ABSTRACT
We present a deterministic distributed algorithm, in the LOCAL model, that computes a (1+o(1))Δ-edge-coloring in polylogarithmic-time, so long as the maximum degree Δ=Ω(logn). For smaller Δ, we give a polylogarithmic-time 3Δ/2-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of 2Δ−1 colors, and they improve significantly on the recent polylogarithmic-time (2Δ−1)(1+o(1))-edge-coloring of Ghaffari and Su [SODA’17] and the (2Δ−1)-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS’17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes.
Supplemental Material
- N. Alon. 2003.Google Scholar
- A Simple Algorithm for Edge-Coloring Bipartite Multigraphs. Inform. Process. Lett. 85, 6 (2003), 301–302. Google ScholarDigital Library
- N. Alon, L. Babai, and A. Itai. 1986.Google Scholar
- A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. J. of Algorithms 7, 4 (1986), 567–583. Google ScholarDigital Library
- L. Barenboim. 2015. Deterministic ( ∆ + 1)-Coloring in Sublinear (in ∆) Time in Static, Dynamic and Faulty Networks. In Proc. 34th ACM Symp. on Principles of Distributed Computing (PODC). 345–354. Google ScholarDigital Library
- L. Barenboim and M. Elkin. 2011.Google Scholar
- Distributed Deterministic Edge Coloring Using Bounded Neighborhood Independence. In Proc. 30th Symp. on Principles of Distributed Computing (PODC). 129–138. Google ScholarDigital Library
- L. Barenboim and M. Elkin. 2013.Google Scholar
- Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers. Google ScholarDigital Library
- L. Barenboim, M. Elkin, and F. Kuhn. 2015. Distributed ( ∆ +1)-Coloring in Linear (in ∆) Time. SIAM J. Computing 43, 1 (2015), 72–95.Google ScholarCross Ref
- L. Barenboim, M. Elkin, and T. Maimon. 2017. Deterministic Distributed (Delta + o(Delta))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity. In Proc. 36th ACM Symp. on Principles of Distributed Computing (PODC). 175–184. Google ScholarDigital Library
- L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. 2012.Google Scholar
- The Locality of Distributed Symmetry Breaking. In Proc. 53th Symp. on Foundations of Computer Science (FOCS). 20:1–20:45.Google Scholar
- B. Bollobas. 1998.Google Scholar
- Modern Graph Theory. Springer.Google Scholar
- Y.-J. Chang, Q. He, W. Li, S. Pettie, and J. Uitto. 2018.Google Scholar
- The Complexity of Distributed Edge Coloring with Small Palettes. In Proc. 29th ACM-SIAM Symp. on Discrete Algorithms (SODA). arXiv:1708.04290. Google ScholarDigital Library
- K.-M. Chung, S. Pettie, and H.-H. Su. 2017. Distributed Algorithms for the Lovász Local Lemma and Graph Coloring. Distributed Computing 30, 4 (2017), 261–280. Google ScholarDigital Library
- Richard Cole and Uzi Vishkin. 1986. Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking. Inf. Control 70, 1 (1986), 32–53. Google ScholarDigital Library
- Andrzej Czygrinow and Michał Hańćkowiak. 2003. Distributed Algorithm for Better Approximation of the Maximum Matching. In International Computing and Combinatorics Conference. 242–251. Google ScholarDigital Library
- A. Czygrinow, M. Hańćkowiak, and M. Karoński. 2001. Distributed O(∆ log n)-Edge-Coloring Algorithm. In Proc. 9th European Symposium on Algorithms (ESA). 345–355. Google ScholarDigital Library
- M. Elkin, S. Pettie, and H.-H. Su. 2015. ( 2 ∆ − 1)-Edge-Coloring is Much Easier Than Maximal Matching in the Distributed Setting. In Proc. 26th ACM-SIAM Symp. on Discrete Algorithms (SODA). 355–370. Google ScholarDigital Library
- M. Fischer. 2017. Deterministic Distributed Matching: Simpler, Faster, Better. In Proc. 30th Symp. on Distributed Computing (DISC). 17:1–17:15.Google Scholar
- M. Fischer, M. Ghaffari, and F. Kuhn. 2017.Google Scholar
- Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching. In Proc. 58th IEEE Symp. on Foundations of Computer Science (FOCS). arXiv:1704.02767.Google Scholar
- J.-C. Fournier. 1973. Colorations des Arêtes d’un Graphe. Cahiers du CERO 15 (1973), 311–314.Google Scholar
- P. Fraigniaud, M. Heinrich, and A. Kosowski. 2016. Local Conflict Coloring. In Proc. 57th IEEE Symp. on Foundations of Computer Science (FOCS). 625–634.Google Scholar
- P. Fraigniaud, A. Korman, and D. Peleg. 2013. Towards a Complexity Theory for Local Distributed Computing. J. of the ACM 60, 5 (2013), 35. Google ScholarDigital Library
- M. Ghaffari, D. G. Harris, and F. Kuhn. 2017. On Derandomizing Local Distributed Algorithms. CoRR (2017). arXiv: 1711.02194Google Scholar
- M. Ghaffari, J. Hirvonen, F. Kuhn, Y. Maus, J. Suomela, and J. Uitto. 2017. Improved Distributed Degree Splitting and Edge Coloring. In Proc. 30th Symp. on Distributed Computing (DISC). 19:1–19:15.Google Scholar
- M. Ghaffari and H.-H. Su. 2017.Google Scholar
- Distributed Degree Splitting, Edge Coloring, and Orientations. In Proc. 28th ACM-SIAM Symp. on Discrete Algorithms (SODA). arXiv:1608.03220. Google ScholarDigital Library
- S. G. Harris, J. Schneider, and H.-H. Su. 2016. Distributed ( ∆ + 1)-Coloring in Sublogarithmic Rounds. In Proc. 48th Symp. on the Theory of Computing (STOC). 465–478 465–478 465–478. Google ScholarDigital Library
- J. E. Hopcroft and R. M. Karp. 1973. An n 5/2 Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput. 2, 4 (1973), 225–231.Google ScholarDigital Library
- S. Hougardy and D. E. Vinkemeier. 2006. Approximating Weighted Matchings in Parallel. Inform. Process. Lett. 99, 3 (2006), 119–123. Google ScholarDigital Library
- N. Linial. 1992. Locality in Distributed Graph Algorithms. SIAM J. Comput. 21, 1 (1992), 193–201. Google ScholarDigital Library
- Z. Lotker, B. Patt-Shamir, and S. Pettie. 2008. Improved Distributed Approximate Matching. In Proc. of the 20th Annual Symposium on Parallelism in Algorithms and Architectures (SPAA). 129–136. Google ScholarDigital Library
- M. Luby. 1986. A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM J. Comput. 15, 4 (1986), 1036–1053. Google ScholarDigital Library
- A. Panconesi and R. Rizzi. 2001. Some Simple Distributed Algorithms for Sparse Networks. Distributed computing 14, 2 (2001), 97–100. Google ScholarDigital Library
- A. Panconesi and A. Srinivasan. 1992.Google Scholar
- Fast Randomized Algorithms for Distributed Edge Coloring. In Proc. 11th ACM Symp. on Principles of Distributed Computing (PODC). 251–262. Google ScholarDigital Library
- A. Panconesi and A. Srinivasan. 1995. The Local Nature of ∆-Coloring and its Algorithmic Applications. Combinatorica 15, 2 (01 Jun 1995), 255–280.Google Scholar
- A. Panconesi and A. Srinivasan. 1995. On the Complexity of Distributed Network Decomposition. Journal of Algorithms 20, 2 (1995), 581–592. Google ScholarDigital Library
Index Terms
- Deterministic distributed edge-coloring with fewer colors
Recommendations
Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation ...
Deterministic distributed vertex coloring in polylogarithmic time
PODC '10: Proceedings of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computingConsider an n-vertex graph G = (V,E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete ...
Improved Distributed Delta-Coloring
PODC '18: Proceedings of the 2018 ACM Symposium on Principles of Distributed ComputingWe present a randomized distributed algorithm that computes a Δ- coloring in any non-complete graph with maximum degree Δ ≥ 4 in O(log Δ) +2O( √ log log n) rounds, as well as a randomized algorithm that computes a Δ-coloring in O((log logn)2) rounds ...
Comments