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Light Orthogonal Networks with Constant Geometric Dilation

  • Adrian Dumitrescu
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

An orthogonal network for a given set of n points in the plane is an axis-aligned planar straight line graph that connects all input points. We show that for any set of n points in the plane, there is an orthogonal network that (i) is short having a total edge length of O(|T|), where |T| denotes the length of a minimum Euclidean spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most constant times longer than the (Euclidean) distance between u and v.

Keywords

Steiner Tree Narrow Channel Steiner Point Base Side Parallel Edge 
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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References

  1. 1.
    Althöfer, I., et al.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81–100 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Aronov, B., et al.: Sparse geometric graphs with small dilation. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 50–59. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Arya, S., et al.: Euclidean spanners: short, thin, and lanky. In: Proc. 27th STOC, pp. 489–498. ACM Press, New York (1995)Google Scholar
  4. 4.
    Bose, P., Gudmundsson, J., Smid, M.: Constructing plane spanners of bounded degree and low weight. Algorithmica 42, 249–264 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Chandra, B., et al.: New sparseness results on graph spanners. Int. J. Comput. Geometry Appl. 5, 125–144 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Computer Sys. Sci. 39, 205–219 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Das, G., Joseph, D.: Which triangulations approximate the complete graph? In: Djidjev, H.N. (ed.) Optimal Algorithms. LNCS, vol. 401, pp. 168–192. Springer, Heidelberg (1989)Google Scholar
  8. 8.
    Das, G., Narasimhan, G., Salowe, J.S.: A new way to weigh malnourished Euclidean graphs. In: Proc. 6th SODA, pp. 215–222. ACM Press, New York (1995)Google Scholar
  9. 9.
    Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5, 399–407 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Dumitrescu, A., et al.: On the geometric dilation of closed curves, graphs, and point sets. Comput. Geom. 36, 16–38 (2006)CrossRefGoogle Scholar
  11. 11.
    Ebbers-Baumann, A., Grüne, A., Klein, R.: On the geometric dilation of finite point sets. Algorithmica 44, 137–149 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Ebbers-Baumann, A., et al.: A fast algorithm for approximating the detour of a polygonal chain. Comput. Geom. 27, 123–134 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Eppstein, D.: Spanning trees and spanners. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  14. 14.
    Keil, M., Gutwin, C.A.: Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7, 13–28 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Langerman, S., Morin, P., Soss, M.: Computing the maximum detour and spanning ratio of planar chains, trees and cycles. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 250–261. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Levcopoulos, C., Lingas, A.: There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees. Algorithmica 8, 251–256 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    MacGregor Smith, J., Winter, P.: Computing in Euclidean geometry. In: Computational geometry and topological network design, pp. 287–385. World Scientific, Singapore (1992)Google Scholar
  18. 18.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Csaba D. Tóth
    • 2
  1. 1.Department of Computer Science, University of Wisconsin-MilwaukeeUSA
  2. 2.Department of Mathematics, MIT, CambridgeUSA

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