Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A $(\tau,\rho)$-LSO is a collection $\Sigma$ of orderings such that for every $x,y\in\mathbb{R}^d$ there is an ordering $\sigma\in\Sigma$, where all the points between $x$ and $y$ w.r.t. $\sigma$ are in the $\rho$-neighborhood of either $x$ or $y$. In essence, LSO allow one to reduce problems to the $1$-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, $\ell_p$, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model.

翻译：Chan、Har-Peled 和 Jones [SICOMP 2020] 为欧几里得空间开发了局部敏感排序（LSO）。一个 $(\tau,\rho)$-LSO 是一个排序集合 $\Sigma$，使得对于任意 $x,y\in\mathbb{R}^d$，存在一个排序 $\sigma\in\Sigma$，其中在 $\sigma$ 意义下位于 $x$ 和 $y$ 之间的所有点都落在 $x$ 或 $y$ 的 $\rho$-邻域内。本质上，LSO 允许将问题简化为一维直线。随后，Filtser 和 Le [STOC 2022] 将 LSO 推广至倍率度量、一般度量空间以及无小图。对于欧几里得和倍率空间，LSO 中排序的数量随维度呈指数增长，这使得它们主要适用于低维场景。本文为欧几里得空间、$\ell_p$ 空间和倍率空间开发了新型 LSO，允许以更少的排序数量换取更大的伸缩比。我们利用这些新型 LSO（以及先前的 LSO）为不同度量空间构造了路径报告低跳伸缩子、容错伸缩子、可靠伸缩子和轻量伸缩子。尽管针对具有隐式距离表示（其中两点间距离可通过其名称计算，例如欧几里得空间）的度量空间已构建了许多最近邻搜索（NNS）数据结构，但对于其他空间几乎一无所知。本文首次研究了标记 NNS 问题，该问题允许人工为度量点分配标签（短名称）。在此模型下，我们利用 LSO 构造了高效的标记 NNS 数据结构。