Fr\'echet Distance in Subquadratic Time

TL;DR

This paper introduces the first subquadratic algorithm for computing the Fréchet distance between polygonal curves, breaking the longstanding quadratic time barrier and achieving expected running time faster than quadratic for certain curve sizes.

Contribution

It presents a novel algorithm that computes the Fréchet distance in subquadratic expected time, improving over the classic quadratic time algorithms for large curves.

Findings

First subquadratic expected time algorithm for Fréchet distance

Breaks the quadratic time barrier for large curves

Achieves faster than quadratic expected running time for certain curve sizes

Abstract

Let $m$ and $n$ be the numbers of vertices of two polygonal curves in $\mathbb{R}^d$ for any fixed $d$ such that $m \leq n$. Since it was known in 1995 how to compute the Fr\'{e}chet distance of these two curves in $O(mn\log (mn))$ time, it has been an open problem whether the running time can be reduced to $o(n^2)$ when $m = \Omega(n)$. In the mean time, several well-known quadratic time barriers in computational geometry have been overcome: 3SUM, some 3SUM-hard problems, and the computation of some distances between two polygonal curves, including the discrete Fr\'{e}chet distance, the dynamic time warping distance, and the geometric edit distance. It is curious that the quadratic time barrier for Fr\'{e}chet distance still stands. We present an algorithm to compute the Fr\'echet distance in $O(mn(\log\log n)^{2+\mu}\log n/\log^{1+\mu} m)$ expected time for some constant $\mu \in (0,1)$. It is the first algorithm that returns the Fr\'{e}chet distance in $o(mn)$ time when $m = \Omega(n^{\varepsilon})$ for any fixed $\varepsilon \in (0,1]$.