Local problems in trees across a wide range of distributed models

TL;DR

This paper investigates the locality of distributed problems in trees across various models, showing that many problems have equivalent complexity in deterministic LOCAL and randomized online-LOCAL models, with no quantum advantage.

Contribution

It provides a near-complete classification of LCL problems in trees, demonstrating equivalence of complexity in several models and identifying cases where global problems remain global.

Findings

Many LCL problems in trees have the same locality in deterministic LOCAL and randomized online-LOCAL models.

The classification covers rooted and unrooted regular trees, including super-logarithmic regions.

Global problems in deterministic LOCAL remain global in the online-LOCAL model.

Abstract

The randomized online-LOCAL model captures a number of models of computing; it is at least as strong as all of these models: - the classical LOCAL model of distributed graph algorithms, - the quantum version of the LOCAL model, - finitely dependent distributions [e.g. Holroyd 2016], - any model that does not violate physical causality [Gavoille, Kosowski, Markiewicz, DISC 2009], - the SLOCAL model [Ghaffari, Kuhn, Maus, STOC 2017], and - the dynamic-LOCAL and online-LOCAL models [Akbari et al., ICALP 2023]. In general, the online-LOCAL model can be much stronger than the LOCAL model. For example, there are locally checkable labeling problems (LCLs) that can be solved with logarithmic locality in the online-LOCAL model but that require polynomial locality in the LOCAL model. However, in this work we show that in trees, many classes of LCL problems have the same locality in deterministic LOCAL and randomized online-LOCAL (and as a corollary across all the above-mentioned models). In particular, these classes of problems do not admit any distributed quantum advantage. We present a near-complete classification for the case of rooted regular trees. We also fully classify the super-logarithmic region in unrooted regular trees. Finally, we show that in general trees (rooted or unrooted, possibly irregular, possibly with input labels) problems that are global in deterministic LOCAL remain global also in the randomized online-LOCAL model.