On parallel time in population protocols

The parallel time of a population protocol is defined as the average number of required interactions in which an agent in the protocol participates, i.e., the quotient between the total number of interactions required by the protocol and the total number n of agents, or just roughly the number of required rounds, where a round stands for a sequence of n consecutive interactions. This naming triggers an intuition that at least the expected number of parallel steps sufficient to implement a round is O(1). In a single parallel step only mutually independent interactions can be involved. We show that when the transition function of a population protocol is treated as a black box then the expected maximum number of parallel steps necessary to implement a round is Ω(log⁡nlog⁡log⁡n). We also provide a combinatorial argument for a matching upper bound on the expected number of parallel steps under additional assumptions. Further, we extend these bounds by showing that the situation changes dramatically for sequences of m=Ω(nlog⁡n) interactions. Then, the expected number of parallel steps required to implement such sequences is Θ(mn) under the aforementioned additional assumptions. Thus, it asymptotically coincides with the notion of parallel time, i.e., O(mn), for sequences of interactions produced by protocols solving any non-trivial problems requiring Ω(nlog⁡n) interactions.