Fast Fréchet Distance Between Curves With Long Edges

Computing Fréchet distance between two curves takes roughly quadratic time. In this paper, we show that for curves with long edges the Fréchet distance computations become easier. Let P and Q be two polygonal curves in R with n and m vertices, respectively. We prove four main results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, (3) a linear-time √ d -approximation algorithm, and (4) a data structure that supports O(m log2 n)-time decision queries, where m is the number of vertices of the query curve and n the number of vertices of the preprocessed curve. 1998 ACM Subject Classification F.2.2 [Nonnumerical Algorithms and Problems] Geometrical problems and computations