Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 17–37 | Cite as

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

  • Pankaj K. Agarwal
  • Rolf Klein
  • Christian Knauer
  • Stefan Langerman
  • Pat Morin
  • Micha Sharir
  • Michael Soss


The detour and spanning ratio of a graph G embedded in \(\mathbb{E}^{d}\) measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(nlog n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain O(nlog 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in \(\mathbb{E}^{3}\) , and show that computing the detour in \(\mathbb{E}^{3}\) is at least as hard as Hopcroft’s problem.


Voronoi Diagram Discrete Comput Geom Hausdorff Distance Voronoi Cell Lower Envelope 
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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Rolf Klein
    • 2
  • Christian Knauer
    • 3
  • Stefan Langerman
    • 4
  • Pat Morin
    • 5
  • Micha Sharir
    • 6
    • 7
  • Michael Soss
    • 8
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Institut für Informatik IUniversität BonnBonnGermany
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany
  4. 4.FNRS, Département d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium
  5. 5.School of Computer ScienceCarleton UniversityOttawaCanada
  6. 6.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  7. 7.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  8. 8.Foreign Exchange Strategy DivisionGoldman SachsNew YorkUSA

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