Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

Dilation of Geometric Networks

2005; Ebbers-Baumann, Grüne, Karpinski, Klein, Kutz, Knauer, Lingas
  • Rolf Klein
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_111

Keywords and Synonyms

Detour; Spanning ratio; Stretch factor              

Problem Definition

Notations

Let \( { G=(V,E) } \)

Keywords

Delaunay Triangulation Straight Line Segment Coffee Shop Stretch Factor Edge Crossing 
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.
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Recommended Reading

  1. 1.
    Aronov, B., de Berg, M., Cheong, O., Gudmundsson, J., Haverkort, H., Vigneron, A.: Sparse Geometric Graphs with Small Dilation. 16th International Symposium ISAAC 2005, Sanya. In: Deng, X., Du, D. (eds.) Algorithms and Computation, Proceedings. LNCS, vol. 3827, pp. 50–59. Springer, Berlin (2005)Google Scholar
  2. 2.
    Das, G., Joseph, D.: Which Triangulations Approximate the Complete Graph? In: Proc. Int. Symp. Optimal Algorithms. LNCS 401, pp. 168–192. Springer, Berlin (1989)CrossRefGoogle Scholar
  3. 3.
    Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay Graphs Are Almost as Good as Complete Graphs. Discret. Comput. Geom. 5, 399–407 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ebbers‐Baumann, A., Gruene, A., Karpinski, M., Klein, R., Knauer, C., Lingas, A.: Embedding Point Sets into Plane Graphs of Small Dilation. Int. J. Comput. Geom. Appl. 17(3), 201–230 (2007)zbMATHCrossRefGoogle Scholar
  5. 5.
    Eppstein, D.: The Geometry Junkyard. http://www.ics.uci.edu/~eppstein/junkyard/dilation-free/
  6. 6.
    Eppstein, D.: Spanning Trees and Spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (1999)Google Scholar
  7. 7.
    Eppstein, D., Wortman, K.A.: Minimum Dilation Stars. In: Proc. 21st ACM Symp. Comp. Geom. (SoCG), Pisa, 2005, pp. 321–326Google Scholar
  8. 8.
    Hillar, C.J., Rhea, D.L. A Result about the Density of Iterated Line Intersections. Comput. Geom.: Theory Appl. 33(3), 106–114 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ismailescu, D., Radoičić, R.: A Dense Planar Point Set from Iterated Line Intersections. Comput. Geom. Theory Appl. 27(3), 257–267 (2004)zbMATHCrossRefGoogle Scholar
  10. 10.
    Keil, J.M., Gutwin, C.A.: The Delaunay Triangulation Closely Approximates the Complete Euclidean Graph. Discret. Comput. Geom. 7, 13–28 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Klein, R., Kutz, M.: The Density of Iterated Plane Intersection Graphs and a Gap Result for Triangulations of Finite Point Sets. In: Proc. 22nd ACM Symp. Comp. Geom. (SoCG), Sedona (AZ), 2006, pp. 264–272Google Scholar
  12. 12.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press (2007)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Rolf Klein
    • 1
  1. 1.Institute for Computer ScienceUniversity of BonnBonnGermany