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