Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

  • Dan Hefetz
  • Fabian Kuhn
  • Yannic Maus
  • Angelika Steger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9888)

Abstract

We show an \(\varOmega \big (\varDelta ^{\frac{1}{3}-\frac{\eta }{3}}\big )\) lower bound on the runtime of any deterministic distributed \(\mathcal {O}\big (\varDelta ^{1+\eta }\big )\)-graph coloring algorithm in a weak variant of the \(\mathsf {LOCAL}\) model.

In particular, given a network graph \(G=(V,E)\), in the weak \(\mathsf {LOCAL}\) model nodes communicate in synchronous rounds and they can use unbounded local computation. The nodes have no identifiers, but instead, the computation starts with an initial valid vertex coloring. A node can broadcast a single message of unbounded size to its neighbors and receives the set of messages sent to it by its neighbors.

The proof uses neighborhood graphs and improves their understanding in general such that it might help towards finding a lower (runtime) bound for distributed graph coloring in the standard \(\mathsf {LOCAL}\) model.

Keywords

Lower bound Distributed graph coloring Color reduction Neighborhood graphs LOCAL model Distributed symmetry breaking 

References

  1. 1.
    Hefetz, D., Kuhn, F., Maus, Y., Steger, A.: A polynomial lower bound for distributed graph coloring in a weak LOCAL model. CoRR, abs/1607.05212 (2016)Google Scholar
  2. 2.
    Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of the Symposium on Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  3. 3.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barenboim, L., Elkin, M.: Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers (2013)Google Scholar
  5. 5.
    Barenboim, L.: Deterministic (\(\Delta \) + 1)-coloring in sublinear (in \(\Delta \)) time in static, dynamic and faulty networks. In: Proceedings of the 34th ACM Symposium on Principles of Distributed Computing (PODC), pp. 345–354 (2015)Google Scholar
  6. 6.
    Fraigniaud, P., Heinrich, M., Kosowski, A.: Local conflict coloring. CoRR, abs/1511.01287 (2015)Google Scholar
  7. 7.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hella, L., Järvisalo, M., Kuusisto, A., Laurinharju, J., Lampiäinen, T., Luosto, K., Suomela, J., Virtema, J.: Weak models of distributed computing, with connections to modal logic. Distrib. Comput. 28(1), 31–53 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kuhn, F., Wattenhofer, R.: On the complexity of distributed graph coloring. In: Proceedings of the 25th ACM Symposium on Principles of Distributed Computing (PODC), pp. 7–15 (2006)Google Scholar
  10. 10.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control 70(1), 32–53 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.A.: Network decomposition and locality in distributed computation. In: Proceedings of the 30th Symposium on Foundations of Computer Science (FOCS), pp. 364–369 (1989)Google Scholar
  12. 12.
    Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. J. Algorithms 20(2), 581–592 (1995)MathSciNetGoogle Scholar
  13. 13.
    Barenboim, L., Elkin, M., Kuhn, F.: Distributed (Delta+1)-coloring in linear (in Delta) time. SIAM J. Comput. 43(1), 72–95 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Barenboim, L., Elkin, M.: Deterministic distributed vertex coloring in polylogarithmic time. In: Proceedings of the 29th Symposium on Principles of Distributed Computing (PODC) (2010)Google Scholar
  15. 15.
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4), 567–583 (1986)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 1036–1053 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetry breaking. In Proceedings of the 53rd Symposium on Foundations of Computer Science (FOCS) (2012)Google Scholar
  18. 18.
    Harris, S.G., Schneider, J., Su, H.-H.: Distributed (\(\Delta +1\))-coloring in sublogarithmic rounds. In: Proceedings of the 48th Symposium on the Theory of Computing (STOC) (2016)Google Scholar
  19. 19.
    Chang, Y.-J., Kopelowitz, T., Pettie, S.: An exponential separation between randomized and deterministic complexity in the LOCAL model. CoRR, abs/1602.08166 (2016)Google Scholar
  20. 20.
    Brand, S., Fischer, O., Hirvonen, J., Keller, B., Lempiäinen, T., Rybicki, J., Suomela, J., Uitto, J.: A lower bound for the distributed Lovász local lemma. In: Proceedings of the 48th Symposium on the Theory of Computing (STOC) (2016)Google Scholar
  21. 21.
    Göös, M., Suomela, J.: No sublogarithmic-time approximation scheme for bipartite vertex cover. Distrib. Comput. 27(6), 435–443 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Göös, M., Hirvonen, J., Suomela, J.: Linear-in-Delta lower bounds in the LOCAL model. In: Proceedings of the 33rd Symposium on Principles of Distributed Computing (PODC), pp. 86–95 (2014)Google Scholar
  23. 23.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proceedings of the 23rd Symposium on Principles of Distributed Computing (PODC), pp. 300–309 (2004)Google Scholar
  24. 24.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local computation: lower and upper bounds. J. ACM 63(2) (2016). http://dl.acm.org/citation.cfm?id=2742012. Article No. 17
  25. 25.
    Alon, N.: On constant time approximation of parameters of bounded degree graphs. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 234–239. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  26. 26.
    Barenboim, L., Elkin, M.: Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition. Distr. Comput. 22(5), 363–379 (2010)CrossRefMATHGoogle Scholar
  27. 27.
    Kuhn, F.: Local multicoloring algorithms: computing a nearly-optimal TDMA schedule in constant time. In: Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS), pp. 613–624 (2009)Google Scholar
  28. 28.
    Szegedy, M., Vishwanathan, S.: Locality based graph coloring. In: Proceedings of the 25th ACM Symposium on Theory of Computing (STOC), pp. 201–207 (1993)Google Scholar
  29. 29.
    Gavoille, C., Klasing, R., Kosowski, A., Kuszner, Ł., Navarra, A.: On the complexity of distributed graph coloring with local minimality constraints. Technical report 6399, INRIA (2007)Google Scholar
  30. 30.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Dan Hefetz
    • 1
    • 2
  • Fabian Kuhn
    • 3
  • Yannic Maus
    • 3
  • Angelika Steger
    • 4
  1. 1.Hebrew UniversityJerusalemIsrael
  2. 2.Tel Aviv UniversityTel AvivIsrael
  3. 3.University of FreiburgFreiburg im BreisgauGermany
  4. 4.ETH ZurichZürichSwitzerland

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