Fr\'echet 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 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 $\varepsilon \in (0,1]$.