On the Central Path Problem

  • Yongding Zhu
  • Jinhui Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

Abstract

In this paper we consider the following Central Path Problem (CPP): Given a set of m arbitrary (i.e., non-simple) polygonal curves Q = {P 1,P 2,…,P m } in 2D space, find a curve P, called central path, that best represents all curves in Q. In order for P to best represent Q, P is required to minimize the maximum distance (measured by the directed Hausdorff distance) to all curves in Q and is the locus of the center of minimal spanning disk of Q. For the CPP problem, a direct approach is to first construct the farthest-path Voronoi diagram FPVD(Q) of Q and then derive the central path from it, which could be rather costly. In this paper, we present a novel approach which computes the central path in an “output-sensitive” fashion. Our approach sweeps a minimal spanning disk through Q and computes only a partial structure of the FPVD(Q) directly related to P. The running time of our approach is thus O((H + mk + n + s)logmlog2 n) and the worst case running time is O(n 22 α(n)logn), where n is the size of Q, s is the total number of self-intersecting points of each individual curve in Q, k is the size of the visited portion of FPVD(Q) by the central path algorithm, and H is the number of intersections between the visited portion of FPVD(Q) and VD(P i )(i = 1,2,…, m).

Keywords

Forward Direction Voronoi Diagram Voronoi Cell Central Path Binary Search Tree 
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.

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References

  1. 1.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Sacristán, V.: The farthest color voronoi diagram and related problems (extended abstract). In: 17th European Workshop Computational Geometry, pp. 113–116 (2001)Google Scholar
  2. 2.
    Aurenhammer, F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Computing Surveys 23(3), 345–405 (1991)CrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment voronoi diagrams. Information Processing Letters 100(6), 220–225 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buchin, K., Buchin, M., Van Kreveld, M., Löffler, M., Silveira, R.I., Wenk, C., Wiratma, L.: Median trajectories. In: Proceedings of the 18th Annual European Conference on Algorithms: Part I, pp. 463–474 (2010)Google Scholar
  5. 5.
    Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.S.: Farthest-polygon voronoi diagrams. In: Proceedings of the 15th Annual European Symposium on Algorithms, pp. 407–418 (2007)Google Scholar
  6. 6.
    de Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. (2000)Google Scholar
  7. 7.
    Fortune, S.: A sweepline algorithm for voronoi diagrams. In: Proceedings of the 2nd Annual Symposium on Computational Geometry, pp. 313–322 (1986)Google Scholar
  8. 8.
    Har-Peled, S., Raichel, B.: The frechet distance revisited and extended. In: Proc. of the 27th Annual ACM Symposium on Computational Geometry, SoCG 2011, pp. 448–457 (2011)Google Scholar
  9. 9.
    Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of voronoi surfaces and its applications. In: Proceedings of the 7th Annual Symposium on Computational Geometry, pp. 194–203 (1991)Google Scholar
  10. 10.
    Zhu, Y., Xu, J.: Improved Algorithms for Farthest Colored Voronoi Diagram of Segments. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 372–386. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yongding Zhu
    • 1
  • Jinhui Xu
    • 1
  1. 1.Department of Computer Science and EngineeringState University of New York at BuffaloBuffaloUSA

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