Deterministic Distributed Algorithms and Measurable Combinatorics on $\Delta$-Regular Forests

TL;DR

This paper establishes a precise correspondence between deterministic distributed algorithms and measurable combinatorics on regular forests, revealing that complexity classes in distributed computing match well-studied classes in measurable combinatorics.

Contribution

It proves that continuous solutions correspond to $O( ext{log}^* n)$ algorithms and Baire measurable solutions correspond to $O( ext{log} n)$ algorithms, linking distributed complexity with measurable combinatorics.

Findings

Continuous solutions correspond to $O( ext{log}^* n)$ algorithms.

Baire measurable solutions correspond to $O( ext{log} n)$ algorithms.

Membership in these classes is decidable.

Abstract

We investigate the connections between the fields of distributed computing and measurable combinatorics by considering complexity classes of locally checkable labeling problems on regular forests. We show that the most important deterministic complexity classes from the LOCAL model of distributed computing exactly coincide with well-studied classes in measurable combinatorics. Namely, first we show that a locally checkable labeling problem admits a continuous solution if and only if it can be solved by a deterministic local algorithm with complexity $O(\log^* n)$. Second, our main result states that, surprisingly, a locally checkable labeling problem admits a Baire measurable solution if and only if it can be solved by a local algorithm with complexity $O(\log n)$. These theorems suggest the existence of deeper connections between the two frameworks. Furthermore, the latter result relies on a complete combinatorial characterization of the classes in question, and as a by-product, it shows that membership in these classes is decidable.