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Constructing Delaunay Triangulations along Space-Filling Curves

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Algorithms - ESA 2009 (ESA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5757))

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Abstract

Incremental construction con BRIO using a space-filling curve order for insertion is a popular algorithm for constructing Delaunay triangulations. So far, it has only been analyzed for the case that a worst-case optimal point location data structure is used which is often avoided in implementations. In this paper, we analyze its running time for the more typical case that points are located by walking. We show that in the worst-case the algorithm needs quadratic time, but that this can only happen in degenerate cases. We show that the algorithm runs in O(n logn) time under realistic assumptions. Furthermore, we show that it runs in expected linear time for many random point distributions.

This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program ’Combinatorics, Geometry, and Computation’ (No. GRK 588/2) and by the Netherlands’ Organisation for Scientific Research (NWO) under BRICKS/FOCUS grant number 642.065.503 and project no. 639.022.707.

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References

  1. Amenta, N., Choi, S., Rote, G.: Incremental constructions con BRIO. In: Proc. 19th Annu. ACM Sympos. Comput. Geom., pp. 211–219. ACM Press, New York (2003)

    Google Scholar 

  2. Attali, D., Boissonnat, J.-D.: A linear bound on the complexity of the Delaunay triangulation of points on polyhedral surfaces. Discrete Comput. Geom. 31(3), 369–384 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bentley, J.L., Weide, B.W., Yao, A.C.: Optimal expected-time algorithms for closest-point problems. ACM Trans. Math. Softw. 6, 563–580 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buchin, K.: Organizing Point Sets: Space-Filling Curves, Delaunay Tessellations of Random Point Sets, and Flow Complexes. PhD thesis, Free University Berlin (2007), http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000003494

  5. Buchin, K., Mulzer, W.: Delaunay triangulations in \({O}(\text{sort}(n))\) and other transdichotomous and hereditary algorithms in computational geometry. In: Proc. 50th Annu. IEEE Sympos. Found. Comput. Sci. (to appear, 2009)

    Google Scholar 

  6. Damerow, V., Meyer auf der Heide, F., Räcke, H., Scheideler, C., Sohler, C.: Smoothed motion complexity. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 161–171. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Delage, C.: Spatial sorting. In: CGAL Editorial Board (eds), CGAL User and Reference Manual (2007)

    Google Scholar 

  8. Devillers, O.: Randomization yields simple O(n log\(^{\mbox{*}}\) n) algorithms for difficult Omega(n) problems. Int. J. Comput. Geometry Appl. 2(1), 97–111 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Devroye, L., Mücke, E., Zhu, B.: A note on point location in Delaunay triangulations of random points. Algorithmica 22, 477–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dwyer, R.A.: Higher-dimensional Voronoi diagrams in linear expected time. Discrete Comput. Geom. 6(4), 343–367 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Erickson, J.: Dense point sets have sparse Delaunay triangulations: or but not too nasty. In: Proc. 13th Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. 125–134 (2002)

    Google Scholar 

  12. Hilbert, D.: Ueber die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann. 38, 459–460 (1891)

    Article  MathSciNet  MATH  Google Scholar 

  13. Katajainen, J., Koppinen, M.: Constructing Delaunay triangulations by merging buckets in quadtree order. Fundam. Inform. 11, 275–288 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, Y., Snoeyink, J.: A comparison of five implementations of 3d Delaunay tesselation. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 439–458. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  16. Mücke, E.P., Saias, I., Zhu, B.: Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations. Comput. Geom. Theory Appl. 12(1-2), 63–83 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Platzman, L.K., Bartholdi III, J.J.: Spacefilling curves and the planar travelling salesman problem. J. ACM 36(4), 719–737 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Su, P., Drysdale, R.: A comparison of sequential Delaunay triangulation algorithms. Comput. Geom. Theory Appl. 7, 361–386 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhou, S., Jones, C.B.: HCPO: an efficient insertion order for incremental Delaunay triangulation. Inf. Process. Lett. 93(1), 37–42 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, Netherlands

    Kevin Buchin

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  1. Kevin Buchin
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Editor information

Editors and Affiliations

  1. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Amos Fiat

  2. Fakultät für Informatik, Universität Karlsruhe (TH), Am Fasanengarten 5, 76131, Karlsruhe, Germany

    Peter Sanders

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Buchin, K. (2009). Constructing Delaunay Triangulations along Space-Filling Curves. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_11

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  • DOI: https://doi.org/10.1007/978-3-642-04128-0_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04127-3

  • Online ISBN: 978-3-642-04128-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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