Distributed Computing

, Volume 31, Issue 4, pp 257–271 | Cite as

Stable leader election in population protocols requires linear time

  • David DotyEmail author
  • David Soloveichik


A population protocol stably elects a leader if, for all n, starting from an initial configuration with n agents each in an identical state, with probability 1 it reaches a configuration \(\mathbf {y}\) that is correct (exactly one agent is in a special leader state \(\ell \)) and stable (every configuration reachable from \(\mathbf {y}\) also has a single agent in state \(\ell \)). We show that any population protocol that stably elects a leader requires \(\varOmega (n)\) expected “parallel time”—\(\varOmega (n^2)\) expected total pairwise interactions—to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.


Population protocols Leader election Time lower bound Chemical reaction network 



The authors thank Anne Condon and Monir Hajiaghayi for several insightful discussions. We also thank the attendees of the 2014 Workshop on Programming Chemical Reaction Networks at the Banff International Research Station, where the first incursions were made into the solution of the problem of PP stable leader election. We are also grateful to anonymous reviewers whose comments have significantly improved the presentation. David Doty was supported by NSF Grants CCF-1619343, CCF-1219274, and CCF-1162589 and the Molecular Programming Project under NSF Grant 1317694. David Soloveichik was supported by an NIGMS Systems Biology Center Grant P50 GM081879 and NSF Grant CCF-1618895.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of California, DavisDavisUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of Texas, AustinAustinUSA

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