Document Open Access Logo

Local Conflict Coloring Revisited: Linial for Lists

Authors Yannic Maus, Tigran Tonoyan

PDF
Thumbnail PDF

File

LIPIcs.DISC.2020.16.pdf
  • Filesize: 0.57 MB
  • 18 pages

Document Identifiers

Author Details

Yannic Maus
  • Technion - Israel Institute of Technology, Haifa, Israel
Tigran Tonoyan
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

This project was supported by the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 755839.

Cite As Get BibTex

Yannic Maus and Tigran Tonoyan. Local Conflict Coloring Revisited: Linial for Lists. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.DISC.2020.16

Abstract

Linial’s famous color reduction algorithm reduces a given m-coloring of a graph with maximum degree Δ to a O(Δ²log m)-coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m-coloring in a directed graph of maximum outdegree β, if every node has a list of size Ω(β² (log β+log log m + log log |𝒞|)) from a color space 𝒞 then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial’s color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local (deg+1)-list coloring algorithm from Barenboim et al. [PODC'18] by slightly reducing the runtime to O(√{ΔlogΔ})+log^* n and significantly reducing the message size (from huge to roughly Δ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. [FOCS'16].

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • distributed graph coloring
  • list coloring
  • low intersecting set families

Metrics

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

Related Versions

References

  1. Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network decomposition and locality in distributed computation. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 364-369, 1989. Google Scholar
  2. Alkida Balliu, Sebastian Brandt, Yuval Efron, Juho Hirvonen, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Classification of distributed binary labeling problems. In Proceedings of the International Symposium on Distributed Computing (DISC), 2020. Google Scholar
  3. Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower bounds for maximal matchings and maximal independent sets. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 481-497, 2019. Google Scholar
  4. Alkida Balliu, Sebastian Brandt, and Dennis Olivetti. Distributed lower bounds for ruling sets. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), 2020. Google Scholar
  5. Alkida Balliu, Juho Hirvonen, Christoph Lenzen, Dennis Olivetti, and Jukka Suomela. Locality of not-so-weak coloring. In Proceedings of the International Colloquium on Structural Information and Communication Complexity (SIROCCO), pages 37-51, 2019. Google Scholar
  6. Alkida Balliu, Fabian Kuhn, and Dennis Olivetti. Distributed edge coloring in time quasi-polylogarithmic in delta. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), 2020. Google Scholar
  7. Philipp Bamberger, Fabian Kuhn, and Yannic Maus. Efficient deterministic distributed coloring with small bandwidth. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), 2020. Google Scholar
  8. Leonid Barenboim. Deterministic (Δ + 1)-coloring in sublinear (in Δ) time in static, dynamic, and faulty networks. Journal of the ACM, 63(5):47:1-47:22, 2016. Google Scholar
  9. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. Distributed Comput., 22(5-6):363-379, 2010. Google Scholar
  10. Leonid Barenboim and Michael Elkin. Deterministic distributed vertex coloring in polylogarithmic time. Journal of the ACM, 58(5):23:1-23:25, 2011. Google Scholar
  11. Leonid Barenboim and Michael Elkin. Distributed deterministic edge coloring using bounded neighborhood independence. In Proceedings of the 30th Annual ACM Symposium on Principles of Distributed Computing, PODC 2011, San Jose, CA, USA, June 6-8, 2011, pages 129-138, 2011. URL: https://doi.org/10.1145/1993806.1993825.
  12. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, 2013. Google Scholar
  13. Leonid Barenboim, Michael Elkin, and Uri Goldenberg. Locally-Iterative Distributed (Δ+ 1)-Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 437-446, 2018. Google Scholar
  14. Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (Delta+1)-Coloring in Linear (in Delta) Time. SIAM J. Comput., 43(1):72-95, 2014. Google Scholar
  15. 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. Google Scholar
  16. Sebastian Brandt. An automatic speedup theorem for distributed problems. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 379-388, 2019. Google Scholar
  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 Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 479-488, 2016. Google Scholar
  18. Sebastian Brandt and Dennis Olivetti. Truly tight-in-δ bounds for bipartite maximal matching and variants. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), 2020. Google Scholar
  19. 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. Google Scholar
  20. 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. Google Scholar
  21. Paul Erdös, Peter Frankl, and Zóltan Füredi. Families of finite sets in which no set is covered by the union of r others. Israel Journal of Mathematics, 51:79-89, 1985. Google Scholar
  22. Manuela Fischer, Mohsen Ghaffari, and Fabian Kuhn. Deterministic distributed edge-coloring via hypergraph maximal matching. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 180-191, 2017. Google Scholar
  23. Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 625-634, 2016. Google Scholar
  24. Pierre Fraigniaud and Ami Paz. The topology of local computing in networks. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 128:1-128:18, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.128.
  25. Cyril Gavoille, Ralf Klasing, Adrian Kosowski, Lukasz Kuszner, and Alfredo Navarra. On the complexity of distributed graph coloring with local minimality constraints. Networks, 54(1):12-19, 2009. Google Scholar
  26. Mohsen Ghaffari, David G. Harris, and Fabian Kuhn. On derandomizing local distributed algorithms. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 662-673, 2018. Google Scholar
  27. Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, and Yannic Maus. Improved distributed delta-coloring. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 427-436, 2018. Google Scholar
  28. Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, Yannic Maus, Jukka Suomela, and Jara Uitto. Improved distributed degree splitting and edge coloring. In Proceedings of the International Symposium on Distributed Computing (DISC), pages 19:1-19:15, 2017. Google Scholar
  29. Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 784-797, 2017. Google Scholar
  30. Mohsen Ghaffari, Fabian Kuhn, Yannic Maus, and Jara Uitto. Deterministic distributed edge-coloring with fewer colors. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 418-430, 2018. Google Scholar
  31. Mohsen Ghaffari and Hsin-Hao Su. Distributed degree splitting, edge coloring, and orientations. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2505-2523, 2017. Google Scholar
  32. David G. Harris. Distributed local approximation algorithms for maximum matching in graphs and hypergraphs. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 700-724, 2019. Google Scholar
  33. David G. Harris, Johannes Schneider, and Hsin-Hao Su. Distributed (Δ +1)-coloring in sublogarithmic rounds. Journal of the ACM, 65(4):19:1-19:21, 2018. Google Scholar
  34. Dan Hefetz, Fabian Kuhn, Yannic Maus, and Angelika Steger. Polynomial lower bound for distributed graph coloring in a weak LOCAL model. In Proceedings of the International Symposium on Distributed Computing (DISC), pages 99-113, 2016. Google Scholar
  35. Frank K. Hwang and Vera T. Sós. Non-adaptive hypergeometric group testing. Studia scient. Math. Hungaria, 22:257-263, 1987. Google Scholar
  36. Fabian Kuhn. Weak graph colorings: distributed algorithms and applications. In Proceedings of the ACM Symposium on Parallelism in Algorithms and Architecture (SPAA), pages 138-144, 2009. Google Scholar
  37. Fabian Kuhn. Faster deterministic distributed coloring through recursive list coloring. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1244-1259, 2020. Google Scholar
  38. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. Journal of the ACM, 63(2), 2016. Google Scholar
  39. Fabian Kuhn and Roger Wattenhofer. On the complexity of distributed graph coloring. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 7-15, 2006. Google Scholar
  40. Juhana Laurinharju and Jukka Suomela. Brief announcement: Linial’s lower bound made easy. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 377-378, 2014. Google Scholar
  41. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992. Google Scholar
  42. Yannic Maus and Tigran Tonoyan. Local conflict coloring revisited: Linial for lists, 2020. URL: http://arxiv.org/abs/2007.15251.
  43. Moni Naor. A lower bound on probabilistic algorithms for distributive ring coloring. SIAM J. Discret. Math., 4(3):409-412, 1991. Google Scholar
  44. Alessandro Panconesi and Romeo Rizzi. Some simple distributed algorithms for sparse networks. Distributed Comput., 14(2):97-100, 2001. Google Scholar
  45. Alessandro Panconesi and Aravind Srinivasan. Improved distributed algorithms for coloring and network decomposition problems. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 581-592, 1992. Google Scholar
  46. David Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. Google Scholar
  47. Václav Rozhon 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
  48. Hsin-Hao Su and Hoa T. Vu. Towards the locality of vizing’s theorem. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 355-364, 2019. Google Scholar
  49. Mario Szegedy and Sundar Vishwanathan. Locality based graph coloring. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 201-207, 1993. 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