Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

  • Sergey Bereg
  • Kevin Buchin
  • Maike Buchin
  • Marina Gavrilova
  • Binhai Zhu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the (continuous/discrete) Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Fréchet distance. Given a set \({\cal C}\) of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (\({\cal C}\)). Our main results are summarized as follows.
  • The combinatorial complexity of VD \(_F({\cal C})\) is at most O(n dk + ε).

  • The combinatorial complexity of VD \(_F({\cal C})\) is at least Ω(n dk ) for dimension d = 1,2; and Ω(n d(k − 1) + 2) for dimension d > 2.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Kevin Buchin
    • 2
  • Maike Buchin
    • 2
  • Marina Gavrilova
    • 3
  • Binhai Zhu
    • 4
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of Information and Computing SciencesUniversiteit UtrechtThe Netherlands
  3. 3.Department of Computer ScienceUniversity of CalgaryCalgaryCanada
  4. 4.Department of Computer ScienceMontana State UniversityBozemanUSA

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