Stable leader election in population protocols requires linear time

• Published in: Distributed computing Vol. 31; no. 4; pp. 257 - 271
• Main Authors: Doty, David, Soloveichik, David
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2018; Springer Nature B.V
• Subjects: Computer Communication Networks; Computer Hardware; Computer Science; Computer Systems Organization and Communication Networks; Configurations; Molecular chains; Organic chemistry; Software Engineering/Programming and Operating Systems; Theory of Computation; Time lower bound; Population protocols; Chemical reaction network; Leader election
• ISSN: 0178-2770; 1432-0452
• DOI: 10.1007/s00446-016-0281-z

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 y that is correct (exactly one agent is in a special leader state ℓ ) and stable (every configuration reachable from y also has a single agent in state ℓ ). We show that any population protocol that stably elects a leader requires Ω ( n ) expected “parallel time”— Ω ( 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.