Global Versus Local Computations: Fast Computing with Identifiers

TL;DR

This paper explores the computational power of community protocols with identifiers in polylogarithmic parallel time, establishing what global and local predicates can be computed efficiently and identifying inherent limitations.

Contribution

It introduces the class CPPL for protocols with polylogarithmic expected interactions and characterizes its boundaries and limitations.

Findings

Protocols for size computation are feasible within CPPL.

Certain local computations, like Linearly Local languages, are impossible within the class.

The class CPPL captures the power and limits of identifier-based local computations in polylogarithmic time.

Abstract

This paper studies what can be computed by using probabilistic local interactions with agents with a very restricted power in polylogarithmic parallel time. It is known that if agents are only finite state (corresponding to the Population Protocol model by Angluin et al.), then only semilinear predicates over the global input can be computed. In fact, if the population starts with a unique leader, these predicates can even be computed in a polylogarithmic parallel time. If identifiers are added (corresponding to the Community Protocol model by Guerraoui and Ruppert), then more global predicates over the input multiset can be computed. Local predicates over the input sorted according to the identifiers can also be computed, as long as the identifiers are ordered. The time of some of those predicates might require exponential parallel time. In this paper, we consider what can be computed with Community Protocol in a polylogarithmic number of parallel interactions. We introduce the class CPPL corresponding to protocols that use $O(n\log^k n)$, for some k, expected interactions to compute their predicates, or equivalently a polylogarithmic number of parallel expected interactions. We provide some computable protocols, some boundaries of the class, using the fact that the population can compute its size. We also prove two impossibility results providing some arguments showing that local computations are no longer easy: the population does not have the time to compare a linear number of consecutive identifiers. The Linearly Local languages, such that the rational language $(ab)^*$, are not computable.