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Local problems in trees across a wide range of distributed models

Anubhav Dhar, Eli Kujawa, Henrik Lievonen, Augusto Modanese, Mikail Muftuoglu, Jan Studený, Jukka Suomela

TL;DR

This work investigates the locality of LCL problems across a broad spectrum of distributed models on trees, exploiting a sublogarithmic to super-logarithmic classification to show that online-LOCAL and LOCAL are asymptotically equivalent for many LCL families. The authors develop depth-based classifications and log-star certificates to translate o(log n) solvability into O(log^* n) (thus O(1)) solvability in rooted regular trees, and extend such results to unrooted regular trees in the super-logarithmic regime, while establishing speedup mechanisms that transfer sublinear locality improvements to sqrt(n) locality in general trees. They introduce a framework combining classification and speedup techniques, including new depth notions and input-instance constructions, to achieve near-complete classifications for rooted regular trees and full classifications for the super-logarithmic region in unrooted regular trees, along with a general gap result showing that global problems remain global across randomized online-LOCAL in general trees. The findings significantly clarify how much quantum or other model advantages can be realized in tree-structured networks and provide practical implications for designing locality-efficient distributed algorithms in hierarchical networks. The results also lay groundwork for precise model-to-model transfers of locality bounds across a wide family of distributed computing paradigms.

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.

Local problems in trees across a wide range of distributed models

TL;DR

This work investigates the locality of LCL problems across a broad spectrum of distributed models on trees, exploiting a sublogarithmic to super-logarithmic classification to show that online-LOCAL and LOCAL are asymptotically equivalent for many LCL families. The authors develop depth-based classifications and log-star certificates to translate o(log n) solvability into O(log^* n) (thus O(1)) solvability in rooted regular trees, and extend such results to unrooted regular trees in the super-logarithmic regime, while establishing speedup mechanisms that transfer sublinear locality improvements to sqrt(n) locality in general trees. They introduce a framework combining classification and speedup techniques, including new depth notions and input-instance constructions, to achieve near-complete classifications for rooted regular trees and full classifications for the super-logarithmic region in unrooted regular trees, along with a general gap result showing that global problems remain global across randomized online-LOCAL in general trees. The findings significantly clarify how much quantum or other model advantages can be realized in tree-structured networks and provide practical implications for designing locality-efficient distributed algorithms in hierarchical networks. The results also lay groundwork for precise model-to-model transfers of locality bounds across a wide family of distributed computing paradigms.

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.
Paper Structure (13 sections, 7 theorems, 3 figures)

This paper contains 13 sections, 7 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Prior Work on LCL Problems
  4. Contributions
  5. Roadmap
  6. Key Ideas, Technical Overview, and Comparison with Prior Work
  7. Classification Arguments for Regular Trees
  8. Speedup Arguments for General Trees
  9. Preliminaries
  10. Locally Checkable Labeling Problems
  11. Models of Distributed Computing
  12. Sub-logarithmic Gap for LCLs in Rooted Regular Trees
  13. Construction of Input Instances

Key Result

Theorem 1

Let $\Pi$ be an LCL problem on rooted regular trees. If $\Pi$ can be solved with $o(\mathop{\mathrm{log}}\nolimits n)$ locality in , then it can be solved in with $O(\mathop{\mathrm{log}}\nolimits^* n)$ locality. Consequently, $\Pi$ can be solved with $O(1)$ locality in , and thus also with the same

Figures (3)

  • Figure 1: Landscape of models, based on akbari24_online_arxiv. In this work we show that for many families of LCL problems, the two extreme models--- and ---are equally strong, and hence the same holds for all intermediate models in this diagram.
  • Figure 2: Landscape of LCL problems in trees after this work---compare with \ref{['fig:landscape-lcls-before']} that shows the state of the art before this work.
  • Figure 3: Landscape of LCL problems in trees before this work---compare with \ref{['fig:landscape-lcls-after']} to see the impact of our new contributions.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Definition 7: LCL problem in general graphs
  • Definition 8: LCL problem on regular rooted trees
  • Definition 9: LCL problem on regular unrooted trees
  • Definition 10: Deterministic model
  • ...and 5 more