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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
Article

Abstract

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.

Keywords

Voronoi Diagram Discrete Comput Geom Hausdorff Distance Voronoi Cell Lower Envelope 
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.

References

  1. 1.
    Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry. Contemporary Mathematics, vol. 223, pp. 1–56. American Mathematical Society, Providence (1999) Google Scholar
  2. 2.
    Agarwal, P.K., Klein, R., Knauer, C., Sharir, M.: Computing the detour of polygonal curves. Technical Report B 02-03, Freie Universität Berlin, Fachbereich Mathematik und Informatik (2002) Google Scholar
  3. 3.
    Agarwal, P.K., Sharir, M., Toledo, S.: Applications of parametric searching in geometric optimization. J. Algorithms 17, 292–318 (1994) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Aichholzer, O., Aurenhammer, F., Icking, C., Klein, R., Langetepe, E., Rote, G.: Generalized self-approaching curves. Discrete Appl. Math. 109, 3–24 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alt, H., Guibas, L.J.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 121–153. Elsevier, Amsterdam (2000) Google Scholar
  6. 6.
    Alt, H., Knauer, C., Wenk, C.: Comparison of distance measures for planar curves. Algorithmica 38, 45–58 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 201–290. Elsevier, Amsterdam (2000) Google Scholar
  8. 8.
    Bose, P., Morin, P.: Competitive online routing in geometric graphs. Theor. Comput. Sci. 324, 273–288 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete Comput. Geom. 22(4), 547–567 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ebbers-Baumann, A., Klein, R., Langetepe, E., Lingas, A.: A fast algorithm for approximating the detour of a polygonal chain. Comput. Geom. Theory Appl. 27, 123–134 (2004) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Edelsbrunner, H., Guibas, L.J., Sharir, M.: The complexity and construction of many faces in arrangements of lines and of segments. Discrete Comput. Geom. 5, 161–196 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Erickson, J.: New lower bounds for Hopcroft’s problem. Discrete Comput. Geom. 16, 389–418 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fortune, S.J.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153–174 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Grüne, A.: Umwege in Polygonen. Master’s thesis, Institut für Informatik I, Universität Bonn (2002) Google Scholar
  15. 15.
    Guibas, L.J., Sharir, M., Sifrony, S.: On the general motion planning problem with two degrees of freedom. Discrete Comput. Geom. 4, 491–521 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Icking, C., Klein, R.: Searching for the kernel of a polygon: a competitive strategy. In: Proceedings of the 11th Annual Symposium on Computational Geometry, pp. 258–266 (1995) Google Scholar
  18. 18.
    Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Philos. Soc. 125, 441–453 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Koltun, V.: Almost tight upper bounds for vertical decompositions in four dimensions. J. ACM 51, 699–730 (2004) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Langerman, S., Morin, P., Soss, M.: Computing the maximum detour and spanning ratio of planar chains, trees and cycles. In: Proceedings of the 19th International Symposium on Theoretical Aspects of Computer Science (STACS 2002). Lecture Notes in Computer Science, vol. 2285, pp. 250–261. Springer, Berlin (2002) Google Scholar
  21. 21.
    Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993) CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean graphs. SIAM J. Comput. 30(3), 978–989 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rote, G.: Curves with increasing chords. Math. Proc. Camb. Philos. Soc. 115, 1–12 (1994) zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995) zbMATHGoogle Scholar

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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