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 (continuous/discrete) Frechet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Frechet distance. Given a set of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram . Our main results are summarized as follows. • The combinatorial complexity of is at most O(ndk+∊). • The combinatorial complexity of is at least Ω(ndk) for dimension d = 1, 2; and Ω(nd(k-1)+2) for dimension d > 2.

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