How many cooks spoil the soup?

Abstract

In this work, we study the following basic question: “How much parallelism does a distributed task permit?” Our definition of parallelism (or symmetry) here is not in terms of speed, but in terms of identical roles that processes have at the same time in the execution. For example, we may ask: “Can a given task be solved by a protocol that always has at least two processes in the same role at the same time?” (i.e., by a protocol that never elects a unique leader). We choose to initiate this study in population protocols, a very simple model that not only allows for a straightforward definition of what a role is, but also encloses the challenge of isolating the properties that are due to the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes. In particular, we define the role of a process at a given time to be equivalent to the state of the process at that time. Moreover, we isolate the symmetry that is due to the protocol (inherent symmetry) by focusing on those schedules that maximize symmetry for that protocol and observing how much symmetry breaking the protocol is forced to achieve in order to solve the problem. To allow for such symmetry maximizing schedules we consider parallel schedulers that in every step may select a whole collection of pairs of nodes (up to a perfect matching) to interact and not just a single pair. Based on these definitions of symmetric computation, we (i) give a partial characterization of the set of predicates on input assignments that can be stably computed with maximum symmetry, i.e., Θ (Nmin) , where Nmin is the minimum multiplicity of a state in the initial configuration, and (ii) we turn our attention to the remaining predicates (that have some essentially different properties) and prove a strong impossibility result for the parity predicate: the inherent symmetry of any protocol that stably computes it is upper bounded by a constant that depends on the size of the protocol. The latter immediately generalizes to a subset of the predicates that are not closed under doubling.