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On Plane Constrained Bounded-Degree Spanners

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7256)

Abstract

Let P be a set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted \(\mathord{\it Vis}(P,S)\), has vertex set P and an edge for each pair of vertices u,v in P for which no line segment of S properly intersects uv. We show that the constrained half-θ 6-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of \(\mathord{\it Vis}(P,S)\). We then show how to construct a plane 6-spanner of \(\mathord{\it Vis}(P,S)\) with maximum degree 6 + c, where c is the maximum number of segments adjacent to a vertex.

Keywords

  • Induction Hypothesis
  • Positive Cone
  • Visibility Graph
  • Negative Cone
  • Canonical Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported in part by NSERC and the Danish Council for Independent Research. Due to space constraints, some proofs are omitted and available in the full version of this paper.

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References

  1. Bonichon, N., Gavoille, C., Hanusse, N., Ilcinkas, D.: Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 266–278. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  2. Bonichon, N., Gavoille, C., Hanusse, N., Perković, L.: Plane Spanners of Maximum Degree Six. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 19–30. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  3. Bose, P., Keil, J.M.: On the Stretch Factor of the Constrained Delaunay Triangulation. In: Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering, pp. 25–31 (2006)

    Google Scholar 

  4. Clarkson, K.: Approximation Algorithms for Shortest Path Motion Planning. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 56–65 (1987)

    Google Scholar 

  5. Das, G.: The Visibility Graph Contains a Bounded-Degree Spanner. In: Proceedings of the 9th Canadian Conference on Computational Geometry, pp. 70–75 (1997)

    Google Scholar 

  6. Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press (2007)

    Google Scholar 

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© 2012 Springer-Verlag Berlin Heidelberg

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Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S. (2012). On Plane Constrained Bounded-Degree Spanners. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_8

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  • DOI: https://doi.org/10.1007/978-3-642-29344-3_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29343-6

  • Online ISBN: 978-3-642-29344-3

  • eBook Packages: Computer ScienceComputer Science (R0)