Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

TL;DR

This paper explores the deep connections among distributed algorithms, finitary factors of iid processes, and descriptive combinatorics, focusing on locally checkable labellings in grid graphs and establishing hierarchy theorems and resolving open questions.

Contribution

It introduces a unified framework linking three fields, proves time hierarchy theorems for finitary factors, and answers open questions about their relationships.

Findings

Proves time hierarchy theorems for finitary factors.

Answers open questions from prior research.

Establishes a formal connection between distributed algorithms and finitary factors.

Abstract

We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+] .