Locally checkable problems in rooted trees

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 e...

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Vydáno v:Distributed computing Ročník 36; číslo 3; s. 277 - 311
Hlavní autoři: Balliu, Alkida, Brandt, Sebastian, Chang, Yi-Jun, Olivetti, Dennis, Studený, Jan, Suomela, Jukka, Tereshchenko, Aleksandr
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023
Springer Nature B.V
Témata:
Algorithms
Automation
Children & youth
Complexity
Computer Communication Networks
Computer Hardware
Computer networks
Computer Science
Computer Systems Organization and Communication Networks
Distributed processing
Graph coloring
Labels
Nodes
Software Engineering/Programming and Operating Systems
Theory of Computation
ISSN:0178-2770, 1432-0452
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Shrnutí: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.
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ISSN:0178-2770
1432-0452
DOI:10.1007/s00446-022-00435-9