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Efficient Classification of Locally Checkable Problems in Regular Trees

Authors Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studený, Jukka Suomela

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Alkida Balliu
  • Gran Sasso Science Institute, L'Aquila, Italy
Sebastian Brandt
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Yi-Jun Chang
  • National University of Singapore, Singapore
Dennis Olivetti
  • Gran Sasso Science Institute, L'Aquila, Italy
Jan Studený
  • Aalto University, Espoo, Finland
Jukka Suomela
  • Aalto University, Espoo, Finland

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Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studený, and Jukka Suomela. Efficient Classification of Locally Checkable Problems in Regular Trees. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.DISC.2022.8

Abstract

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [Θ(log n), Θ(n)] region, 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • locally checkable labeling
  • locality
  • distributed computational complexity

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