Fréchet Distance in Subquadratic Time

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 Fréchet 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 $m = Ω(n)$. 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 Fréchet distance, the dynamic time warping distance, and the geometric edit distance. It is curious that the quadratic time barrier for Fréchet distance still stands. We present an algorithm to compute the Fréchet distance in $O(mn(\log\log n)^{2+μ}\log n/\log^{1+μ} m)$ expected time for some constant $μ\in (0,1)$. It is the first algorithm that returns the Fréchet distance in $o(mn)$ time when $m = Ω(n^{\varepsilon})$ for any fixed $\varepsilon \in (0,1]$.