Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

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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.