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.

中文翻译：

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线段的近似最近邻搜索

近似最近邻搜索是一个基本的算法问题，由于它在许多情况下都具有重要作用，因此继续激发研究的兴趣。与大多数先前的工作集中于点集相反，对于恒定维$ d $，我们考虑针对$ \ mathbb {R} ^ d $中的一组线段的最近邻查询。给定$ \ mathbb {R} ^ d $中的一组$ S $的$ n $不相交的线段，并且错误参数$ \ varepsilon> 0 $，目标是建立一个数据结构，使得对于任何查询点$ q $，可以返回其距$ q $的欧几里得距离最多为$（1+ \ varepsilon）$乘以$ q $与其最近的线段的距离的线段。我们使用存储$ O（（n ^ 2 / \ varepsilon ^ {d}）\ log（\ Delta / \ varepsilon））$和查询时间$ O（\ log（\ max（n， \ Delta）/ \ varepsilon）），其中$ \ Delta $是细分集$ S $的价差。我们的方法基于各向异性元素覆盖的空间，这些元素根据附近线段的方向对齐。