On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

Matthew Connor,Paul Spirakis,Othon Michail

doi:10.3390/info12060254

Matthew Connor, Paul Spirakis + Show 1 more

Open Access

https://doi.org/10.3390/info12060254

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Journal: Information	Publication Date: Jun 19, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: University of Liverpool, University of Patras

Abstract

We study the class of networks, which can be created in polylogarithmic parallel time by network constructors: groups of anonymous agents that interact randomly under a uniform random scheduler with the ability to form connections between each other. Starting from an empty network, the goal is to construct a stable network that belongs to a given family. We prove that the class of trees where each node has any k≥2 children can be constructed in O(logn) parallel time with high probability. We show that constructing networks that are k-regular is Ω(n) time, but a minimal relaxation to (l,k)-regular networks, where l=k−1, can be constructed in polylogarithmic parallel time for any fixed k, where k>2. We further demonstrate that when the finite-state assumption is relaxed and k is allowed to grow with n, then k=loglogn acts as a threshold above which network construction is, again, polynomial time. We use this to provide a partial characterisation of the class of polylogarithmic time network constructors.

Highlights

Information 2021, 12, 254. https://Passively dynamic networks are an important type of dynamic network in which the network dynamics are external to the algorithm and are a property of the environment in which a given system operates
One line of research assumes the scheduler to be fair in the sense that it can forever conceal potentially reachable configurations of the system. This sub-type of passively dynamic networks are known as population protocols and were introduced in the seminal paper of Angluin et al [1]
We investigate which families of networks can be stably constructed by a distributed computing system in polylogarithmic parallel time

Summary

Introduction

Information 2021, 12, 254. https://Passively dynamic networks are an important type of dynamic network in which the network dynamics are external to the algorithm and are a property of the environment in which a given system operates. In terms of modelling such systems, the network dynamics are usually assumed to be controlled by an adversary scheduler who has exclusive control over the interaction or communication sequence among the computational entities. One line of research assumes the scheduler to be fair in the sense that it can forever conceal potentially reachable configurations of the system. This sub-type of passively dynamic networks are known as population protocols and were introduced in the seminal paper of Angluin et al [1]. The population protocols literature considers extremely weak entities and the goal is to reveal the computational possibilities and limitations under such a challenging interaction scheme.

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