Deterministic Distributed Algorithms and Measurable Combinatorics on $Δ$-Regular Forests

TL;DR

This work builds a rigorous bridge between deterministic LOCAL algorithms in distributed networks and measurable/descriptive combinatorics on Δ-regular forests. It proves two fundamental equalities: LOCAL(O(log^* n)) coincides with CONTINUOUS, and LOCAL(O(log n)) coincides with BAIRE, by translating finite fast-local solutions into continuous/BAIRE solutions and vice versa using carefully constructed continuous forests, hyperaperiodic elements, and LLL techniques. Central to the BAIRE connection is the ℓ-full combinatorial condition, which the authors show is both necessary and sufficient, enabling a TOAST/rake-and-compress framework to realize logarithmic-time local algorithms from measurable decompositions. The paper extends prior results known for Cayley graphs to Δ-regular forests and provides a decidability angle for membership in these complexity classes, thus advancing a unified theory linking locality in distributed computing with descriptive set theory on graphs.

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.

Figures (4)

• Figure 1: Complexity classes on $\Delta$-regular forests considered in the areas of distributed computing and measurable combinatorics. The left part shows deterministic complexity classes of distributed computing. The black arrows with two endpoints represent equalities between the complexity classes. The black arrows with just one endpoint stand for proper inclusions between the classes. We highlight two phenomena. First, we have $\mathsf{LOCAL}(O(\log^* n)) = \mathsf{CONTINUOUS}$, that is, there is a natural class of basic symmetry breaking problems in both areas. Second, we have $\mathsf{LOCAL}(O(\log n)) = \mathsf{BAIRE}$, that is, the most powerful class in measurable combinatorics exactly matches the power of finite local construction with local complexity $O(\log n)$.
• Figure 2: A $3$-regular tree on 15 vertices. Every vertex has $3$ half-edges but not all half-edges lead to a different vertex.
• Figure 3: A decomposition satisfying the requirements in \ref{['lem-rake-and-compress-modified']}.
• Figure 4: The gluing and labeling procedures for $\Delta=3$

Theorems & Definitions (140)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Half-edge
• Definition 2.2: $\Delta$-regular forest
• Definition 2.3: LCL problems on $\Delta$-regular forests
• Remark 2.4
• Definition 2.5: Local algorithm
• Definition 2.6: Local complexity
• Theorem 2.7: Classification of deterministic local complexities of LCL problems on $\Delta$-regular forests naorstockmeyerchang_kopelowitz_pettie2019exp_separationchang_pettie2019time_hierarchy_trees_rand_speedupchang2020n1k_speedupsballiu2020almost_global_problemsballiu2019hardness_homogeneousGrunauRozhonBrandt2022complexity_o(logstar(n))
• Definition 2.8: Continuous forest