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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
A similar model, but for completely anonymous graphs, has been studied in [8].
- 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
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)
Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of the Symposium on Complexity of Computer Computations, pp. 85–103 (1972)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)
Barenboim, L., Elkin, M.: Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers (2013)
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)
Fraigniaud, P., Heinrich, M., Kosowski, A.: Local conflict coloring. CoRR, abs/1511.01287 (2015)
Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)
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)
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)
Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control 70(1), 32–53 (1986)
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)
Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. J. Algorithms 20(2), 581–592 (1995)
Barenboim, L., Elkin, M., Kuhn, F.: Distributed (Delta+1)-coloring in linear (in Delta) time. SIAM J. Comput. 43(1), 72–95 (2015)
Barenboim, L., Elkin, M.: Deterministic distributed vertex coloring in polylogarithmic time. In: Proceedings of the 29th Symposium on Principles of Distributed Computing (PODC) (2010)
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)
Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 1036–1053 (1986)
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)
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)
Chang, Y.-J., Kopelowitz, T., Pettie, S.: An exponential separation between randomized and deterministic complexity in the LOCAL model. CoRR, abs/1602.08166 (2016)
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)
Göös, M., Suomela, J.: No sublogarithmic-time approximation scheme for bipartite vertex cover. Distrib. Comput. 27(6), 435–443 (2014)
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)
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)
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
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)
Barenboim, L., Elkin, M.: Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition. Distr. Comput. 22(5), 363–379 (2010)
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)
Szegedy, M., Vishwanathan, S.: Locality based graph coloring. In: Proceedings of the 25th ACM Symposium on Theory of Computing (STOC), pp. 201–207 (1993)
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)
Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-53426-7_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-53425-0
Online ISBN: 978-3-662-53426-7
eBook Packages: Computer ScienceComputer Science (R0)