Fast Frechet Distance Between Curves With Long Edges

Computing the Fr'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr'echet distance computations become easier. Let P and Q be two polygonal curves in ℝd with n and m vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr'echet distance between them: (1) a linear-time algorithm for deciding the Fr'echet distance between two curves, (2) an algorithm that computes the Fr'echet distance in O((n+m)log (n+m)) time, (3) a linear-time √d-approximation algorithm, and (4) a data structure that supports O(mlog2 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.