On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

TL;DR

This paper investigates how certain stable networks can be constructed efficiently in polylogarithmic parallel time using anonymous agents, identifying classes of networks that are feasible to build quickly and exploring the limits of such constructions.

Contribution

It introduces methods for constructing specific network families in polylogarithmic time and characterizes the boundaries of what can be efficiently built with distributed agents.

Findings

Trees with nodes having k >= 2 children can be constructed in O(log n) time.

k-regular networks require linear time to construct.

Relaxing the finite-state assumption allows constructing networks with k up to log log n in polylogarithmic time.

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 which 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(log n) parallel time with high probability. We show that constructing networks which are k-regular is Omega(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 = log log n 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.