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

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Distributed Computing (DISC 2016)

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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.

A full version of this paper with all proofs is avalaible on arXiv.org [1].

F. Kuhn and Y. Maus—Supported by ERC Grant No. 336495 (ACDC).

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Notes

  1. 1.

    The function \(\log ^{*}x\) denotes the number of iterated logarithms needed to obtain a value at most 1, that is, \(\forall x\le 1: \log ^{*}x=0,\ \forall x>1: \log ^{*}x = 1 + \log ^{*}\log x\).

  2. 2.

    A similar model, but for completely anonymous graphs, has been studied in [8].

  3. 3.

    The algorithm of [6] works for the even more general conflict coloring problem and \(\widetilde{\mathcal {O}}\) ignores polylog factors in \(\log \varDelta \).

References

  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. Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of the Symposium on Complexity of Computer Computations, pp. 85–103 (1972)

    Google Scholar 

  3. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barenboim, L., Elkin, M.: Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers (2013)

    Google Scholar 

  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. Fraigniaud, P., Heinrich, M., Kosowski, A.: Local conflict coloring. CoRR, abs/1511.01287 (2015)

    Google Scholar 

  7. Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control 70(1), 32–53 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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. Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. J. Algorithms 20(2), 581–592 (1995)

    MathSciNet  Google Scholar 

  13. Barenboim, L., Elkin, M., Kuhn, F.: Distributed (Delta+1)-coloring in linear (in Delta) time. SIAM J. Comput. 43(1), 72–95 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 1036–1053 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. Göös, M., Suomela, J.: No sublogarithmic-time approximation scheme for bipartite vertex cover. Distrib. Comput. 27(6), 435–443 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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

    Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Barenboim, L., Elkin, M.: Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition. Distr. Comput. 22(5), 363–379 (2010)

    Article  MATH  Google Scholar 

  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. 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. 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. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)

    Google Scholar 

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Correspondence to Yannic Maus .

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Hefetz, D., Kuhn, F., Maus, Y., Steger, A. (2016). Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_8

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  • DOI: https://doi.org/10.1007/978-3-662-53426-7_8

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