public","oa_version":"Published Version","conference":{"end_date":"2026-07-10","start_date":"2026-07-06","name":"PODC: Symposium on Principles of Distributed Computing","location":"Egham, United Kingdom"},"type":"conference","date_created":"2026-07-19T22:01:47Z","arxiv":1,"corr_author":"1"}]