The Landscape of Distributed Complexities on Trees and Beyond

Abstract

We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below log ˚n. Our contribution is threefold: * The author ordering was randomized using https://www.aeaweb.org/journals/ policies/random-author-order/generator. It is requested that citations of this work list the authors separated by \textcircled{r} instead of commas: Grunau r ○ Rozhoň r ○ Brandt.

Efficient Classification of Local Problems in Regular Trees

“…Moreover, there are problems with complexity Θ(𝑛 1/𝑘 ), for any natural number 𝑘 ≥ 1 [19]. It is known that these are the only possible time complexities in trees [7,11,[17][18][19]28]. In [9], it has been shown that the same results hold also in a more restrictive model of distributed computing, called CONGEST model, and that for any given problem, its complexities in the LOCAL and in the CONGEST model, on trees, are actually the same.…”

“…We give practical, efficient algorithms that automatically determine the distributed round complexity of a given locally checkable graph problem in rooted or unrooted regular trees, for both LOCAL and CONGEST models (see Section 3 for the precise definitions). In these cases, the distributed round complexity of any locally checkable problem is known to fall in one of the classes shown in Figure 1 [7,9,11,[17][18][19]28]. Our algorithms are able to distinguish between all higher complexity classes from Θ(log 𝑛) to Θ(𝑛).…”