ABSTRACT
We consider graph coloring and related problems in the distributed message-passing model. \em Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1 - hop-neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Δ + 1)-coloring algorithm requires Ω(Δ log Δ + log^* n) rounds, unless there exists "a very special type of coloring that can be very efficiently reduced" \citeSV93.
No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms, and to explore other approaches to the coloring problem \citeBE09,K09,B15,FHK16. The latter gave rise to faster algorithms, but their heavy machinery which is of non-locally-iterative nature made them far less suitable to various settings. In this paper we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative (Δ + 1)-coloring algorithm with running time O(Δ + log^* n), i.e., \em below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings. This includes the following results. \beginitemize ıtem We obtain self-stabilizing distributed algorithms for (Δ + 1)-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set and maximal matching with O(Δ + log^* n) time. This significantly improves previously-known results that have O(n) or larger running times \citeGK10. ıtem We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O(Δ + log^* n) time and O(Δ)-edge-coloring in the Bit-Round model with O(Δ + log n) time. The factors of log^* n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously-known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. ıtem We obtain an arbdefective coloring algorithm with running time O(\sqrt Δ + log^* n). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it in order to compute a proper (1 + ε)Δ-coloring within O(√ Δ + log^* n) time, and √(Δ + 1)√-coloring within √O(√ Δ log Δ log^* Δ + log^* n)√ time. This improves the recent state-of-the-art bounds of Barenboim from PODC'15 \citeB15 and Fraigniaud et al. from FOCS'16 \citeFHK16 by polylogarithmic factors. ıtem Our algorithms are applicable to the SET-LOCAL model \citeHKMS15 (also known as the weak LOCAL model). In this model a relatively strong lower bound of √Ω(Δ^1/3 )√ is known for √(Δ + 1)√-coloring. However, most of the coloring algorithms do not work in this model. (In \citeHKMS15 only Linial's √O(Δ^2)√-time algorithm and Kuhn-Wattenhofer √O(Δ log Δ)√-time algorithms are shown to work in it.) We obtain the first linear-in-Δ algorithms that work also in this model. \enditemize
- L. Barenboim. Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks. Journal of the ACM, 63(5) : 47, 2016. Google ScholarDigital Library
- L. Barenboim, and M. Elkin. Distributed (Δ + 1)- coloring in linear (in Δ) time. In Proc. of the 41st ACM Symp. on Theory of Computing, pp. 111--120, 2009. Google ScholarDigital Library
- L. Barenboim, and M. Elkin. Deterministic distributed vertex coloring in polylogarithmic time. In Proc. 29th ACM Symp. on Principles of Distributed Computing, pages 410--419, 2010. Google ScholarDigital Library
- L. Barenboim, and M. Elkin. Distributed deterministic edge coloring using bounded neighborhood independence. In Proc. of the 30th ACM Symp. on Principles of Distributed Computing, pages 129 - 138, 2011. Google ScholarDigital Library
- L. Barenboim, and M. Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan and Claypool, 2013. Google ScholarDigital Library
- L. Barenboim, M. Elkin, and F. Kuhn. Distributed (Delta+1)-Coloring in Linear (in Delta) Time. SIAM Journal on Computing, 43(1): 72--95, 2014.Google ScholarDigital Library
- L. Barenboim, M. Elkin, T. Maimon. Deterministic Distributed (Δ + o(Δ))-EdgeColoring, and Vertex-Coloring of Graphs with Bounded Diversity. In Proc. of the 36th ACM Symp. on Principles of Distributed Computing, pages 175--184, 2017. Google ScholarDigital Library
- L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The locality of distributed symmetry breaking. In Proc. of the 53rd Annual Symp. on Foundations of Computer Science, pages 321--330, 2012. Google ScholarDigital Library
- L. Barenboim, M. Elkin U. Goldenberg. Locally-Iterative Distributed (Δ + 1)-Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models https://arxiv.org/pdf/1712.00285.pdfGoogle Scholar
- R. Cole, and U. Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32--53, 1986. Google ScholarDigital Library
- E. Dijkstra. Self-stabilizing systems in spite of distributed control. Communication of the ACM, 17 (11): 643-644, 1974. Google ScholarDigital Library
- S. Dolev. Self-Stabilization. MIT Press, 2000. Google ScholarDigital Library
- S. Dolev, and T. Herman. Superstabilizing Protocols for Dynamic Distributed Systems. Chicago J. Theor. Comput. Sci. 1997.Google Scholar
- D. Dubhashi, D. Grable, and A. Panconesi. Nearly-optimal distributed edgecolouring via the nibble method. Theoretical Computer Science, a special issue for the best papers of ESA95, 203(2):225--251, 1998. Google ScholarDigital Library
- M. Elkin, S. Pettie, and H. Su. (2Δ - 1)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting. In Proc. of the 26th ACM-SIAM Symp. on Discrete Algorithms, pages 355--370, 2015. Google ScholarDigital Library
- M. Fischer, M. Ghaffari, and F. Kuhn . Deterministic Distributed Edge Coloring via Hypergraph Maximal Matching. To appear in 58th Annual Symp on Foundations of Computer Science, 2017.Google Scholar
- P. Fraigniaud, M. Heinrich, and A. Kosowski. Local Conflict Coloring. In Proc. of the 57th Annual Symp. on Foundations of Computer Science, pages 625 - 634, 2016.Google ScholarCross Ref
- D. Grable, and A. Panconesi. Nearly optimal distributed edge colouring in O(log log n) rounds. Random Structures and Algorithms, 10(3): 385--405, 1997. Google ScholarDigital Library
- A. Goldberg, and S. Plotkin. Parallel (Δ+1)-Coloring of Constant-Degree Graphs. Inf. Process. Lett. 25(4): 241--245, 1987. Google ScholarDigital Library
- A. Goldberg, S. Plotkin, and G. Shannon. Parallel symmetry-breaking in sparse graphs. SIAM Journal on Discrete Mathematics, 1(4):434--446, 1988. Google ScholarDigital Library
- N. Guellati, and H. Kheddouci. A survey on self-stabilizing algorithms for independence, domination, coloring, and matching in graphs. Journal of Parallel and Distributed Computing, 70(4): 406--415, 2010. Google ScholarDigital Library
- G. Hardy, and E. Wright. An introduction to the theory of numbers. Oxford university press, 5th edition, 1980.Google Scholar
- D. Hefetz, F. Kuhn, Y. Maus, and A. Steger. Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model. In Proc. of the 30th International Symp. on DiStributed Computing, pages 99 - 113, 2016.Google ScholarCross Ref
- T. Herman. Self-stabilization bibliography: Access guide. Chicago Journal of Theoretical Computer Science, Working Paper WP-1, 2002.Google Scholar
- S.C. Hsu, S.T. Huang. A self-stabilizing algorithm for maximal matching. Information Processing Letters, 43 (2):7781, 1992 . Google ScholarDigital Library
- M. Ikeda, S. Kamei, and H. Kakugawa. A space-optimal self-stabilizing algorithm for the maximal independent set problem. In Proc. 3rd International Conference on Parallel and Distributed Computing, Applications and Technologies, 2002.Google Scholar
- A. Kosowski, L. Kuszner. Self-stabilizing algorithms for graph coloring with improved performance guarantees. In Proc. 8th International Conference on Artificial Intelligence and Soft Computing, pages 1150--1159 2006. Google ScholarDigital Library
- K. Kothapalli, C. Scheideler, M. Onus, and C. Schindelhauer. Distributed coloring in O(p log n) bit rounds. In Proc. of the 20th International Parallel and Distributed Processing Symp., 2006. Google ScholarDigital Library
- F. Kuhn. Weak graph colorings: distributed algorithms and applications. In Proc. of the 21st ACM Symp. on Parallel Algorithms and Architectures, pages 138--144, 2009. Google ScholarDigital Library
- F. Kuhn, and R. Wattenhofer. On the complexity of distributed graph coloring. In Proc. 25th ACM Symp. Principles of Distributed Computing, pp. 7--15, 2006. Google ScholarDigital Library
- N. Linial. Distributive graph algorithms: Global solutions from local data In Proc. 28th Symp. on Foundation of Computer Science, pp. 331--335, 1987. Google ScholarDigital Library
- L. Lovasz. On decompositions of graphs. Studia Sci. Math. Hungar., 1:237-238, 1966.Google Scholar
- M. Naor, and L. Stockmeyer. What can be computed locally? In Proc. 25th ACM Symp. on Theory of Computing, pages 184--193, 1993. Google ScholarDigital Library
- S. Pai, G. Pandurangan, S. Pemmaraju, T. Riaz, and P. Robinson. Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets. https://arxiv.org/abs/1705.07861Google Scholar
- A. Panconesi, and R. Rizzi. Some simple distributed algorithms for sparse networks. Distributed Computing, 14(2):97--100, 2001. Google ScholarDigital Library
- A. Panconesi, and A. Srinivasan. Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds. SIAM Journal on Computing, 26(2):350--368, 1997. Google ScholarDigital Library
- D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. Google ScholarDigital Library
- S. Sur, and P.K. Srimani. A self-stabilizing algorithm for coloring bipartite graphs. Information Sciences, 69, pages 219--227, 1993 . Google ScholarDigital Library
- M. Szegedy, and S. Vishwanathan. Locality based graph coloring. In Proc. 25th ACM Symp. on Theory of Computing, pages 201--207, 1993. Google ScholarDigital Library
- V. Vizing. On an estimate of the chromatic class of a p-graph. Metody Diskret. Analiz, 3: 25--30, 1964.Google Scholar
Index Terms
- Locally-Iterative Distributed (Δ+ 1): -Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models
Recommendations
Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic, and Faulty Networks
We study the distributed (Δ + 1)-vertex-coloring and (2Δ − 1)-edge-coloring problems. These problems are among the most important and intensively studied problems in distributed computing. Despite very intensive research in the last 30 years, no ...
The Locality of Distributed Symmetry Breaking
FOCS '12: Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer ScienceWe present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include \begin{enumerate} \item A randomized algorithm for computing a maximal matching (MM) in $O(\log\Delta + (\log\...
Comments