The discrete Fr\'echet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fr\'echet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length $n$ in the plane, the fastest known algorithm runs in time $\tilde{\cal O}(n^{5})$ [Ben Avraham, Kaplan, Sharir '15]. This is achieved by constructing an arrangement of disks of size ${\cal O}(n^{4})$, and then traversing its faces while updating reachability in a directed grid graph of size $N := {\cal O}(n^2)$, which can be done in time $\tilde{\cal O}(\sqrt{N})$ per update [Diks, Sankowski '07]. The contribution of this paper is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than $\tilde{\cal O}(\sqrt{N})$, we improve this part of the algorithm: We observe that an offline variant of dynamic $s$-$t$-reachability in directed grid graphs suffices, and we solve this variant in amortized time $\tilde{\cal O}(N^{1/3})$ per update, resulting in an improved running time of $\tilde{\cal O}(n^{4.66...})$ for the discrete Fr\'echet distance under translation. Second, we provide evidence that constructing the arrangement of size ${\cal O}(n^{4})$ is necessary in the worst case, by proving a conditional lower bound of $n^{4 - o(1)}$ on the running time for the discrete Fr\'echet distance under translation, assuming the Strong Exponential Time Hypothesis.

翻译：离散 Fr\'echet 距离是用于比较多边形曲线的流行度量。 一个重要的变量是翻译中的离散 Fr\'echet 距离, 从而能够检测不同空间域的类似运动模式。 对于飞机长度为$n美元的多边形曲线, 最快已知的算法在时间上运行 $\ tilde\ cal O} (n ⁇ 5}) 美元 [Ben Avraham, Kaplan, Sharir'15] 。 这是通过构建一个大小为$\ cal4 (n ⁇ 4} 4} (n} 4}) 的磁盘的配置, 然后在方向型号的直径网图中更新其面部位的可达性 $=n:=\ cal O} (n\\ dal_ dal_ dal_ directral_ directral_ directral_ lax we dreal_ dreal_ dreal_ dreal_ dreal_ droad_ dirmaxn_ dirmaxn_ dirmaxl_ dal_ dirmax_ 美元 rout_ dirmax_ drode) routd_ 美元 美元 美元 美元 美元 routd_ routd_ 美元 routd_ 美元 routt_ dir_ dir_ routdal_ routtal_ routdal_ rode drode dal_ dal_ dal_ dal_ lad_ ladal_ dir_ rodal_ rod_ rodal_ rodal_ rodal_ dal_ dirdaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldald ro) ro) ro) rod rod rod rodaldaldaldaldald rod rod ro ro ro ro ro rod ro ro ro ro