Constant Approximation of Fréchet Distance in Strongly Subquadratic Time

Let $τ$ and $σ$ be two polygonal curves in $\mathbb{R}^d$ for any fixed $d$. Suppose that $τ$ and $σ$ have $n$ and $m$ vertices, respectively, and $m\le n$. While conditional lower bounds prevent approximating the Fréchet distance between $τ$ and $σ$ 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 Fréchet 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} Fréchet distance within a factor of $7+\varepsilon$ in strongly subquadratic time. They are the first algorithms to approximate the Fréchet distance and the discrete Fréchet distance within constant factors in strongly subquadratic time.