Exact Bounds for Distributed Graph Colouring

  • Joel Rybicki
  • Jukka Suomela
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9439)


We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with n colours, by prior work it is known that we can find a proper 3-colouring in \(\frac{1}{2} \log^{*}(n) \pm O(1)\) communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many n the time complexity is precisely \(\frac{1}{2} \log^{*} n\) communication rounds.


Time Complexity Directed Path Chromatic Number SIAM Journal Exact Bound 
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  1. 1.
    Åstrand, M., Suomela, J.: Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks. In: Proc. 22nd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2010), pp. 294–302. ACM Press (2010)Google Scholar
  2. 2.
    Barenboim, L., Elkin, M.: Distributed graph coloring: Fundamentals and recent Developments. Morgan & Claypool (2013)Google Scholar
  3. 3.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control 70(1), 32–53 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  5. 5.
    Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast distributed approximations in planar graphs. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 78–92. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Fich, F.E., Ramachandran, V.: Lower bounds for parallel computation on linked structures. In: Proc. 2nd Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 1990), pp. 109–116. ACM Press (1990)Google Scholar
  7. 7.
    Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed computing with advice: Information sensitivity of graph coloring. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 231–242. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM Journal on Computing 27(1), 302–316 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldberg, A.V., Plotkin, S.A., Shannon, G.E.: Parallel symmetry-breaking in sparse graphs. SIAM Journal on Discrete Mathematics 1(4), 434–446 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kuhn, F., Wattenhofer, R.: On the complexity of distributed graph coloring. In: Proc. 25th Annual ACM Symposium on Principles of Distributed Computing (PODC 2006), pp. 7–15. ACM Press (2006)Google Scholar
  11. 11.
    Laurinharju, J., Suomela, J.: Brief announcement: Linial’s lower bound made easy. In: Proc. 33rd ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2014), pp. 377–378. ACM Press (2014)Google Scholar
  12. 12.
    Lenzen, C., Patt-Shamir, B.: Improved distributed Steiner forest construction. In: Proc. 33rd ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2014), pp. 262–271. ACM Press (2014)Google Scholar
  13. 13.
    Linial, N.: Locality in distributed graph algorithms. SIAM Journal on Computing 21(1), 193–201 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Naor, M.: A lower bound on probabilistic algorithms for distributive ring coloring. SIAM Journal on Discrete Mathematics 4(3), 409–412 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM Journal on Computing 24(6), 1259–1277 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distributed Computing 14(2), 97–100 (2001)CrossRefGoogle Scholar
  17. 17.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rybicki, J.: Exact bounds for distributed graph colouring. Master’s thesis, University of Helsinki, May 2011.
  19. 19.
    Suomela, J.: Distributed Algorithms (2014).
  20. 20.
    Szegedy, M., Vishwanathan, S.: Locality based graph coloring. In: Proc. 25th Annual ACM Symposium on Theory of Computing (STOC 1993), pp. 201–207. ACM Press (1993)Google Scholar
  21. 21.
    Wattenhofer, R.: Lecture notes on principles of distributed computing (2013).

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Joel Rybicki
    • 1
    • 2
  • Jukka Suomela
    • 1
  1. 1.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceAalto UniversitySaarbrückenGermany
  2. 2.Department of Algorithms and ComplexityMax Planck Institute for InformaticsSaarbrückenGermany

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