Fast Frechet Distance Between Curves With Long Edges

Abstract

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in ${\mathbb{R}}^d$ with $n$ and $m$ vertices, respectively. We prove four 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 $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m{\log }^{2}n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve.