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arXiv · 2007.00625

On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

Matthew Connor·Othon Michail·Paul Spirakis
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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.

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Matthew Connor, Othon Michail, Paul Spirakis. 2020-07-01. On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time. https://arxiv.org/abs/2007.00625

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