Stable leader election in population protocols requires linear time

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 $$\mathbf {y}$$y that is correct (exactly one agent is in a special leader state $$\ell $$ℓ) and stable (every configuration reachable from $$\mathbf {y}$$y also has a single agent in state $$\ell $$ℓ). We show that any population protocol that stably elects a leader requires $$\varOmega (n)$$Ω(n) expected “parallel time”—$$\varOmega (n^2)$$Ω(n2) 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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