Distance-Sensitive Planar Point Location

  • Boris Aronov
  • Mark de Berg
  • Marcel Roeloffzen
  • Bettina Speckmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


Let \(\mathcal{S}\) be a connected planar polygonal subdivision with n edges and of total area 1. We present a data structure for point location in \(\mathcal{S}\) where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time \(O(1+\min(\log \frac{1}{\Delta_{p}}, \log n))\), where Δp is the distance of the query point p to the boundary of the region containing p. Our structure is based on the following result: any simple polygon P can be decomposed into a linear number of convex quadrilaterals with the following property: for any point p ∈ P, the quadrilateral containing p has area \(\Omega(\Delta_{p}^2)\).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arya, S., Malamatos, T., Mount, D.M.: A simple entropy-based algorithm for planar point location. ACM Trans. Algorithms 3, article 17 (2007)Google Scholar
  2. 2.
    Arya, S., Malamatos, T., Mount, D.M., Wong, K.C.: Optimal expected-case planar point location. SIAM J. Comput. 37, 584–610 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)Google Scholar
  4. 4.
    Bern, M.: Triangulations and mesh generation. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., ch. 25. Chapman & Hall/CRC (2004)Google Scholar
  5. 5.
    Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. J. of Computer and System Sciences 48(3), 384–409 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bern, M., Mitchell, S., Ruppert, J.: Linear-size nonobtuse triangulation of polygons. Discrete & Computational Geometry 14(1), 411–428 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete & Computational Geometry 6, 485–524 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Collette, S., Dujmović, V., Iacono, J., Langerman, S., Morin, P.: Entropy, triangulation, and point location in planar subdivisions. ACM Trans. Algorithms 8(3), 1–18 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer (1987)Google Scholar
  10. 10.
    Fortune, S.: A fast algorithm for polygon containment by translation. In: Brauer, W. (ed.) ICALP 1985. LNCS, vol. 194, pp. 189–198. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  11. 11.
    Iacono, J.: Expected asymptotically optimal planar point location. Computational Geometry 29(1), 19–22 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Iacono, J., Mulzer, W.: A static optimality transformation with applications to planar point location. Int. J. of Comput. Geom. and Appl. 22(4), 327–340 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Knuth, D.E.: Sorting and Searching, 2nd edn. The Art of Computer Programming, vol. 3. Addison-Wesley (1998)Google Scholar
  15. 15.
    Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms 18(3), 548–585 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Shannon, C.E.: A mathematical theory of communication. Bell Sys. Tech. Journal 27, 379–423, 623–656 (1948)MathSciNetMATHGoogle Scholar
  17. 17.
    Snoeyink, J.: Point location. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., ch. 34. Chapman & Hall/CRC (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Boris Aronov
    • 1
  • Mark de Berg
    • 2
  • Marcel Roeloffzen
    • 2
  • Bettina Speckmann
    • 2
  1. 1.Dept. of Computer Science and EngineeringPolytechnic Institute of NYUUSA
  2. 2.Dept. of Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations