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A distributed low tree-depth decomposition algorithm for bounded expansion classes

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Abstract

We study the distributed low tree-depth decomposition problem for graphs restricted to a bounded expansion class. Low tree-depth decomposition have been introduced in 2006 and have found quite a few applications. For example it yields a linear-time model checking algorithm for graphs in a bounded expansion class. Recall that bounded expansion classes cover classes of graphs of bounded degree, of planar graphs, of graphs of bounded genus, of graphs of bounded treewidth, of graphs that exclude a fixed minor, and many other graphs. There is a sequential algorithm to compute low tree-depth decomposition (with bounded number of colors) in linear time. In this paper, we give the first efficient distributed algorithm for this problem. As it is usual for a symmetry breaking problem, we consider a synchronous model, and as we are interested in a deterministic algorithm, we use the usual assumption that each vertex has a distinct identity number. We consider the distributed message-passing \(\mathcal {CONGEST}_\mathrm{BC}\) model, in which messages have logarithmic length and only local broadcast are allowed. In this model, we present a logarithmic time distributed algorithm for computing a low tree-depth decomposition of graphs in a fixed bounded expansion class. In the sequential centralized case low tree-depth decomposition linear time algorithm are used as a core procedure in several non-trivial linear time algorithms. We believe that, similarly, low tree-depth decomposition could be at the heart of several non-trivial logarithmic time algorithms.

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Notes

  1. that is half of a positive integer, like 5 / 2.

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Acknowledgments

The authors would like to thank the referees for their many remarks, which allowed to improve the quality of the presentation of this paper.

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Authors and Affiliations

  1. Computer Science Institute of Charles University (IUUK and ITI), Malostranské nám.25, 11800, Praha 1, Czech Republic

    J. Nešetřil & P. Ossona de Mendez

  2. Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557), 190-198 avenue de France, 75013, Paris, France

    P. Ossona de Mendez

Authors
  1. J. Nešetřil
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  2. P. Ossona de Mendez
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Correspondence to P. Ossona de Mendez.

Additional information

Supported by grant ERCCZ LL-1201 of the Czech Ministry of Education and LEA STRUCO.

J. Nešetřil: Supported by grant CE-ITI P202/12/G061 of GAČR.

P. Ossona de Mendez: Supported by ANR STINT.

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Nešetřil, J., de Mendez, P.O. A distributed low tree-depth decomposition algorithm for bounded expansion classes. Distrib. Comput. 29, 39–49 (2016). https://doi.org/10.1007/s00446-015-0251-x

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