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Algorithmica

, Volume 61, Issue 1, pp 207–225 | Cite as

Geometric Spanners for Weighted Point Sets

  • Mohammad Ali AbamEmail author
  • Mark de Berg
  • Mohammad Farshi
  • Joachim Gudmundsson
  • Michiel Smid
Open Access
Article

Abstract

Let (S,d) be a finite metric space, where each element pS has a non-negative weight w (p). We study spanners for the set S with respect to the following weighted distance function:
$$\mathbf{d}_{\omega}(p,q)=\left\{\begin{array}{ll}0&\mbox{ if $p=q$,}\\ \operatorname {w}(p)+\mathbf{d}(p,q)+ \operatorname {w}(q)&\mbox{ if $p\neq q$.}\end{array}\right.$$
We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d ω -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 weighted point 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(nlog n). 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.

Keywords

Computational geometry Geometric spanners Doubling dimension Well-separated pair decomposition Semi-separated pair decomposition Geodesic metric 

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Mohammad Ali Abam
    • 1
    Email author
  • Mark de Berg
    • 2
  • Mohammad Farshi
    • 3
  • Joachim Gudmundsson
    • 4
  • Michiel Smid
    • 5
  1. 1.Department of Computer ScienceTechnische Universität DortmundDortmundGermany
  2. 2.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands
  3. 3.Department of Computer ScienceYazd UniversityYazdIran
  4. 4.NICTASydneyAustralia
  5. 5.School of Computer ScienceCarleton UniversityOttawaCanada

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