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On the Node-Averaged Complexity of Locally Checkable Problems on Trees

Authors Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Gustav Schmid

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Author Details

Alkida Balliu
  • Gran Sasso Science Institute, L'Aquila, Italy
Sebastian Brandt
  • Helmholtz Center for Information Security, Saarbrücken, Germany
Fabian Kuhn
  • University of Freiburg, Germany
Dennis Olivetti
  • Gran Sasso Science Institute, L'Aquila, Italy
Gustav Schmid
  • University of Freiburg, Germany

Funding

This work has been partially funded by the European Union - NextGenerationEU under the Italian Ministry of University and Research (MUR) National Innovation Ecosystem grant ECS00000041 - VITALITY – CUP: D13C21000430001, and by the German Research Foundation (DFG), Grant 491819048.

Cite AsGet BibTex

Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, and Gustav Schmid. On the Node-Averaged Complexity of Locally Checkable Problems on Trees. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.DISC.2023.7

Abstract

Over the past decade, a long line of research has investigated the distributed complexity landscape of locally checkable labeling (LCL) problems on bounded-degree graphs, culminating in an almost-complete classification on general graphs and a complete classification on trees. The latter states that, on bounded-degree trees, any LCL problem has deterministic worst-case time complexity O(1), Θ(log^* n), Θ(log n), or Θ(n^{1/k}) for some positive integer k, and all of those complexity classes are nonempty. Moreover, randomness helps only for (some) problems with deterministic worst-case complexity Θ(log n), and if randomness helps (asymptotically), then it helps exponentially. In this work, we study how many distributed rounds are needed on average per node in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic node-averaged complexity landscape for LCL problems. As our main result, we show that every problem with worst-case round complexity O(log n) has deterministic node-averaged complexity O(log^* n). We further establish bounds on the node-averaged complexity of problems with worst-case complexity Θ(n^{1/k}): we show that all these problems have node-averaged complexity Ω̃(n^{1 / (2^k - 1)}), and that this lower bound is tight for some problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • distributed graph algorithms
  • locally checkable labelings
  • node-averaged complexity
  • trees

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References

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