Abstract
This paper studies what can be computed by using probabilistic local interactions between agents with a very restricted power in polylogarithmic parallel time.
If agents have a finite internal state (corresponding to the Population Protocol model by Angluin et al.), only semilinear predicates over the global input can be computed. If identifiers are added (corresponding to the Community Protocol model by Guerraoui and Ruppert), 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 \(O(n\log ^kn)\), for some k, expected interactions to compute their predicates, or equivalently a polylogarithmic number of parallel expected interactions.
We provide some computable protocols, including one computing the size of the population. We also explore the limits of this model of computation by 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.
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References
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Rabie, M. (2016). Global Versus Local Computations: Fast Computing with Identifiers (Short Paper). In: Bonakdarpour, B., Petit, F. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2016. Lecture Notes in Computer Science(), vol 10083. Springer, Cham. https://doi.org/10.1007/978-3-319-49259-9_24
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DOI: https://doi.org/10.1007/978-3-319-49259-9_24
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