We study the Fr\'echet queries problem. It is a data structure problem, where we are given a set $S$ of $n$ polygonal curves and a distance threshold $\rho$. The data structure should support queries with a polygonal curve $q$ for the elements of $S$, for which the continuous Fr\'echet distance to $q$ is at most $\rho$. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point $q$ should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with $\mathcal{O}(n \log ^{t-1} n)$ storage and $\mathcal{O}(\log^{t-1} n+k)$ query time, where $k$ denotes the output size and $t$ can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter $d= \lfloor t/2 \rfloor$ can be applied to our problem via reduction. .

翻译：我们研究Fréchet查询问题。这是一个数据结构问题，给定一组包含$n$条多边形曲线的集合$S$和一个距离阈值$\rho$。该数据结构应支持以多边形曲线$q$进行查询，返回$S$中与$q$的连续Fréchet距离不超过$\rho$的元素。Afshani和Driemel在2018年针对二维多边形曲线研究了该问题，并给出了空间与查询时间权衡的上界和下界。我们研究曲线所在空间为一维的情况，并揭示其与已深入研究的矩形穿刺问题之间的密切关联。在此问题中，输入是一组超矩形，查询点$q$应返回所有包含该点的输入矩形。利用已知的矩形穿刺或正交范围搜索数据结构，可直接得到具有$\mathcal{O}(n \log ^{t-1} n)$存储空间和$\mathcal{O}(\log^{t-1} n+k)$查询时间的数据结构，其中$k$表示输出规模，$t$可选取为（a）存储曲线或（b）查询曲线的最大顶点数。所得结果在存储空间和查询时间两方面均优于Afshani和Driemel的界限。此外，我们证明当维度参数$d= \lfloor t/2 \rfloor$时，矩形穿刺和正交范围报告问题的已知下界可通过归约应用于我们的问题。