The θ5-Graph is a Spanner

  • Prosenjit Bose
  • Pat Morin
  • André van Renssen
  • Sander Verdonschot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)

Abstract

Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most \(\sqrt{50 + 22\sqrt{5}}\) ≈ 9.960 . This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument, giving a, possibly self-intersecting, path between any two vertices, whose length is at most \(\sqrt{50 + 22 \sqrt{5}}\) times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of \(1\over2\) (11\(\sqrt{5}\)-17) ≈ 3.798.

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References

  1. 1.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press (2007)Google Scholar
  2. 2.
    Clarkson, K.: Approximation algorithms for shortest path motion planning. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC 1987), pp. 56–65 (1987)Google Scholar
  3. 3.
    Keil, J.: Approximating the complete Euclidean graph. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 208–213. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  4. 4.
    Ruppert, J., Seidel, R.: Approximating the d-dimensional complete Euclidean graph. In: Proceedings of the 3rd Canadian Conference on Computational Geometry (CCCG 1991), pp. 207–210 (1991)Google Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    El Molla, N.M.: Yao spanners for wireless ad hoc networks. Master’s thesis, Villanova University (2009)Google Scholar
  7. 7.
    Barba, L., Bose, P., De Carufel, J.-L., van Renssen, A., Verdonschot, S.: On the stretch factor of the theta-4 graph. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 109–120. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Rysavy, P.: Wireless broadband and other fixed-wireless systems, http://www.networkcomputing.com/netdesign/bb1.html (accessed December 3, 2012)
  9. 9.
    SATEL Oy. What is a radio modem?, http://www.satel.com/products/what-is-a-radio-modem (accessed December 3, 2012)
  10. 10.
    Morin, P., Verdonschot, S.: On the average number of edges in theta graphs. CoRR, abs/1304.3402 (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Pat Morin
    • 1
  • André van Renssen
    • 1
  • Sander Verdonschot
    • 1
  1. 1.School of Computer ScienceCarleton UniversityCanada

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