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.

翻译：我们描述第一个强烈的亚赤道时间算法,其中含有两个多边形链之间大约计算Fr\'echet距离的亚爆炸近似近似率。 具体地说, 美元和Q美元是两个以美元为单位的顶点的多边形链, 以美元为单位的 Euclidean 空间为单位, 以美元为单位的顶点, 以美元为单位的顶点, 以美元为单位的顶点, 以美元为单位的顶点, 以美元为单位的顶点。 我们的算法在时间上找到一个美元( ALpha) $- 接近Fr\ echet 通信的准点决定程序。 我们还描述了如何将Fr\ echet 距离的准点转换为近点的近点优化算法, 其近点比率为以美元为单位的顶点的顶点 。