Fast Fréchet Distance Between Curves With Long Edges

10/28/2017

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by   Joachim Gudmundsson, et al.

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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^d 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) (n+m)) time, (3) a linear-time √(d) -approximation algorithm, and (4) a data structure that supports O(m^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.