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.


Agate Transportation 


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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

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