Constant Approximation of Fr\'echet Distance in Strongly Subquadratic Time

Let $\tau$ and $\sigma$ be two polygonal curves in $\mathbb{R}^d$ for any fixed $d$. Suppose that $\tau$ and $\sigma$ have $n$ and $m$ vertices, respectively, and $m\le n$. While conditional lower bounds prevent approximating the Fr\'echet distance between $\tau$ and $\sigma$ within a factor of 3 in strongly subquadratic time, the current best approximation algorithm attains a ratio of $n^c$ in strongly subquadratic time, for some constant $c\in(0,1)$. We present a randomized algorithm with running time $O(nm^{0.99}\log(n/\varepsilon))$ that approximates the Fr\'echet distance within a factor of $7+\varepsilon$, with a success probability at least $1-1/n^6$. We also adapt our techniques to develop a randomized algorithm that approximates the \emph{discrete} Fr\'echet distance within a factor of $7+\varepsilon$ in strongly subquadratic time. They are the first algorithms to approximate the Fr\'echet distance and the discrete Fr\'echet distance within constant factors in strongly subquadratic time.