Document Open Access Logo

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

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

PDF
Thumbnail PDF

File

LIPIcs.DISC.2023.7.pdf
  • Filesize: 0.86 MB
  • 21 pages

Document Identifiers

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 As Get 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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, and Gustav Schmid. On the node-averaged complexity of locally checkable problems on trees. CoRR, abs/2308.04251, 2023. URL: https://doi.org/10.48550/arXiv.2308.04251.
  2. Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. How much does randomness help with locally checkable problems? In Proc. 39th ACM Symposium on Principles of Distributed Computing (PODC 2020), pages 299-308. ACM Press, 2020. URL: https://doi.org/10.1145/3382734.3405715.
  3. Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. Almost global problems in the LOCAL model. Distributed Comput., 34(4):259-281, 2021. URL: https://doi.org/10.1007/s00446-020-00375-2.
  4. Alkida Balliu, Keren Censor-Hillel, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Locally checkable labelings with small messages. In 35th International Symposium on Distributed Computing, DISC 2021, pages 8:1-8:18, 2021. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.8.
  5. Alkida Balliu, Mohsen Ghaffari, Fabian Kuhn, and Dennis Olivetti. Node and edge averaged complexities of local graph problems. In Proc. 41st ACM Symp. on Principles of Distributed Computing (PODC), pages 4-14, 2022. Google Scholar
  6. Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, and Jukka Suomela. New classes of distributed time complexity. In Proc. 50th ACM Symposium on Theory of Computing (STOC 2018), pages 1307-1318. ACM Press, 2018. URL: https://doi.org/10.1145/3188745.3188860.
  7. Alkida Balliu, Juho Hirvonen, Dennis Olivetti, and Jukka Suomela. Hardness of minimal symmetry breaking in distributed computing. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 369-378. ACM Press, 2019. URL: https://doi.org/10.1145/3293611.3331605.
  8. Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (Δ+1)-coloring in linear (in Δ) time. SIAM J. on Computing, 43(1):72-95, 2015. Google Scholar
  9. Leonid Barenboim and Yaniv Tzur. Distributed symmetry-breaking with improved vertex-averaged complexity. In Proc. 20th Int. Conf. on Distributed Computing and Networking (ICDCN), pages 31-40, 2019. Google Scholar
  10. Yi-Jun Chang. The complexity landscape of distributed locally checkable problems on trees. In Proc. 34th International Symposium on Distributed Computing (DISC 2020), volume 179 of LIPIcs, pages 18:1-18:17. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.18.
  11. Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. Distributed edge coloring and a special case of the constructive lovász local lemma. ACM Trans. Algorithms, 16(1):8:1-8:51, 2020. Google Scholar
  12. Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. An exponential separation between randomized and deterministic complexity in the LOCAL model. SIAM J. Comput., 48(1):122-143, 2019. URL: https://doi.org/10.1137/17M1117537.
  13. Yi-Jun Chang and Seth Pettie. A time hierarchy theorem for the LOCAL model. SIAM J. Comput., 48(1):33-69, 2019. URL: https://doi.org/10.1137/17M1157957.
  14. Soumyottam Chatterjee, Robert Gmyr, and Gopal Pandurangan. Sleeping is efficient: MIS in O(1)-rounds node-averaged awake complexity. In Proc. 39th ACM Symp. on on Principles of Distributed Computing (PODC), pages 99-108, 2020. Google Scholar
  15. Laurent Feuilloley. How long it takes for an ordinary node with an ordinary ID to output? In Proc. 24th Int. Coll. on Structural Information and Communication Complexity (SIROCCO), volume 10641, pages 263-282, 2017. Google Scholar
  16. Manuela Fischer and Mohsen Ghaffari. Sublogarithmic distributed algorithms for Lovász local lemma, and the complexity hierarchy. In Proc. 31st International Symposium on Distributed Computing (DISC 2017), volume 91 of LIPIcs, pages 18:1-18:16, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.18.
  17. Christoph Grunau, Václav Rozhon, and Sebastian Brandt. The landscape of distributed complexities on trees and beyond. In Proc. 41st ACM Symposium on Principles of Distributed Computing (PODC 2022), pages 37-47, 2022. URL: https://doi.org/10.1145/3519270.3538452.
  18. Öjvind Johansson. Simple distributed Delta+1-coloring of graphs. Inf. Process. Lett., 70(5):229-232, 1999. Google Scholar
  19. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. J. ACM, 63(2):17:1-17:44, 2016. Google Scholar
  20. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992. Google Scholar
  21. Gary L. Miller and John H. Reif. Parallel tree contraction and its application. In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21-23 October 1985, pages 478-489. IEEE Computer Society, 1985. URL: https://doi.org/10.1109/SFCS.1985.43.
  22. Moni Naor and Larry J. Stockmeyer. What can be computed locally? SIAM J. Comput., 24(6):1259-1277, 1995. URL: https://doi.org/10.1137/S0097539793254571.
  23. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proc. 52nd ACM Symp. on Theory of Computing (STOC), pages 350-363, 2020. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact