A Subquadratic $n^\epsilon$-approximation for the Continuous Fr\'echet Distance

TL;DR

This paper presents a new algorithm that approximates the continuous Fréchet distance between curves more efficiently, achieving a subquadratic runtime with adjustable approximation quality in fixed dimensions.

Contribution

It introduces a subquadratic-time approximation algorithm for the continuous Fréchet distance with a flexible tradeoff between accuracy and efficiency.

Findings

Achieves an $O( ext{approximation factor})$-approximate algorithm with subquadratic runtime.

Improves upon previous tradeoff results by reducing the dependency on $n$ and $m$.

Works in fixed dimension with assumptions $m eq n$ and $d$ constant.

Abstract

The Fr\'echet distance is a commonly used similarity measure between curves. It is known how to compute the continuous Fr\'echet distance between two polylines with $m$ and $n$ vertices in $\mathbb{R}^d$ in $O(mn (\log \log n)^2)$ time; doing so in strongly subquadratic time is a longstanding open problem. Recent conditional lower bounds suggest that it is unlikely that a strongly subquadratic algorithm exists. Moreover, it is unlikely that we can approximate the Fr\'echet distance to within a factor $3$ in strongly subquadratic time, even if $d=1$. The best current results establish a tradeoff between approximation quality and running time. Specifically, Colombe and Fox (SoCG, 2021) give an $O(\alpha)$-approximate algorithm that runs in $O((n^3 / \alpha^2) \log n)$ time for any $\alpha \in [\sqrt{n}, n]$, assuming $m = n$. In this paper, we improve this result with an $O(\alpha)$-approximate algorithm that runs in $O((n + mn / \alpha) \log^3 n)$ time for any $\alpha \in [1, n]$, assuming $m \leq n$ and constant dimension $d$.