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Combinatorial algorithms for distributed graph coloring

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Abstract

Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just few examples of this phenomenon. The quest for combinatorial algorithms that do not use heavy algebraic machinery, but are roughly as efficient, has become a central field of study in this area. Combinatorial algorithms are often simpler than their algebraic counterparts. Moreover, in many cases, combinatorial algorithms and proofs provide additional understanding of studied problems. In this paper we initiate the study of combinatorial algorithms for Distributed Graph Coloring problems. In a distributed setting a communication network is modeled by a graph \(G=(V,E)\) of maximum degree \(\varDelta \). The vertices of \(G\) host the processors, and communication is performed over the edges of \(G\). The goal of distributed vertex coloring is to color \(V\) with \((\varDelta + 1)\) colors such that any two neighbors are colored with distinct colors. Currently, efficient algorithms for vertex coloring that require \(O(\varDelta + \log ^* n)\) time are based on the algebraic algorithm of Linial (SIAM J Comput 21(1):193–201, 1992) that employs set-systems. The best currently-known combinatorial set-system free algorithm, due to Goldberg et al. (SIAM J Discret Math 1(4):434–446, 1988), requires \(O(\varDelta ^2+\log ^*n)\) time. We significantly improve over this by devising a combinatorial \((\varDelta + 1)\)-coloring algorithm that runs in \(O(\varDelta + \log ^* n)\) time. This exactly matches the running time of the best-known algebraic algorithm. In addition, we devise a tradeoff for computing \(O(\varDelta \cdot t)\)-coloring in \(O(\varDelta /t + \log ^* n)\) time, for almost the entire range \(1 < t < \varDelta \). We also compute a Maximal Independent Set in \(O(\varDelta + \log ^* n)\) time on general graphs, and in \(O(\log n/ \log \log n)\) time on graphs of bounded arboricity. Prior to our work, these results could be only achieved using algebraic techniques. We believe that our algorithms are more suitable for real-life networks with limited resources, such as sensor networks.

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Notes

  1. See Sect. 2 for its definition.

  2. Linial’s [37] and Kuhn’s [32] algorithms require each vertex to store locally \(O(\log n + \varDelta )\) words, of \(O(\log n)\) bits each. Our combinatorial algorithms have lower space requirement of \(O(\varDelta )\) words, of \(O(\log n)\) bits each.

  3. The algorithm of [27] is based on an earlier algorithm of Cole and Vishkin [16].

  4. We refer to a procedure as generic method if it accepts another procedure as input.

  5. Line 3 of Algorithm 4 invokes the algorithm of Linial. The latter algorithm employs the coloring \(\vartheta \) for a faster computation of the colorings \(\varphi _i\).

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Acknowledgments

We are grateful to an anonymous reviewer of the Distributed Computing Journal, whose numerous comments helped us to improve the presentation in this paper.

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Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva, 84105, Israel

    Leonid Barenboim & Michael Elkin

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  1. Leonid Barenboim
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  2. Michael Elkin
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Correspondence to Leonid Barenboim.

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This research is supported by the Binational Science Foundation, Grant No. 2008390. Leonid Barenboim is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

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Barenboim, L., Elkin, M. Combinatorial algorithms for distributed graph coloring. Distrib. Comput. 27, 79–93 (2014). https://doi.org/10.1007/s00446-013-0203-2

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