Voronoi diagram of polygonal chains under the discrete Frechet 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 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(dk+epsilon)). The combinatorial complexity of VDF(C) is at least Omega(n(dk)) for dimension d = 1, 2; and Omega(n(d(k-l)+2)) for dimension d > 2.