Document Open Access Logo

Efficient Classification of Locally Checkable Problems in Regular Trees

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

PDF
Thumbnail PDF

File

LIPIcs.DISC.2022.8.pdf
  • Filesize: 0.7 MB
  • 19 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

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

Metrics

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

Related Versions

References

  1. Yehuda Afek, Shay Kutten, and Moti Yung. The local detection paradigm and its application to self-stabilization. Theor. Comput. Sci., 186(1-2):199-229, 1997. URL: https://doi.org/10.1016/S0304-3975(96)00286-1.
  2. Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. The distributed complexity of locally checkable problems on paths is decidable. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 262-271. ACM Press, 2019. URL: https://doi.org/10.1145/3293611.3331606.
  3. Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studenỳ, and Jukka Suomela. Efficient classification of local problems in regular trees. arXiv preprint, 2022. URL: http://arxiv.org/abs/2202.08544.
  4. Alkida Balliu, Sebastian Brandt, Yuval Efron, Juho Hirvonen, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Classification of distributed binary labeling problems. In Proc. 34th International Symposium on Distributed Computing (DISC 2020), volume 179 of LIPIcs, pages 17:1-17:17. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.17.
  5. Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower bounds for maximal matchings and maximal independent sets. In Proc. 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2019), pages 481-497. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00037.
  6. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Improved distributed lower bounds for MIS and bounded (out-)degree dominating sets in trees. In PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26-30, 2021, pages 283-293. ACM, 2021. URL: https://doi.org/10.1145/3465084.3467901.
  7. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Deterministic δ-coloring plays hide-and-seek. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022). ACM, 2022. Google Scholar
  8. Alkida Balliu, Sebastian Brandt, and Dennis Olivetti. Distributed lower bounds for ruling sets. In Proc. 61st IEEE Symp. on Foundations of Computer Science (FOCS), pages 365-376, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00042.
  9. Alkida Balliu, Sebastian Brandt, Dennis Olivetti, Jan Studený, Jukka Suomela, and Aleksandr Tereshchenko. Locally checkable problems in rooted trees. In Proc. 40th ACM Symposium on Principles of Distributed Computing (PODC 2021), pages 263-272. ACM Press, 2021. URL: https://doi.org/10.1145/3465084.3467934.
  10. 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.
  11. Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. Almost global problems in the LOCAL model. Distributed Computing, 34:259-281, 2021. URL: https://doi.org/10.1007/s00446-020-00375-2.
  12. 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, volume 209 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.8.
  13. 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.
  14. 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.
  15. Sebastian Brandt. An automatic speedup theorem for distributed problems. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 379-388. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331611.
  16. Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky. Local problems on trees from the perspectives of distributed algorithms, finitary factors, and descriptive combinatorics. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, volume 215 of LIPIcs, pages 29:1-29:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.29.
  17. Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, and Jara Uitto. A lower bound for the distributed Lovász local lemma. In Proc. 48th ACM Symposium on Theory of Computing (STOC 2016), pages 479-488. ACM Press, 2016. URL: https://doi.org/10.1145/2897518.2897570.
  18. Sebastian Brandt, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Patric R. J. Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. LCL problems on grids. In Proc. 36th ACM Symposium on Principles of Distributed Computing (PODC 2017), pages 101-110. ACM Press, 2017. URL: https://doi.org/10.1145/3087801.3087833.
  19. Sebastian Brandt and Dennis Olivetti. Truly tight-in-Δ bounds for bipartite maximal matching and variants. In Proc. 39th ACM Symp. on Principles of Distributed Computing (PODC), pages 69-78, 2020. URL: https://doi.org/10.1145/3382734.3405745.
  20. 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.
  21. 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.
  22. 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.
  23. Yi-Jun Chang, Jan Studený, and Jukka Suomela. Distributed graph problems through an automata-theoretic lens. In Proc. 28th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2021), LNCS. Springer, 2021. URL: http://arxiv.org/abs/2002.07659.
  24. Kai-Min Chung, Seth Pettie, and Hsin-Hao Su. Distributed algorithms for the lovász local lemma and graph coloring. Distributed Comput., 30(4):261-280, 2017. URL: https://doi.org/10.1007/s00446-016-0287-6.
  25. Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control., 70(1):32-53, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80023-7.
  26. Manuela Fischer. Improved deterministic distributed matching via rounding. In Proceedings of the 31st International Symposium on Distributed Computing (DISC 2017), pages 17:1-17:15, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.17.
  27. 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. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.18.
  28. Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Proc. 57th IEEE Symp. on Foundations of Computer Science (FOCS), pages 625-634, 2016. URL: https://doi.org/10.1109/FOCS.2016.73.
  29. Mohsen Ghaffari and Fabian Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In Proc. 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2021), 2021. Google Scholar
  30. Mohsen Ghaffari and Hsin-Hao Su. Distributed Degree Splitting, Edge Coloring, and Orientations. In Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 2505-2523. Society for Industrial and Applied Mathematics, 2017. URL: https://doi.org/10.1137/1.9781611974782.166.
  31. Christoph Grunau, Václav Rozhoň, and Sebastian Brandt. The landscape of distributed complexities on trees, 2021. URL: http://arxiv.org/abs/2202.04724.
  32. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  33. Michael Luby. A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 15(4):1036-1053, 1986. URL: https://doi.org/10.1137/0215074.
  34. Yannic Maus and Tigran Tonoyan. Local conflict coloring revisited: Linial for lists. In 34th International Symposium on Distributed Computing, DISC 2020, volume 179 of LIPIcs, pages 16:1-16:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.16.
  35. Gary L. Miller and John H. Reif. Parallel tree contraction and its application. In Proc. 26th Annual Symposium on Foundations of Computer Science (FOCS 1985), pages 478-489. IEEE, 1985. URL: https://doi.org/10.1109/SFCS.1985.43.
  36. Moni Naor. A lower bound on probabilistic algorithms for distributive ring coloring. SIAM J. Discret. Math., 4(3):409-412, 1991. URL: https://doi.org/10.1137/0404036.
  37. 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.
  38. Dennis Olivetti. Round Eliminator: a tool for automatic speedup simulation, 2020. URL: https://github.com/olidennis/round-eliminator.
  39. Alessandro Panconesi and Romeo Rizzi. Some simple distributed algorithms for sparse networks. Distributed Computing, 14(2):97-100, 2001. URL: https://doi.org/10.1007/PL00008932.
  40. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), pages 350-363. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384298.
  41. Jan Studený and Aleksandr Tereshchenko. Rooted tree classifier, 2021. URL: https://github.com/jendas1/rooted-tree-classifier.
  42. J. J. Sylvester. On subvariants, i.e. semi-invariants to binary quantics of an unlimited order. American Journal of Mathematics, 5(1):79-136, 1882. URL: http://www.jstor.org/stable/2369536.
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 Imprint Privacy Contact