Distributed Computing

, Volume 26, Issue 2, pp 119–137 | Cite as

Distributed minimum dominating set approximations in restricted families of graphs

  • Christoph Lenzen
  • Yvonne-Anne Pignolet
  • Roger Wattenhofer
Article

Abstract

A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity, whose edge sets can be decomposed into a small number of forests. We give two algorithms, a randomized one excelling in its approximation ratio and a uniform deterministic one which is faster and simpler. Finally, we show that in planar graphs, which can be drawn in the Euclidean plane without intersecting edges, a constant approximation factor can be ensured within a constant number of communication rounds.

Keywords

Upper bound Lower bound Unit disk Bounded arboricity Planar 

Notes

Acknowledgments

We would like to thank Topi Musto, Jukka Suomela, and the anonymous reviewers who helped in improving this article and its predecessors in many ways. This work has been partly supported by the Swiss National Fund (SNF) and the Society of Swiss Friends of the Weizmann Institute.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christoph Lenzen
    • 1
  • Yvonne-Anne Pignolet
    • 2
  • Roger Wattenhofer
    • 3
  1. 1.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.ABB Corporate ResearchDättwil-BadenSwitzerland
  3. 3.Computer Engineering and Networks Laboratory (TIK)ETH ZurichSwitzerland

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