Brief Announcement: On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

Abstract

We study the class of networks which can be created in polylogarithmic parallel time by network constructors [O. Michail, P. Spirakis, Distributed Computing 29(3), 207–237 (2016)]. We prove that the class of trees where each node has any $k \ge 2$ children can be constructed in $O(\log {n})$ parallel time with high probability. We show that constructing networks which are k-regular is $\varOmega (n)$ 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.

This reports on joint work with O. Michail and P. Spirakis which has appeared in [3].