On Truly Parallel Time in Population Protocols

TL;DR

This paper investigates the true parallel time complexity of population protocols, revealing a logarithmic lower bound and providing conditions for a matching upper bound, thus clarifying the efficiency limits of parallel implementations.

Contribution

It establishes a fundamental lower bound on the parallel steps needed in population protocols and offers a combinatorial upper bound under certain assumptions.

Findings

Expected maximum parallel steps is  (rac{\u2212  rac{ ext{log} n}{ ext{log} ext{log} n}})

Lower bound applies when transition function is a black box

Matching upper bound provided under additional assumptions

Abstract

The {\em parallel time} of a population protocol is defined as the average number of required interactions that 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 with $n$ interactions. This naming triggers an intuition that at least on the average a round of $n$ interactions can be implemented in $O(1)$ parallel steps. 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 of $n$ interactions is $\Omega (\frac {\log n}{\log \log n})$. We also provide a combinatorial argument for a matching upper bound on the number of parallel steps in the average case under additional assumptions.