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On Plane Constrained Bounded-Degree Spanners

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Abstract

Let P be a finite 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 \(\mathord { Vis}(P,S)\), 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-\(\theta _6\)-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of \(\mathord { Vis}(P,S)\). We then show how to construct a plane 6-spanner of \(\mathord { Vis}(P,S)\) with maximum degree \(6+c\), where c is the maximum number of segments of S incident to a vertex.

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References

  1. Bonichon, N., Gavoille, C., Hanusse, N., Ilcinkas, D.: Connections between theta-graphs, Delaunay triangulations, and orthogonal surfaces. In: Proceedings of the International Conference on Graph Theoretic Concepts in Computer Science, pp. 266–278 (2010)

  2. Bonichon, N., Gavoille, C., Hanusse, N., Perkovic, L.: Plane spanners of maximum degree six. In: Proceedings of the International Colloquium on Automata, Languages and Programming, pp. 19–30 (2010)

  3. Bonichon, N., Kanj, I., Perković, L., Xia, G.: There are plane spanners of degree 4 and moderate stretch factor. Discrete Comput Geom 53(3), 514–546 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bose, P., De Carufel, J.L., van Renssen, A.: Constrained generalized Delaunay graphs are plane spanners. In: Proceedings of the Computational Intelligence in Information Systems (CIIS 2016), Advances in Intelligent Systems and Computing, vol. 532, pp. 281–293 (2016)

  5. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: On plane constrained bounded-degree spanners. In: Proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012), Lecture Notes in Computer Science, vol. 7256, pp. 85–96 (2012)

  6. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM J. Comput. 44(6), 1626–1649 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bose, P., Keil, J.M.: On the stretch factor of the constrained Delaunay triangulation. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering, pp. 25–31 (2006)

  8. Bose, P., van Renssen, A.: Upper bounds on the spanning ratio of constrained theta-graphs. In: Proceedings of the 11th Latin American Symposium on Theoretical Informatics (LATIN 2014), Lecture Notes in Computer Science, vol. 8392, pp. 108–119 (2014)

  9. Bose, P., Smid, M.: On plane geometric spanners: a survey and open problems. Comput. Geom. Theory Appl. CGTA 46(7), 818–830 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clarkson, K.: Approximation algorithms for shortest path motion planning. In: Proceedings of the Annual ACM Symposium on Theory of Computing, pp. 56–65 (1987)

  11. Das, G.: The visibility graph contains a bounded-degree spanner. In: Proceedings of the Canadian Conference on Computational Geometry, pp. 70–75 (1997)

  12. Kanj, I., Perkovic, L., Turkoglu, D.: Degree four plane spanners: simpler and better. J. Comput. Geom. 8(2), 3–31 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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Acknowledgements

This work is supported in part by the Natural Science and Engineering Research Council of Canada, Carleton University’s President’s 2010 Doctoral Fellowship, the Ontario Ministry of Research and Innovation, and the Danish Council for Independent Research, Natural Sciences, Grant DFF-1323-00247, and JST ERATO Grant No. JPMJER1201, Japan.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, 5302 Herzberg Laboratories, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Prosenjit Bose & Sander Verdonschot

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Rolf Fagerberg

  3. School of Information Technologies, University of Sydney, Sydney, 2006, NSW, Australia

    André van Renssen

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  1. Prosenjit Bose
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  2. Rolf Fagerberg
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  3. André van Renssen
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  4. Sander Verdonschot
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Correspondence to André van Renssen.

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An extended abstract of this paper appeared in the proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012) [5].

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Bose, P., Fagerberg, R., van Renssen, A. et al. On Plane Constrained Bounded-Degree Spanners. Algorithmica 81, 1392–1415 (2019). https://doi.org/10.1007/s00453-018-0476-8

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  • DOI: https://doi.org/10.1007/s00453-018-0476-8

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