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Geometric Spanners for Weighted Point Sets

  • Mohammad Ali Abam
  • Mark de Berg
  • Mohammad Farshi
  • Joachim Gudmundsson
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

Let (S,d) be a finite metric space, where each element p ∈ S has a non-negative weight w(p). We study spanners for the set S with respect to weighted distance function d w , where d w (p,q) is w(p) + d(p,q) + wq if p ≠ q and 0 otherwise. We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d w -metric. For any given ε> 0, we can apply our method to obtain (5 + ε)-spanners with a linear number of edges for three cases: points in Euclidean space ℝ d , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝ d where d is the geodesic distance function.

We also describe an alternative method that leads to (2 + ε)-spanners for points in ℝ d and for points on the boundary of a convex body in ℝ d . The number of edges in these spanners is O(nlogn). This bound on the stretch factor is nearly optimal: in any finite metric space and for any ε> 0, it is possible to assign weights to the elements such that any non-complete graph has stretch factor larger than 2 − ε.

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References

  1. 1.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Har-Peled, S., Mendel, M.: Fast construction of nets in low-dimensional metrics and their applications. SIAM J. on Computing 35, 1148–1184 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: STOC 2004, pp. 281–290 (2004)Google Scholar
  4. 4.
    Bose, P., Carmi, P., Couture, M.: Spanners of additively weighted point sets. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 367–377. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Abam, M.A., de Berg, M., Farshi, M., Gudmundsson, J.: Region-fault tolerant geometric spanners. In: SODA 2007, pp. 1–10 (2007)Google Scholar
  6. 6.
    Varadarajan, K.R.: A divide-and-conquer algorithm for min-cost perfect matching in the plane. In: FOCS 1998, pp. 320–331 (1998)Google Scholar
  7. 7.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. of the ACM 42, 67–90 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.: An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. J. of the ACM 45, 891–923 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gottlieb, L.A., Roditty, L.: An optimal dynamic spanner for doubling metric spaces. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 478–489. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Cole, R., Gottlieb, L.A.: Searching dynamic point sets in spaces with bounded doubling dimension. In: STOC 2006, pp. 574–583 (2006)Google Scholar
  11. 11.
    Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: SODA 1993, pp. 291–300 (1993)Google Scholar
  12. 12.
    Agarwal, P.K., Har-Peled, S., Sharir, M., Varadarajan, K.R.: Approximate shortest paths on a convex polytope in three dimensions. J. of the ACM 44, 567–584 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hansel, G.: Nombre minimal de contacts de fermeture nécessaires pour réaliser une fonction booléenne symétrique de n variables. Comptes Rendus de l’Académie des Sciences 258, 6037–6040 (1964)MathSciNetMATHGoogle Scholar
  14. 14.
    Bollobás, B., Scott, A.D.: On separating systems. European J. of Combinatorics 28, 1068–1071 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Mohammad Ali Abam
    • 1
  • Mark de Berg
    • 2
  • Mohammad Farshi
    • 3
  • Joachim Gudmundsson
    • 4
  • Michiel Smid
    • 3
  1. 1.MADALGO CenterAarhus UniversityDenmark
  2. 2.Department of Computer ScienceTU EindhovenThe Netherlands
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada
  4. 4.NICTASydneyAustralia

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