Faster Fr\'echet Distance under Transformations

TL;DR

This paper introduces faster algorithms for computing the Fréchet distance between polygonal curves under various transformations, significantly improving efficiency over previous methods for translations and more complex transformations.

Contribution

It presents an improved algorithm for translation-based Fréchet distance computation and generalizes to rationally parameterized transformations with multiple degrees of freedom.

Findings

Translation case solved in rac{1}{3}n^{7+rac{1}{3}}) time

General transformations handled in rac{1}{3}n^{3k+rac{4}{3}}) time

Significant speedup over previous rac{1}{3}n^{8}) algorithm

Abstract

We study the problem of computing the Fr\'echet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves $\pi$ and $\sigma$ of total complexity $n$ and a threshold $\delta \geq 0$, we present an $\tilde{\mathcal{O}}(n^{7 + \frac{1}{3}})$ time algorithm to determine whether there exists a translation $t \in \mathbb{R}^2$ such that the Fr\'echet distance between $\pi$ and $\sigma + t$ is at most $\delta$. This improves on the previous best result, which is an $\mathcal{O}(n^8)$ time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class $\mathcal T$ of rationally parametrized transformations with $k$ degrees of freedom, we show that one can determine whether there is a transformation $\tau \in \mathcal T$ such that the Fr\'echet distance between $\pi$ and $\tau(\sigma)$ is at most $\delta$ in $\tilde{\mathcal{O}}(n^{3k+\frac{4}{3}})$ time.