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

Jan Grebík, Václav Rozhoň|arXiv (Cornell University)|Mar 15, 2021

Data Management and Algorithms5 citations

TL;DR

This paper establishes a unified framework connecting distributed algorithms, finitary factors of i.i.d. processes, and descriptive combinatorics by studying local problems on d-dimensional grids. It proves time hierarchy theorems in the finitary factors setting, resolves open questions about ffiid complexity, and shows that uniform local complexity classes like UL0CAL(O(log* 1/ε)) are strictly contained within finitely dependent processes, advancing the theory of locality across these fields.

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+] .

Motivation & Objective

• To unify three perspectives—distributed algorithms, finitary factors of i.i.d. processes, and descriptive combinatorics—under a common theory of locality.
• To resolve open questions on the complexity of finitary factor solutions, particularly those posed by Holroyd, Schramm, and Wilson (2017) and Brandt et al. (2017).
• To establish formal connections between distributed computing and finitary factor constructions, especially through the TOAST and RTOAST frameworks.
• To analyze uniform local complexities in distributed and centralized models, showing that average-case complexity can be constant even when worst-case complexity is logarithmic.

Proposed method

• Adapts techniques from distributed algorithms—particularly round elimination and locality hierarchy results—to analyze finitary factor solutions on infinite grids.
• Introduces the TOAST and RTOAST frameworks as randomized constructions that generate solutions with finite dependence, enabling complexity analysis.
• Uses the concept of uniform local complexity to analyze algorithms that do not depend on global knowledge like n, linking to average-case and amortized complexity.
• Applies results from descriptive combinatorics, such as Borel and measurable solutions, to characterize the complexity of local problems on Z^d.
• Employs the notion of finitely dependent processes to show that certain problems (e.g., 3-coloring) cannot be in FINDEP, implying strict hierarchies.
• Leverages the equivalence between distributed algorithms and finitary factors to transfer complexity results across domains, especially via the t-hop neighborhood mapping.

Experimental results

Research questions

• RQ1 Is there a formal connection between distributed algorithms and finitary factors of i.i.d. processes, as conjectured by Brandt et al. (2017)?
• RQ2 Can time hierarchy theorems be established in the finitary factor setting, analogous to those in the LOCAL model?
• RQ3 Is the class of finitely dependent processes strictly contained within the class of TOAST processes, or do they coincide?
• RQ4 What is the uniform local complexity of 3-coloring on d-dimensional grids, and can it be improved beyond (1/ε) · 2^{O(√log(1/ε))}?
• RQ5 Does the uniform complexity of local problems on grids exhibit intermediate growth rates, such as Θ((1/ε)^{1/r}) for non-integer r?

Key findings

• The paper proves a time hierarchy theorem in the finitary factor setting, showing that for every t, there exists a local problem solvable in t rounds but not in o(t) rounds.
• It establishes that UL0CAL(O(log* 1/ε)) ⊆ FINDEP ⊆ UL0CAL((1/ε)^{1+o(1)}), demonstrating a strict hierarchy within finitary factor solutions.
• It resolves an open question by showing that 3-coloring on d-dimensional grids (for d > 1) is not in FINDEP, confirming a result of Holroyd, Schramm, and Wilson (2017).
• The uniform complexity of 3-coloring on grids is shown to be (1/ε) · 2^{O(√log(1/ε))}, and this bound may be improvable to polylogarithmic in 1/ε.
• The paper demonstrates that average-case complexity in distributed and LCA models can be constant even when worst-case complexity is logarithmic, due to uniformity in algorithm design.
• It shows that the amortized query complexity in the LCA model is constant for problems like Δ+1 coloring and LLL, using uniform versions of Linial’s algorithm.