Higher Order City Voronoi Diagrams

  • Andreas Gemsa
  • D. T. Lee
  • Chih-Hung Liu
  • Dorothea Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L 1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k th-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n + kc), where c is the complexity of the transportation network. This is quite different from the bound O(k(n − k)) in the Euclidean metric [12]. For the special case where k = n − 1 the complexity in the Euclidean metric is O(n), while that in the city metric is Θ(nc). Furthermore, we develop an O(k 2(n + c)log(n + c))-time iterative algorithm to compute the k th-order city Voronoi diagram and an O(nclog2(n + c)logn)-time divide-and-conquer algorithm to compute the farthest-site city Voronoi diagram.


Voronoi Diagram Transportation Network Point Site Voronoi Region Voronoi Edge 
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.
    Aichholzer, O., Aurenhammer, F., Palop, B.: Quickest paths, straight skeletons, and the city Voronoi diagram. Discrete Comput. Geom. 31, 17–35 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agarwal, P.K., de Berg, M., Matoušek, J., Schwarzkopf, O.: Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27(3), 654–667 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F., Schwarzkopf, O.: A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. Internat. J. Comput. Geom. Appl. 2, 363–381 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bae, S.W., Chwa, K.-Y.: Shortest Paths and Voronoi Diagrams with Transportation Networks Under General Distances. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1007–1018. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Boissonnat, J.D., Devillers, O., Teillaud, M.: A semidynamic construction of higher-order Voronoi diagrams and its randomized analysis. Algorithmica 9, 329–356 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bae, S.W., Kim, J.-H., Chwa, K.-Y.: Optimal construction of the city Voronoi diagram. Internat. J. Comput. Geom. Appl. 19(2), 95–117 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chazelle, B., Edelsbrunner, H.: An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Transactions on Computers 36(11), 1349–1454 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.-S.: Farthest-polygon Voronoi diagrams. Comput. Geom. Theory Appl. 44, 234–247 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15(2), 317–340 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gemsa, A., Lee, D.T., Liu, C.-H., Wagner, D.: Higher order city Voronoi diagrams. ArXiv e-prints, arXiv:1204.4374 (April 2012)Google Scholar
  11. 11.
    Görke, R., Shin, C.-S., Wolff, A.: Constructing the city Voronoi diagram faster. Internat. J. Comput. Geom. Appl. 18(4), 275–294 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lee, D.T.: On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput. 31(6), 478–487 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Mulmuley, K.: A fast planar partition algorithm, I. J. Symbolic Comput. 10(3-4), 253–280 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Mulmuley, K.: On levels in arrangements and Voronoi diagrams. Discrete Comput. Geom. 6, 307–338 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Gemsa
    • 1
  • D. T. Lee
    • 2
    • 3
  • Chih-Hung Liu
    • 1
    • 2
  • Dorothea Wagner
    • 1
  1. 1.Karlsruhe Institute of TechnologyGermany
  2. 2.Academia SinicaTaiwan
  3. 3.National Chung Hsing UniversityTaiwan

Personalised recommendations