Fast Frechet Distance Between Curves With Long Edges

TL;DR

This paper introduces efficient algorithms for computing the Fréchet distance between polygonal curves with long edges, significantly improving computation time for this special case.

Contribution

It presents four novel algorithms and data structures that enable faster Fréchet distance computations when all edges are long relative to the distance.

Findings

Linear-time decision algorithm for Fréchet distance

O((n+m) log(n+m)) time algorithm for computing the distance

Linear-time sqrt(d)-approximation algorithm

Abstract

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 $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the 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 $\sqrt{d}$-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.