Computing the Fr\'echet Distance with a Retractable Leash

TL;DR

This paper introduces a novel quadratic time algorithm for computing the Fréchet distance between polygonal curves in Euclidean space, bypassing traditional decision-based methods, and also provides a near-linear approximation under Euclidean metrics.

Contribution

It presents the first quadratic time algorithm for Fréchet distance in certain metric spaces and offers a new approach that avoids the decision oracle framework.

Findings

Quadratic time algorithm for Fréchet distance in polyhedral metrics.

A (1+ε)-approximation algorithm for Euclidean Fréchet distance.

Potential for faster exact algorithms in Euclidean space.

Abstract

All known algorithms for the Fr\'echet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fr\'echet distance between polygonal curves in $R^d$ under polyhedral distance functions (e.g., $L_1$ and $L_\infty$). We also get a $(1+\varepsilon)$-approximation of the Fr\'echet distance under the Euclidean metric, in quadratic time for any fixed $\varepsilon > 0$. For the exact Euclidean case, our framework currently yields an algorithm with running time $O(n^2 \log^2 n)$. However, we conjecture that it may eventually lead to a faster exact algorithm.