The Fréchet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fréchet distance between walks in the $d$-dimensional grid graphs, i.e. $\mathbb{Z}^d$ where points share an edge if they differ by one in one coordinate. We give an algorithm, that for two simple paths on $n$ vertices, $(1+\varepsilon)$-approximates the Fréchet distance in time $\widetilde{O}((\frac{n}{\varepsilon})^{2-2/d} +n)$. We complement this by a near-matching fine-grained lower bound: for constant dimensions $d \geq 3$, there is no $O((\varepsilon^{2/d}(\frac{n}{\varepsilon})^{2-2/d})^{1-δ})$ algorithm for any $δ>0$ unless the Orthogonal Vector Hypothesis fails. Thus, our results are tight up to a factor $\varepsilon^{2/d}$ and $\log(n)$-factors. We extend our results to imbalanced lower and upper bounds, where the curves have $n$ and $m$ vertices respectively, and also obtain near-tight bounds. Driemel, Har-Peled and Wenk [DCG'12] studied \emph{realistic assumptions} for curves to speed up Fréchet distance computation. One of these assumptions is $λ$-low density and they can compute a $(1+\varepsilon)$-approximation between $λ$-low dense curves in time $\widetilde{O}( \varepsilon^{-2} λ^2 n^{2(1-1/d)})$. By adapting our lower bound, we show that their algorithm has a tight dependency on $n$ and a tight dependency on $\varepsilon$ as $d$ goes to infinity. A gap remains in terms of $λ$.

翻译：弗赖歇距离是空间中轨迹或曲线之间，以及图中路径之间一种流行的距离度量。我们研究在$d$维网格图（即$\mathbb{Z}^d$，其中当两个点在某一坐标上相差1时共享一条边）中计算两条路径之间弗赖歇距离的问题。我们提出一种算法，对于包含$n$个顶点的两条简单路径，能在$\widetilde{O}((\frac{n}{\varepsilon})^{2-2/d} +n)$时间内实现$(1+\varepsilon)$-近似的弗赖歇距离。作为补充，我们给出了一个几乎匹配的细粒度下界：对于常数维度$d \geq 3$，除非正交向量假设不成立，否则不存在任何$δ>0$使得算法复杂度为$O((\varepsilon^{2/d}(\frac{n}{\varepsilon})^{2-2/d})^{1-δ})$。因此，我们的结果在$\varepsilon^{2/d}$因子和$\log(n)$因子范围内是最优的。我们将结果扩展到不平衡的上下界情形，即曲线分别包含$n$和$m$个顶点时，同样得到了近紧界。Driemel、Har-Peled和Wenk [DCG'12] 研究了加速弗赖歇距离计算的*现实假设*。其中一个假设是$λ$-低密度，他们能在$\widetilde{O}( \varepsilon^{-2} λ^2 n^{2(1-1/d)})$时间内计算$λ$-低密度曲线之间的$(1+\varepsilon)$-近似。通过调整我们的下界，我们证明他们的算法在$n$上具有紧的依赖性，并且当$d$趋于无穷大时在$\varepsilon$上也具有紧的依赖性。但在$λ$方面仍存在一个差距。