Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider nearest-neighbor queries against a set of line segments in $\mathbb{R}^d$, for constant dimension $d$. Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^d$ and an error parameter $\varepsilon > 0$, the objective is to build a data structure such that for any query point $q$, it is possible to return a line segment whose Euclidean distance from $q$ is at most $(1+\varepsilon)$ times the distance from $q$ to its nearest line segment. We present a data structure for this problem with storage $O((n^2/\varepsilon^{d}) \log (\Delta/\varepsilon))$ and query time $O(\log (\max(n,\Delta)/\varepsilon))$, where $\Delta$ is the spread of the set of segments $S$. Our approach is based on a covering of space by anisotropic elements, which align themselves according to the orientations of nearby segments.

翻译：近邻搜索是一个根本性的算法问题,它继续激励人们研究其在许多情况下的重要作用。 与以往大多数侧重于点数组的工作相比,我们考虑对固定维度以$mathbb{R ⁇ d$计的一组线段提出近邻查询。 鉴于一个固定维度以$mathbb{R ⁇ d$计的固定美元线段和一个错误参数 $\varepsilon > 0, 目标是建立一个数据结构, 对任何查询点 $ q$ 来说, 可以返回一个线段段, 该线段的Euclidean距离$最多为$(1 ⁇ varepslon) 乘以美元到其最接近的线段段的距离。 我们用存储$(n_2/\varepslon ⁇ d} (\ delta/\ varepslon) 美元和查询时间 $(\log) $(n, Delx, delta\\\\\\\ vareplon) max asion asion asion a roveloplections) subleges ro ro ro ro ro ro robismismism $x rommmm $x $x 。 我们的版段段系以我们的空间段/ dism $=sm ropal_ 美元/ dispalmlations_ 美元/ dislationsl) 。