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Stable Leader Election in Population Protocols Requires Linear Time

Abstract

A population protocol *stably elects a leader* if, for all nn, starting from an initial configuration with nn agents each in an identical state, with probability 1 it reaches a configuration y\mathbf{y} that is correct (exactly one agent is in a special leader state \ell) and stable (every configuration reachable from y\mathbf{y} also has a single agent in state \ell). We show that any population protocol that stably elects a leader requires Ω(n)\Omega(n) expected "parallel time" --- Ω(n2)\Omega(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.

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