The Fr\'{e}chet distance is one of the most studied distance measures between curves $P$ and $Q$. The data structure variant of the problem is a longstanding open problem: Efficiently preprocess $P$, so that for any $Q$ given at query time, one can efficiently approximate their Fr\'{e}chet distance. There exist conditional lower bounds that prohibit $(1 + \varepsilon)$-approximate Fr\'{e}chet distance computations in subquadratic time, even when preprocessing $P$ using any polynomial amount of time and space. As a consequence, the problem has been studied under various restrictions: restricting $Q$ to be a (horizontal) segment, or requiring $P$ and $Q$ to be so-called \emph{realistic} input curves. We give a data structure for $(1+\varepsilon)$-approximate discrete Fr\'{e}chet distance in any metric space $\mathcal{X}$ between a realistic input curve $P$ and any query curve $Q$. After preprocessing the input curve $P$ (of length $|P|=n$) in $O(n \log n)$ time, we may answer queries specifying a query curve $Q$ and an $\varepsilon$, and output a value $d(P,Q)$ which is at most a $(1+\varepsilon)$-factor away from the true Fr\'{e}chet distance between $Q$ and $P$. Our query time is asymptotically linear in $|Q|=m$, $\frac{1}{\varepsilon}$, $\log n$, and the realism parameter $c$ or $\kappa$. Our data structure is the first to: adapt to the approximation parameter $\varepsilon$ at query time, handle query curves with arbitrarily many vertices, work for any ambient space of the curves, or be dynamic. The method presented in this paper simplifies and generalizes previous contributions to the static problem variant. We obtain efficient queries (and therefore static algorithms) for Fr\'{e}chet distance computation in high-dimensional spaces and other ambient metric spaces.

翻译：Fréchet距离是曲线$P$和$Q$之间被研究最多的距离度量之一。该问题的数据结构变体是一个长期存在的开放问题：高效预处理$P$，使得对于查询时给定的任意$Q$，能够高效近似计算其Fréchet距离。存在条件性下界表明，即使使用多项式时间和空间对$P$进行预处理，也无法在次二次时间内计算$(1+\varepsilon)$-近似的Fréchet距离。因此，该问题在多种限制条件下被研究：限制$Q$为（水平）线段，或要求$P$和$Q$为所谓的“真实”输入曲线。我们提出了一种数据结构，用于在任何度量空间$\mathcal{X}$中，对真实输入曲线$P$与任意查询曲线$Q$之间的$(1+\varepsilon)$-近似离散Fréchet距离进行计算。在$O(n \log n)$时间内预处理输入曲线$P$（长度$|P|=n$）后，我们可响应指定查询曲线$Q$和参数$\varepsilon$的查询，并输出一个值$d(P,Q)$，该值最多与$Q$和$P$之间真实Fréchet距离相差$(1+\varepsilon)$倍。我们的查询时间在$|Q|=m$、$\frac{1}{\varepsilon}$、$\log n$以及真实参数$c$或$\kappa$方面呈渐近线性。我们的数据结构是首个能够：在查询时自适应逼近参数$\varepsilon$、处理具有任意数量顶点的查询曲线、适用于曲线的任意环境空间，或是动态的数据结构。本文提出的方法简化并推广了先前对静态问题变体的贡献。我们为高维空间及其他环境度量空间中的Fréchet距离计算实现了高效查询（进而实现静态算法）。