Keywords and Synonyms
Detour; Spanning ratio; Stretch factor
Problem Definition
Notations
Let \( { G=(V,E) } \) be a plane geometric network, whose vertex set V is a finite set of point sites in \( { {\mathbb R}^2 } \), connected by an edge set E of non-crossing straight line segments with endpoints in V. For two points \( { p \not= q \in V } \) let \( { \xi_G(p,q) } \) denote a shortest path from p to q in G. Then
is the detour one encounters when using network G, in order to get from p to q, instead of walking straight. Here, \( { |.| } \) denotes the Euclidean length.
The dilation of G is defined by
This value is also known as the spanning ratio or the stretch factor of G. It should, however, not be confused with the geometric dilation of a network, where the points on the edges are also being considered, in addition to the vertices.
Given a finite set Sof points in the...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Recommended Reading
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)
Das, G., Joseph, D.: Which Triangulations Approximate the Complete Graph? In: Proc. Int. Symp. Optimal Algorithms. LNCS 401, pp. 168–192. Springer, Berlin (1989)
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)
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)
Eppstein, D.: The Geometry Junkyard. http://www.ics.uci.edu/~eppstein/junkyard/dilation-free/
Eppstein, D.: Spanning Trees and Spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (1999)
Eppstein, D., Wortman, K.A.: Minimum Dilation Stars. In: Proc. 21st ACM Symp. Comp. Geom. (SoCG), Pisa, 2005, pp. 321–326
Hillar, C.J., Rhea, D.L. A Result about the Density of Iterated Line Intersections. Comput. Geom.: Theory Appl. 33(3), 106–114 (2006)
Ismailescu, D., Radoičić, R.: A Dense Planar Point Set from Iterated Line Intersections. Comput. Geom. Theory Appl. 27(3), 257–267 (2004)
Keil, J.M., Gutwin, C.A.: The Delaunay Triangulation Closely Approximates the Complete Euclidean Graph. Discret. Comput. Geom. 7, 13–28 (1992)
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. 264–272
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
Klein, R. (2008). Dilation of Geometric Networks. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_111
Download citation
DOI: https://doi.org/10.1007/978-0-387-30162-4_111
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-30770-1
Online ISBN: 978-0-387-30162-4
eBook Packages: Computer ScienceReference Module Computer Science and Engineering