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Locally Checkable Problems in Rooted Trees

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posted on 2023-11-29, 18:06 authored by Alkida Balliu, Sebastian BrandtSebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studený, Jukka Suomela, Aleksandr Tereshchenk
Consider any locally checkable labeling problem Π in rooted regular trees: there is a finite set of labels Σ, and for each label x ∈ Σ we specify what are permitted label combinations of the children for an internal node of label x (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set. We show that the distributed computational complexity of any such problem Π falls in one of the following classes: it is O(1), Θ(log^* n), Θ(log n), or n^{Θ(1)} rounds in trees with n nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic LOCAL, randomized LOCAL, deterministic CONGEST, and randomized CONGEST model. In particular, we show that randomness does not help in this setting, and the complexity class Θ(log log n) does not exist (while it does exist in the broader setting of general trees). We also show how to systematically determine the complexity class of any such problem Π, i.e., whether Π takes O(1), Θ(log^* n), Θ(log n), or n^{Θ(1)} rounds. While the algorithm may take exponential time in the size of the description of Π, it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.


Preferred Citation

Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studený, Jukka Suomela and Aleksandr Tereshchenk. Locally Checkable Problems in Rooted Trees. In: Distributed Computing. 2022.

Primary Research Area

  • Algorithmic Foundations and Cryptography

Legacy Posted Date



Distributed Computing

Open Access Type

  • Gold

Sub Type

  • Article


@article{cispa_all_3763, title = "Locally Checkable Problems in Rooted Trees", author = "Balliu, Alkida and Brandt, Sebastian and Chang, Yi-Jun and Olivetti, Dennis and Studený, Jan and Suomela, Jukka and Tereshchenk, Aleksandr", journal="{Distributed Computing}", year="2022", }

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