Approximating the (Continuous) Fréchet Distance

We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fr\'echet distance between two polygonal chains. Specifically, let $P$ and $Q$ be two polygonal chains with $n$ vertices in $d$-dimensional Euclidean space, and let $\alpha \in [\sqrt{n}, n]$. Our algorithm deterministically finds an $O(\alpha)$-approximate Fr\'echet correspondence in time $O((n^3 / \alpha^2) \log n)$. In particular, we get an $O(n)$-approximation in near-linear $O(n \log n)$ time, a vast improvement over the previously best know result, a linear time $2^{O(n)}$-approximation. As part of our algorithm, we also describe how to turn any approximate decision procedure for the Fr\'echet distance into an approximate optimization algorithm whose approximation ratio is the same up to arbitrarily small constant factors. The transformation into an approximate optimization algorithm increases the running time of the decision procedure by only an $O(\log n)$ factor.