Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

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 famous (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 C of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VDF (C). Our main results are summarized as follows. • The combinatorial complexity of VDF (C) is at most O(n ). • The combinatorial complexity of VDF (C) is at least Ω(n) for dimension d = 1, 2; and Ω(n) for dimension d > 2.