Document Open Access Logo

Distributed Delta-Coloring Under Bandwidth Limitations

Authors Magnús M. Halldórsson , Yannic Maus

PDF
Thumbnail PDF

File

LIPIcs.DISC.2024.31.pdf
  • Filesize: 0.94 MB
  • 22 pages

Document Identifiers

Author Details

Magnús M. Halldórsson
  • Reykjavik University, Iceland
Yannic Maus
  • TU Graz, Austria

Funding

  • Halldórsson, Magnús M.: Supported in part by Icelandic Research Fund grant 217965.
  • Maus, Yannic: Supported by the Austrian Science Fund (FWF), Grant P36280-N.

Acknowledgements

We thank Saku Peltonen for valuable input and discussions.

Cite As Get BibTex

Magnús M. Halldórsson and Yannic Maus. Distributed Delta-Coloring Under Bandwidth Limitations. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.DISC.2024.31

Abstract

We consider the problem of coloring graphs of maximum degree Δ with Δ colors in the distributed setting with limited bandwidth. Specifically, we give a polylog log n-round randomized algorithm in the CONGEST model. This is close to the lower bound of Ω(log log n) rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lovász local lemma (LLL) and the deg+1-list coloring problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Graph problems
  • Graph coloring
  • Lovász local lemma
  • LOCAL model
  • CONGEST model
  • Distributed computing

Metrics

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

Related Versions

References

  1. Sepehr Assadi, Pankaj Kumar, and Parth Mittal. Brooks’ theorem in graph streams: a single-pass semi-streaming algorithm for Δ-coloring. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 234-247, 2022. Google Scholar
  2. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Distributed Δ-coloring plays hide-and-seek. In Proc. 54th ACM Symp. on Theory of Computing (STOC), 2022. Google Scholar
  3. Alkida Balliu, Keren Censor-Hillel, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Locally checkable labelings with small messages. In Seth Gilbert, editor, 35th International Symposium on Distributed Computing, DISC 2021, October 4-8, 2021, Freiburg, Germany (Virtual Conference), 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.
  4. Étienne Bamas and Louis Esperet. Distributed coloring of graphs with an optimal number of colors. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), volume 126 of LIPIcs, pages 10:1-10:15. LZI, 2019. URL: https://doi.org/10.4230/LIPICS.STACS.2019.10.
  5. Philipp Bamberger, Fabian Kuhn, and Yannic Maus. Efficient deterministic distributed coloring with small bandwidth. In PODC '20: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, August 3-7, 2020, pages 243-252, 2020. URL: https://doi.org/10.1145/3382734.3404504.
  6. L. Barenboim. Deterministic (Δ + 1)-coloring in sublinear (in Δ) time in static, dynamic and faulty networks. In Proc. 34th ACM Symposium on Principles of Distributed Computing (PODC), pages 345-354, 2015. Google Scholar
  7. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, 2013. URL: https://doi.org/10.2200/S00520ED1V01Y201307DCT011.
  8. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. Journal of the ACM, 63(3):20:1-20:45, 2016. URL: https://doi.org/10.1145/2903137.
  9. 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, 2016. URL: https://doi.org/10.1145/2897518.2897570.
  10. R. Leonard Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37(2):194-197, 1941. URL: https://doi.org/10.1017/S030500410002168X.
  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, 2020. URL: https://doi.org/10.1145/3365004.
  12. Yi-Jun Chang, Wenzheng Li, and Seth Pettie. An optimal distributed (Δ+1)-coloring algorithm? In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 445-456, 2018. URL: https://doi.org/10.1145/3188745.3188964.
  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. Shiri Chechik and Doron Mukhtar. Optimal distributed coloring algorithms for planar graphs in the LOCAL model. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 787-804. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.49.
  15. 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.
  16. Sam Coy, Artur Czumaj, Peter Davies, and Gopinath Mishra. Parallel derandomization for coloring, 2024. Note: https://arxiv.org/abs/2302.04378v1 contains the Delta-coloring algorithm.
  17. Peter Davies. Improved distributed algorithms for the Lovász local lemma and edge coloring. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4273-4295. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH163.
  18. Michael Elkin, Seth Pettie, and Hsin-Hao Su. (2Δ-1)-edge-coloring is much easier than maximal matching in the distributed setting. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 355-370, 2015. URL: https://doi.org/10.1137/1.9781611973730.26.
  19. Paul Erdös and László Lovász. Problems and Results on 3-chromatic Hypergraphs and some Related Questions. Colloquia Mathematica Societatis János Bolyai, pages 609-627, 1974. Google Scholar
  20. Manuela Fischer. Improved deterministic distributed matching via rounding. In Andréa W. Richa, editor, 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria, volume 91 of LIPIcs, pages 17:1-17:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.17.
  21. Manuela Fischer and Mohsen Ghaffari. Sublogarithmic Distributed Algorithms for Lovász Local Lemma, and the Complexity Hierarchy. In the Proceedings of the 31st International Symposium on Distributed Computing (DISC), pages 18:1-18:16, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.18.
  22. Manuela Fischer, Magnús M. Halldórsson, and Yannic Maus. Fast distributed Brooks' theorem. In Proceedings of the SIAM-ACM Symposium on Discrete Algorithms (SODA), pages 2567-2588, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch98.
  23. Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 625-634, 2016. URL: https://doi.org/10.1109/FOCS.2016.73.
  24. Marc Fuchs and Fabian Kuhn. List defective colorings: Distributed algorithms and applications. In Rotem Oshman, editor, 37th International Symposium on Distributed Computing, DISC 2023, October 10-12, 2023, L'Aquila, Italy, volume 281 of LIPIcs, pages 22:1-22:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.DISC.2023.22.
  25. Mohsen Ghaffari. Distributed maximal independent set using small messages. In Proc. 30th Symp. on Discrete Algorithms (SODA), pages 805-820, 2019. URL: https://doi.org/10.1137/1.9781611975482.50.
  26. Mohsen Ghaffari, David G. Harris, and Fabian Kuhn. On derandomizing local distributed algorithms. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 662-673, 2018. URL: https://doi.org/10.1109/FOCS.2018.00069.
  27. Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, and Yannic Maus. Improved distributed delta-coloring. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, Egham, United Kingdom, July 23-27, 2018, pages 427-436, 2018. URL: https://dl.acm.org/citation.cfm?id=3212764.
  28. Mohsen Ghaffari and Fabian Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 1009-1020, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00101.
  29. Magnús M. Halldórsson, Fabian Kuhn, Yannic Maus, and Tigran Tonoyan. Efficient randomized distributed coloring in CONGEST. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 1180-1193, 2021. Full version at CoRR abs/2105.04700. URL: https://doi.org/10.1145/3406325.3451089.
  30. Magnús M. Halldórsson, Fabian Kuhn, Alexandre Nolin, and Tigran Tonoyan. Near-optimal distributed degree+1 coloring. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 450-463. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520023.
  31. Magnús M. Halldórsson, Yannic Maus, and Alexandre Nolin. Fast distributed vertex splitting with applications. In Christian Scheideler, editor, 36th International Symposium on Distributed Computing, DISC 2022, October 25-27, 2022, Augusta, Georgia, USA, volume 246 of LIPIcs, pages 26:1-26:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.DISC.2022.26.
  32. Magnús M. Halldórsson and Alexandre Nolin. Superfast coloring in CONGEST via efficient color sampling. In Tomasz Jurdzinski and Stefan Schmid, editors, Structural Information and Communication Complexity - 28th International Colloquium, SIROCCO 2021, Wrocław, Poland, June 28 - July 1, 2021, Proceedings, volume 12810 of Lecture Notes in Computer Science, pages 68-83. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-79527-6_5.
  33. Magnús M. Halldórsson, Yannic Maus, and Saku Peltonen. Distributed Lovász local lemma under bandwidth limitations, 2024. https://arxiv.org/abs/2405.07353, URL: https://doi.org/10.48550/arXiv.2405.07353.
  34. Magnús M. Halldórsson, Alexandre Nolin, and Tigran Tonoyan. Overcoming congestion in distributed coloring. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 26-36. ACM, 2022. URL: https://doi.org/10.1145/3519270.3538438.
  35. David G. Harris, Johannes Schneider, and Hsin-Hao Su. Distributed (Δ + 1)-coloring in sublogarithmic rounds. Journal of the ACM, 65:19:1-19:21, 2018. URL: https://doi.org/10.1145/3178120.
  36. Öjvind Johansson. Simple distributed Δ+1-coloring of graphs. Inf. Process. Lett., 70(5):229-232, 1999. Google Scholar
  37. Nati Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  38. Yannic Maus, Saku Peltonen, and Jara Uitto. Distributed symmetry breaking on power graphs via sparsification. In Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing, PODC '23, pages 157-167, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3583668.3594579.
  39. Yannic Maus and Tigran Tonoyan. Local conflict coloring revisited: Linial for lists. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, 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.
  40. Yannic Maus and Jara Uitto. Efficient CONGEST algorithms for the Lovász local lemma. In Seth Gilbert, editor, Proceedings of the International Symposium on Distributed Computing (DISC), volume 209 of LIPIcs, pages 31:1-31:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.DISC.2021.31.
  41. Alessandro Panconesi and Aravind Srinivasan. The local nature of Δ-coloring and its algorithmic applications. Combinatorica, 15(2):255-280, 1995. URL: https://doi.org/10.1007/BF01200759.
  42. Luke Postle. Linear-time and efficient distributed algorithms for list coloring graphs on surfaces. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 929-941. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00060.
  43. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proceedings of the ACM Symposium 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