Locally Checkable Labelings with Small Messages

Balliu, Alkida; Censor-Hillel, Keren; Maus, Yannic; Olivetti, Dennis; Suomela, Jukka

doi:10.4230/lipics.disc.2021.8

Cited by 4 publications

(19 citation statements)

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“…It is known that these are the only possible time complexities in trees [7,11,[17][18][19]28]. In [9], it has been shown that the same results hold also in a more restrictive model of distributed computing, called CONGEST model, and that for any given problem, its complexities in the LOCAL and in the CONGEST model, on trees, are actually the same.…”

Section: Related Workmentioning

confidence: 83%

“…We give practical, efficient algorithms that automatically determine the distributed round complexity of a given locally checkable graph problem in rooted or unrooted regular trees, for both LOCAL and CONGEST models (see Section 3 for the precise definitions). In these cases, the distributed round complexity of any locally checkable problem is known to fall in one of the classes shown in Figure 1 [7,9,11,[17][18][19]28]. Our algorithms are able to distinguish between all higher complexity classes from Θ(log 𝑛) to Θ(𝑛).…”

Section: Introductionmentioning

confidence: 99%

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Efficient Classification of Local Problems in Regular Trees

Balliu¹,

Brandt²,

Chang³

et al. 2022

Preprint

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We give practical, efficient algorithms that automatically determine the distributed round complexity of a given locally checkable graph problem, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees.The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in 𝑂 (log 𝑛) rounds. If not, it is known that the complexity has to be Θ(𝑛 1/𝑘 ) for some 𝑘 = 1, 2, . . . , and in this case the algorithms also output the right value of the exponent 𝑘.In rooted trees in the 𝑂 (log 𝑛) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the 𝑂 (log 𝑛) region remains an open question.

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Section: Related Workmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Efficient Classification of Local Problems in Regular Trees

Balliu¹,

Brandt²,

Chang³

et al. 2022

Preprint

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“…Even more recently, a paper [10] showed that the asymptotic complexity of any LCL problem on trees is the same in the LOCAL and the CONGEST model. This implies that the complexity landscape of LCLs on trees in CONGEST is precisely the same as in LOCAL, and in particular extends our gap between ωp1q and oplog ˚nq to the CONGEST model.…”

Section: Main Contribution: Finishing the Classification Of Lcls On T...mentioning

confidence: 99%

The Landscape of Distributed Complexities on Trees and Beyond

Grunau¹,

Rozhoň²,

Brandt³

2022

Preprint

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We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below log ˚n. Our contribution is threefold: * The author ordering was randomized using https://www.aeaweb.org/journals/ policies/random-author-order/generator. It is requested that citations of this work list the authors separated by \textcircled{r} instead of commas: Grunau r ○ Rozhoň r ○ Brandt.

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“…For the fine-grained structure inside the n (1) class we refer to the prior work [6,14,16]; while these papers study the case of unrooted trees, we note that the orientation can be encoded as a locally checkable input, and the results are also applicable here. It follows that there are only classes O(1), (log * n), (log n), and (n 1/k ) for k = 1, 2, .…”

Section: Road Mapmentioning

confidence: 99%

Locally checkable problems in rooted trees

et al. 2022

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Consider any locally checkable labeling problem $$\Pi $$ Π in rooted regular trees: there is a finite set of labels $$\Sigma $$ Σ , and for each label $$x \in \Sigma $$ 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 $$\Pi $$ Π falls in one of the following classes: it is O(1), $$\Theta (\log ^* n)$$ Θ ( log ∗ n ) , $$\Theta (\log n)$$ Θ ( log n ) , or $$n^{\Theta (1)}$$ 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 $$\mathsf {LOCAL}$$ LOCAL , randomized $$\mathsf {LOCAL}$$ LOCAL , deterministic $$\mathsf {CONGEST}$$ CONGEST , and randomized $$\mathsf {CONGEST}$$ CONGEST model. In particular, we show that randomness does not help in this setting, and the complexity class $$\Theta (\log \log n)$$ 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 $$\Pi $$ Π , i.e., whether $$\Pi $$ Π takes O(1), $$\Theta (\log ^* n)$$ Θ ( log ∗ n ) , $$\Theta (\log n)$$ Θ ( log n ) , or $$n^{\Theta (1)}$$ n Θ ( 1 ) rounds. While the algorithm may take exponential time in the size of the description of $$\Pi $$ 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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Locally Checkable Labelings with Small Messages

Cited by 4 publications

References 0 publications

Efficient Classification of Local Problems in Regular Trees

Efficient Classification of Local Problems in Regular Trees

The Landscape of Distributed Complexities on Trees and Beyond

Locally checkable problems in rooted trees

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