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) 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 $\mathcal{C}$ of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram $VD_F (\mathcal{C})$. Our main results are summarized as follows. • The combinatorial complexity of $VD_F (\mathcal{C})$ is at most O(ndk+∊). • The combinatorial complexity of $VD_F (\mathcal{C})$ is at least Ω(ndk) for dimension d = 1, 2; and Ω(nd(k-1)+2) for dimension d > 2. [ABSTRACT FROM AUTHOR]