Dynamic Planar Point Location with Sub-logarithmic Local Updates

  • Maarten Löffler
  • Joseph A. Simons
  • Darren Strash
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


We study planar point location in a collection of disjoint fat regions, and investigate the complexity of local updates: replacing any region by a different region that is “similar” to the original region. (i.e., the size differs by at most a constant factor, and distance between the two regions is a constant times that size).We show that it is possible to create a linear size data structure that allows for insertions, deletions, and queries in logarithmic time, and allows for local updates in sub-logarithmic time on a pointer machine. We also give results parameterized by the fatness and similarity of the objects considered.


Search Tree Query Point Representative Point Local Update Query 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alstrup, S., Husfeldt, T., Rauhe, T.: Marked ancestor problems. In: Proc. 39th Symp. on Foundations of Computer Science, pp. 534–543 (1998)Google Scholar
  2. 2.
    Arge, L., Brodal, G.S., Georgiadis, L.: Improved dynamic planar point location. In: Proc. 47th Symp. on Foundations of Computer Science, pp. 305–314 (2006)Google Scholar
  3. 3.
    Bentley, J.L.: Solutions to Klee’s rectangle problems. Technical report, Carnegie-Mellon Univ., Pittsburgh, PA (1977)Google Scholar
  4. 4.
    Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. J. Comput. Syst. Sci. 48(3), 384–409 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cheng, S.W., Janardan, R.: New results on dynamic planar point location. SIAM J. Comput. 21(5), 972–999 (1992)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Berg, M.d., Gray, C.: Vertical ray shooting and computing depth orders for fat objects. In: SODA, pp. 494–503. ACM Press (2006)Google Scholar
  7. 7.
    Dobkin, D.P., Lipton, R.J.: Multidimensional searching problems. SIAM J. Comput. 5(2), 181–186 (1976)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Finkel, R.A., Bentley, J.L.: Quad trees: A data structure for retrieval on composite keys. Acta Inform. 4, 1–9 (1974)MATHCrossRefGoogle Scholar
  9. 9.
    Fleischer, R.: A simple balanced search tree with o(1) worst-case update time. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 138–146. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  10. 10.
    Giora, Y., Kaplan, H.: Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Trans. Algorithms 5(3), 28:1–28:51 (2009)Google Scholar
  11. 11.
    Husfeldt, T., Rauhe, T.: Lower bounds for dynamic transitive closure, planar point location, and parentheses matching. Nordic J. Computing 3 (1996)Google Scholar
  12. 12.
    Löffler, M., Simons, J.A., Strash, D.: Dynamic planar point location with sub-logarithmic local updates. Arxiv report, arXiv:1204.4714 [cs.CG] (April 2012)Google Scholar
  13. 13.
    Nekrich, Y.: Data structures with local update operations. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 138–147. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Sarnak, N., Tarjan, R.E.: Planar point location using persistent search trees. Commun. ACM 29, 669–679 (1986), http://doi.acm.org/10.1145/6138.6151, doi:10.1145/6138.6151MathSciNetCrossRefGoogle Scholar
  15. 15.
    van Kreveld, M.: On fat partitioning, fat covering, and the union size of polygons. Comput. Geom. Theory Appl. 9(4), 197–210 (1998)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Joseph A. Simons
    • 2
  • Darren Strash
    • 2
  1. 1.Dept. of Information and Computing SciencesUtrecht UniversityItaly
  2. 2.Dept. of Computer ScienceUniversity of CaliforniaIrvineUSA

Personalised recommendations