Distributed Partial Coloring via Gradual Rounding

Authors Avinandan Das, Pierre Fraigniaud , Adi Rosén



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Author Details

Avinandan Das
  • Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Université Paris Cité, France
Pierre Fraigniaud
  • Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Université Paris Cité, France
Adi Rosén
  • Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Université Paris Cité, France

Acknowledgements

We thank Alkida Balliu and Dennis Olivetti for useful discussions, and for pointing to us an error in a first draft of our result. We also thank Baruch Schieber for useful discussion in the early stage of this work. We finally thank the anonymous reviewers for their detailed comments.

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Avinandan Das, Pierre Fraigniaud, and Adi Rosén. Distributed Partial Coloring via Gradual Rounding. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.OPODIS.2023.30

Abstract

For k ≥ 0, k-partial (k+1)-coloring asks to color the nodes of an n-node graph using a palette of k+1 colors such that every node v has at least min{k,deg(v)} neighbors colored with colors different from its own color. Hence, proper (Δ+1)-coloring is the special case of k-partial (k+1)-coloring when k = Δ. Ghaffari and Kuhn [FOCS 2021] recently proved that there exists a deterministic distributed algorithm that solves proper (Δ+1)-coloring of n-node graphs with maximum degree Δ in O(log n ⋅ log²Δ) rounds under the LOCAL model of distributed computing. This breakthrough result is achieved via an original iterated rounding approach. Using the same technique, Ghaffari and Kuhn also showed that there exists a deterministic algorithm that solves proper O(a)-coloring of n-node graphs with arboricity a in O(log n ⋅ log³a) rounds. It directly follows from this latter result that k-partial O(k)-coloring can be solved deterministically in O(log n ⋅ log³k) rounds. We develop an extension of the Ghaffari and Kuhn algorithm for proper (Δ+1)-coloring, and show that it solves k-partial (k+1)-coloring, thus generalizing their main result. Our algorithm runs in O(log n ⋅ log³k) rounds, like the algorithm that follows from Ghaffari and Kuhn’s algorithm for graphs with bounded arboricity, but uses only k+1 color, i.e., the smallest number c of colors such that every graph has a k-partial c-coloring. Like all the previously mentioned algorithms, our algorithm actually solves the general list-coloring version of the problem. Specifically, every node v receives as input an integer demand d(v) ≤ deg(v), and a list of at least d(v)+1 colors. Every node must then output a color from its list such that the resulting coloring satisfies that every node v has at least d(v) neighbors with colors different from its own. Our algorithm solves this problem in O(log n ⋅ log³k) rounds where k = max_v d(v). Moreover, in the specific case where all lists of colors given to the nodes as input share a common colors c^* known to all nodes, one can save one log k factor. In particular, for standard k-partial (k+1)-coloring, which corresponds to the case where all nodes are given the same list {1,… ,k+1}, one can modify our algorithm so that it runs in O(log n ⋅ log²k) rounds, and thus matches the complexity of Ghaffari and Kuhn’s algorithm for (Δ+1)-coloring for k = Δ.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Distributed graph coloring
  • partial coloring
  • weak coloring

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References

  1. Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network decomposition and locality in distributed computation. In 30th IEEE Symposium on Foundations of Computer Science (FOCS), pages 364-369, 1989. URL: https://doi.org/10.1109/SFCS.1989.63504.
  2. Alkida Balliu, Juho Hirvonen, Christoph Lenzen, Dennis Olivetti, and Jukka Suomela. Locality of not-so-weak coloring. In 26th International Colloquium on Structural Information and Communication Complexity (SIROCCO), volume 11639 of LNCS, pages 37-51. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24922-9_3.
  3. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, 2013. URL: https://doi.org/10.2200/S00520ED1V01Y201307DCT011.
  4. 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 37th ACM Symposium on Principles of Distributed Computing (PODC), pages 437-446, 2018. URL: https://dl.acm.org/citation.cfm?id=3212769.
  5. Sebastian Brandt. An automatic speedup theorem for distributed problems. In 38th ACM Symposium on Principles of Distributed Computing (PODC), pages 379-388, 2019. URL: https://doi.org/10.1145/3293611.3331611.
  6. Salwa Faour, Mohsen Ghaffari, Christoph Grunau, Fabian Kuhn, and Václav Rozhon. Local distributed rounding: Generalized to mis, matching, set cover, and beyond. In 34th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4409-4447, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH168.
  7. Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Irit Dinur, editor, 57th IEEE Symposium on Foundations of Computer Science (FOCS), pages 625-634, 2016. URL: https://doi.org/10.1109/FOCS.2016.73.
  8. Mohsen Ghaffari and Christoph Grunau. Faster deterministic distributed MIS and approximate matching. In 55th ACM Symposium on Theory of Computing (STOC), pages 1777-1790, 2023. URL: https://doi.org/10.1145/3564246.3585243.
  9. Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, and Václav Rozhon. Improved distributed network decomposition, hitting sets, and spanners, via derandomization. In 34th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2532-2566, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH97.
  10. Mohsen Ghaffari and Fabian Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In 62nd IEEE Symposium on Foundations of Computer Science (FOCS), pages 1009-1020, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00101.
  11. Fabian Kuhn. Weak graph colorings: distributed algorithms and applications. In 21st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 138-144, 2009. URL: https://doi.org/10.1145/1583991.1584032.
  12. Nathan Linial. Distributive graph algorithms-global solutions from local data. In 28th IEEE Symp. on Foundations of Computer Science (FOCS), pages 331-335, 1987. URL: https://doi.org/10.1109/SFCS.1987.20.
  13. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  14. Yannic Maus and Tigran Tonoyan. Local conflict coloring revisited: Linial for lists. In 34th International Symposium on Distributed Computing (DISC), 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.
  15. 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.
  16. Alessandro Panconesi and Aravind Srinivasan. On the complexity of distributed network decomposition. J. Algorithms, 20(2):356-374, 1996. URL: https://doi.org/10.1006/JAGM.1996.0017.
  17. David Peleg. Distributed computing: a locality-sensitive approach. SIAM, 2000. Google Scholar
  18. Václav Rozhon and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In 52nd ACM Symposium on Theory of Computing (STOC), pages 350-363, 2020. URL: https://doi.org/10.1145/3357713.3384298.