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On Lawson’s Oriented Walk in Random Delaunay Triangulations

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Fundamentals of Computation Theory (FCT 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2751))

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Abstract

In this paper we study the performance of Lawson’s Oriented Walk, a 25-year old randomized point location algorithm without any preprocessing and extra storage, in 2-dimensional Delaunay triangulations. Given n pseudo-random points drawn from a convex set C with unit area and their Delaunay triangulation \(\mathcal{D}\), we prove that the algorithm locates a query point q in \(\mathcal{D}\) in expected \(O(\sqrt{n {\rm log }n})\) time. We also present an improved version of this algorithm, Lawson’s Oriented Walk with Sampling, which takes expected O(n 1/3) time. Our technique is elementary and the proof is in fact to relate Lawson’s Oriented Walk with Walkthrough, another well-known point location algorithm without preprocessing. Finally, we present empirical results to compare these two algorithms with their siblings, Walkthrough and Jump&Walk.

The research is partially supported by NSF CARGO grant DMS-0138065 and a MONTS grant.

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References

  1. Asano, T., Edahiro, M., Imai, H., Iri, M., Murota, K.: Practical use of bucketing techniques in computational geometry. In: Toussaint, G.T. (ed.) Computational Geometry, pp. 153–195. North-Holland, Amsterdam (1985)

    Google Scholar 

  2. Aronov, B., Fortune, S.: Average-case ray shooting and minimum weight triangulations. In: Proceedings of the 13th Symposium on Computational Geometry, pp. 203–212 (1997)

    Google Scholar 

  3. Arya, S., Cheng, S.W., Mount, D., Ramesh, H.: Efficient expected-case algorithms for planar point location. In: Proceedings of the 7th Scand. Workshop on Algorithm Theory, pp. 353–366 (2000)

    Google Scholar 

  4. Arya, S., Malamatos, T., Mount, D.: Nearly optimal expected-case planar point location. In: Proceedings of the 41th IEEE Symp on Foundation of Computer Science, pp. 208–218 (2000)

    Google Scholar 

  5. Arya, S., Malamatos, T., Mount, D.: A simple entropy-based algorithm for planar point location. In: Proceedings of the 12th ACM/SIAM Symp on Discrete Algorithms, pp. 262–268 (January 2001)

    Google Scholar 

  6. Arya, S., Malamatos, T., Mount, D.: Entropy-preserving cuttings and space-efficient planar point location. In: Proceedings of the 12th ACM/SIAM Symp on Discrete Algorithms, January 2001, pp. 256–261. SIAM, Philadelphia (2001)

    Google Scholar 

  7. Bose, P., Devroye, L.: Intersections with random geometric objects. Comp. Geom. Theory and Appl. 10, 139–154 (1998)

    MATH  MathSciNet  Google Scholar 

  8. Boissonnat, J., Devillers, O., Teillaud, M., Yvinc, M.: Triangulations in CGAL triangulation. In: Proc. 16th Symp. On Computational Geometry, pp. 11–18 (2000)

    Google Scholar 

  9. Bern, M., Eppstein, D., Yao, F.: The expected extremes in a Delaunay triangulation. International Journal of Computational Geometry & Applications 1, 79–91 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bowyer, A.: Computing Dirichlet tessellations. The Computer Journal 24, 162–166 (1981)

    Article  MathSciNet  Google Scholar 

  11. Brisson, E.: Representing geometric structures in d dimensions: Topology and Order. Discrete & Computational Geometry 9(4), 387–426 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Devillers, O.: Improved incremental randomized Delaunay triangulation. In: Proceedings of the 14th Symposium on Computational Geometry, pp. 106–115 (1998)

    Google Scholar 

  13. De Floriani, L., Falcidieno, B., Nagy, G., Pienovi, C.: On sorting triangles in a Delaunay tessellation. Algorithmica 6, 522–532 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Devroye, L., Lemaire, C., Moreau, J.-M.: Fast Delaunay point location with search structures. In: Proceedings of the 11th Canadian Conf. on Computational Geometry, pp. 136–141 (1999)

    Google Scholar 

  15. Devroye, L., Mücke, E.P., Zhu, B.: A note on point location in Delaunay triangulations of random points. Algorithmica, Special Issue on Average Case Analysis of Algorithms 22(4), 477–482 (1998)

    MATH  Google Scholar 

  16. Dobkin, D.P., Laszlo, M.J.: Primitives for the manipulation of three dimensional subdivisions. Algorithmica 4(1), 3–32 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Devillers, O., Pion, S., Teillaud, M.: Walking in a triangulation. In: Proceedings of 17th ACM Symposium on Computational Geometry (SCG 2001), pp. 106–114 (2001)

    Google Scholar 

  18. Edelsbrunner, H.: An acyclicity theorem for cell complexes in d dimensions. Combinatorica 10(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goodrich, M.T., Orletsky, M., Ramaiyer, K.: Methods for achieving fast query times in point location data structures. In: Proceedings of Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1997), pp. 757–766 (1997)

    Google Scholar 

  20. Green, P.J., Sibson, R.: Computing Dirichlet tessellations in the plane. The Computer Journal 21, 168–173 (1978)

    MATH  MathSciNet  Google Scholar 

  21. Guibas, L.J., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphics 4(2), 74–123 (1985)

    Article  MATH  Google Scholar 

  22. Hershberger, J., Suri, S.: A pedestrian approach to ray shootings: shoot a ray, take a walk. J. Algorithms 18, 403–431 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lawson, C.L.: Software for C1 surface interpolation. In: Rice, J.R. (ed.) Mathematical Software III, pp. 161–194. Academic Press, New York (1977)

    Google Scholar 

  24. Mücke, E.P.: Shapes and Implementations in Three-Dimensional Geometry. Ph.D. thesis. Technical Report UIUCDCS-R-93-1836. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois (1993)

    Google Scholar 

  25. Mücke, E.P., Saias, I., Zhu, B.: Fast randomized point location without preprocessing in two and three-dimensional Delaunay triangulations. Comp. Geom. Theory and Appl. Special Issue for SoCG 1996 12(1/2), 63–83 (1999)

    MATH  Google Scholar 

  26. Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)

    Google Scholar 

  27. Shewchuk, J.R.: Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Proceedings of the First ACM Workshop on Applied Computational Geometry, pp. 124–133 (1996)

    Google Scholar 

  28. Snoeyink, J.: Point location. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 559–574. CRC Press, Boca Raton (1997)

    Google Scholar 

  29. Trease, H., George, D., Gable, C., Fowler, J., Linnbur, E., Kuprat, A., Khamayseh, A.: The X3D Grid Generation System. In: Proceedings of the 5th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 239–244 (1996)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Montana State University, Bozeman, MT, 59717-3880, USA

    Binhai Zhu

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  1. Binhai Zhu
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Lund University, 22100, Lund, Sweden

    Andrzej Lingas

  2. School of Technology and Society, Malmö University, SE-205 06, Malmö, Sweden

    Bengt J. Nilsson

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© 2003 Springer-Verlag Berlin Heidelberg

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Zhu, B. (2003). On Lawson’s Oriented Walk in Random Delaunay Triangulations. In: Lingas, A., Nilsson, B.J. (eds) Fundamentals of Computation Theory. FCT 2003. Lecture Notes in Computer Science, vol 2751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45077-1_21

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  • DOI: https://doi.org/10.1007/978-3-540-45077-1_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40543-6

  • Online ISBN: 978-3-540-45077-1

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