Advertisement

Dynamic Connectivity for Axis-Parallel Rectangles

  • Peyman Afshani
  • Timothy M. Chan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

Abstract

In this paper we give a fully dynamic data structure to maintain the connectivity of the intersection graph of n axis-parallel rectangles. The amortized update time (insertion and deletion of rectangles) is O(n 10/11polylog n) and the query time (deciding whether two given rectangles are connected) is O(1). It slightly improves the update time (O(n 0.94)) of the previous method while drastically reducing the query time (near O(n 1/3)). Our method does not use fast matrix multiplication results and supports a wider range of queries.

Keywords

Intersection Graph Query Time Connectivity Query Class Vertex Dynamic Data Structure 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Advances in Discrete and Computational Geometry, pp. 1–56. AMS Press (1999)Google Scholar
  2. 2.
    Chan, T.M.: Dynamic subgraph connectivity with geometric applications. In: Proc. 34th ACM Sympos. on Theory of Comput., pp. 7–13 (2002)Google Scholar
  3. 3.
    Chan, T.M.: Semi-online maintenance of geometric optima and measures. SIAM J. Comput. 32, 700–716 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9, 251–280 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edelsbrunner, H., Maurer, H.A.: On the intersection of orthogonal objects. Inform. Process. Lett. 13, 177–181 (1981)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fredman, M., Henzinger, M.: Lower bounds for fully dynamic connectivity problems in graphs. Algorithmica 22, 351–362 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gupta, P., Janardan, R., Smid, M.: Computational geometry: generalized intersection searching. In: Handbook of Data Structures and Applications, pp. 64–1–64–17. Chapman & Hall/CRC, Boca Raton (2005)Google Scholar
  9. 9.
    Henzinger, M.R., King, V.: Randomized dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46, 76–103 (2000)MathSciNetGoogle Scholar
  10. 10.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48, 723–760 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4(4), 310–323 (1983)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, Heidelberg (2002)MATHGoogle Scholar
  13. 13.
    Pǎtraşcu, M., Demaine, E.D.: Logarithmic lower bounds in the cell-probe model. SIAM J. Comput. 35(4), 932–963 (2006)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Thorup, M.: Decremental dynamic connectivity. J. Algorithms 33(2), 229–243 (1999)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proc. 32nd ACM Sympos. on Theory of Comput., pp. 343–350 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Peyman Afshani
    • 1
  • Timothy M. Chan
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations