Near-tight Bounds for Computing the Fr\'echet Distance in d-Dimensional Grid Graphs and the Implications for {\lambda}-low Dense Curves

The Fr\'echet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fr\'echet 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 Fr\'echet 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 $O((\varepsilon^{2/d}(\frac{n}{\varepsilon})^{2-2/d})^{1-\delta})$ algorithm for any $\delta>0$ 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 Fr\'echet distance computation. One of these assumptions is $\lambda$-low density and they can compute a $(1+\varepsilon)$-approximation between $\lambda$-low dense curves in time $\widetilde{O}( \varepsilon^{-2} \lambda^2 n^{2(1-1/d)})$. 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 $\lambda$.