Chapter

Computing and Combinatorics

Volume 5092 of the series Lecture Notes in Computer Science pp 352-362

Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

  • Sergey BeregAffiliated withDepartment of Computer Science, University of Texas at Dallas
  • , Kevin BuchinAffiliated withDepartment of Information and Computing Sciences, Universiteit Utrecht
  • , Maike BuchinAffiliated withDepartment of Information and Computing Sciences, Universiteit Utrecht
  • , Marina GavrilovaAffiliated withDepartment of Computer Science, University of Calgary
  • , Binhai ZhuAffiliated withDepartment of Computer Science, Montana State University

* Final gross prices may vary according to local VAT.

Get Access

Abstract

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.