Deterministic distributed vertex coloring in polylogarithmic time

ABSTRACT

Consider 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 rounds.

In the distributed vertex coloring problem the objective is to color G with Δ + 1, or slightly more than Δ + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time. Efficient randomized algorithms for this problem are known for more than twenty years [1, 19]. Specifically, these algorithms produce a Δ + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Δ²) colors. This algorithm was devised in a seminal FOCS'87 paper by Linial [16]. Its running time is O(log* n). In the same paper Linial asked whether one can color with significantly less than Δ² colors in deterministic polylogarithmic time. By now this question of Linial became one of the most central long-standing open questions in this area.

In this paper we answer this question in the affirmative, and devise a deterministic algorithm that employs Δ^1+o(1) colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Δ) log Δ log n, for an arbitrarily slow-growing function f(Δ) = ω(1). We can also produce O(Δ^{1 + η})-coloring in O(log Δ log n)-time, for an arbitrarily small constant η > 0, and O(Δ)-coloring in O(Δ^ε log n) time, for an arbitrarily small constant ε > 0. Our results are, in fact, far more general than this. In particular, for a graph of arboricity a, our algorithm produces an O(a^{1 + η})-coloring, for an arbitrarily small constant η > 0, in time O(log a log n).

References

1. N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567--583, 1986.
2. B. Awerbuch, A. V. Goldberg, M. Luby, and S. Plotkin. Network decomposition and locality in distributed computation. In Proc. of the 30th Symposium on Foundations of Computer Science, pages 364--369, 1989.
3. L. Barenboim, and M. Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. In Proc. of the 27th ACM Symp. on Principles of Distributed Computing, pages 25--34, 2008.
4. L. Barenboim, and M. Elkin. Distributed (Δ + 1)-coloring in linear (in Δ) time. In Proc. of the 41th ACM Symp. on Theory of Computing, pages 111--120, 2009.
5. L. Barenboim, and M. Elkin. Distributed (Δ + 1)-coloring in linear (in Δ) time. http://arXiv.org/abs/0812.1379v2, 2008.
6. L. Cowen, R. Cowen, and D. Woodall. Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valence. Journal of Graph Theory, 10:187--195, 1986.
7. R. Cole, and U. Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32--53, 1986.
8. D. Eppstein. Arboricity and bipartite subgraph listing algorithms. Information Processing Letters, 51(4):207--211, 1994.
9. A. Goldberg, and S. Plotkin. Efficient parallel algorithms for (Δ + 1)- coloring and maximal independent set problem. In Proc. 19th ACM Symposium on Theory of Computing, pages 315--324, 1987.
10. A. Goldberg, S. Plotkin, and G. Shannon. Parallel symmetry-breaking in sparse graphs. SIAM Journal on Discrete Mathematics, 1(4):434--446, 1988.
11. F. Harary, and K. Jones. Conditional colorability II: Bipartite variations. Congressus Numer, 50:205--218, 1985.
12. Ö. Johansson. Simple distributed (Δ + 1)-coloring of graphs. Information Processing Letters, 70(5):229--232, 1999.
13. K. Kothapalli, C. Scheideler, M. Onus, and C. Schindelhauer. Distributed coloring in O(&38730;log n) bit rounds. In Proc. of the 20th International Parallel and Distributed Processing Symposium, 2006.
14. F. Kuhn. Weak graph colorings: distributed algorithms and applications. In proc. of the 21st ACM Symposium on Parallel Algorithms and Architectures, pages (138--144) February 2009.
15. F. Kuhn, and R. Wattenhofer. On the complexity of distributed graph coloring. In Proc. of the 25th ACM Symp. on Principles of Distributed Computing, pages 7--15, 2006.
16. N. Linial. Distributive graph algorithms: Global solutions from local data In Proc. of the 28th Annual Symp. on Foundation of Computer Science, pages 331--335, 1987.
17. N. Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193--201, 1992.
18. N. Linial and M. Saks. Low diameter graph decomposition. Combinatorica 13: 441--454, 1993.
19. M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 15:1036--1053, 1986.
20. C. Nash-Williams. Decompositions of finite graphs into forests. J. London Math, 39:12, 1964.
21. D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000.
22. A. Panconesi, and A. Srinivasan. On the complexity of distributed network decomposition. Journal of Algorithms, 20(2):581--592, 1995.

Index Terms

• Mathematics of computing

  • Discrete mathematics

    • Graph theory

      • Network flows

• Theory of computation

  • Design and analysis of algorithms

    • Graph algorithms analysis

      • Network flows

  • Randomness, geometry and discrete structures