Efficient Size Estimation and Impossibility of Termination in Uniform Dense Population Protocols

Abstract

We study uniform population protocols: networks of anonymous agents whose pairwise interactions are chosen at random, where each agent uses an identical transition algorithm that does not depend on the population size n. Many existing polylog(n) time protocols for leader election and majority computation are nonuniform: to operate correctly, they require all agents to be initialized with an approximate estimate of n (specifically, the value łfloorłog n\rfloor). Our first main result is a uniform protocol for calculating łog(n) \pm O(1) with high probability in O(łog^2 n) time and O(łog^4 n) states (O(łog łog n) bits of memory). The protocol is not terminating : it does not signal when the estimate is close to the true value of łog n. If it could be made terminating with high probability, this would allow composition with protocols requiring a size estimate initially. We do show how our main protocol can be indirectly composed with others in a simple and elegant way, based on leaderless phase clocks, demonstrating that those protocols can in fact be made uniform. However, our second main result implies that the protocol cannot be made terminating, a consequence of a much stronger result: a uniform protocol for any task requiring more than constant time cannot be terminating even with probability bounded above 0, if infinitely many initial configurations are dense : any state present initially occupies Ømega(n) agents. (In particular no leader is allowed.) Crucially, the result holds no matter the memory or time permitted.