Approximating the Average Stretch Factor of Geometric Graphs

  • Siu-Wing Cheng
  • Christian Knauer
  • Stefan Langerman
  • Michiel Smid
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6506)

Abstract

Let G be a geometric graph whose vertex set S is a set of n points in ℝd. The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the sum of all \(n \choose 2\) stretch factors determined by all pairs of points in S. We show that for paths, cycles, and trees, this sum can be approximated, within a factor of 1 + ε, in O(npolylog(n)) time. For plane graphs, we present a (2 + ε)-approximation algorithm with running time O(n5/3polylog(n)), and a (4 + ε)-approximation algorithm with running time O(n3/2polylog(n)).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Klein, R., Knauer, C., Langerman, S., Morin, P., Sharir, M., Soss, M.: Computing the detour and spanning ratio of paths, trees, and cycles in 2D and 3D. Discrete & Computational Geometry 39, 17–37 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arikati, S., Chen, D.Z., Chew, L.P., Das, G., Smid, M., Zaroliagis, C.D.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 514–528. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  3. 3.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM 42, 67–90 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Farshi, M., Giannopoulos, P., Gudmundsson, J.: Improving the stretch factor of a geometric graph by edge augmentation. SIAM Journal on Computing 38, 226–240 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing 16, 1004–1022 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Klein, R., Knauer, C., Narasimhan, G., Smid, M.: On the dilation spectrum of paths, cycles, and trees. Computational Geometry: Theory and Applications 42, 923–933 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean graphs. SIAM Journal on Computing 30, 978–989 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Wulff-Nilsen, C.: Wiener index and diameter of a planar graph in subquadratic time. In: EuroCG, pp. 25–28 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Siu-Wing Cheng
    • 1
  • Christian Knauer
    • 2
  • Stefan Langerman
    • 3
  • Michiel Smid
    • 4
  1. 1.Department of Computer Science and EngineeringHKUSTHong Kong
  2. 2.Institute of Computer ScienceUniversität BayreuthGermany
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBelgium
  4. 4.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations