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] 针对欧几里得空间提出了局部敏感排序（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$\sigma\in\Sigma$，其中所有在$\sigma$下介于$x$和$y$之间的点都位于$x$或$y$的$\rho$邻域内。本质上，LSO允许将问题简化为$1$维直线。后来，Filtser和Le [STOC 2022] 针对加倍度量空间、一般度量空间以及无小图子图的空间发展了LSO。对于欧几里得空间和加倍空间，LSO中的排序数量随维度呈指数增长，这使得它们主要适用于低维场景。在本文中，我们为欧几里得空间、$\ell_p$空间和加倍空间开发了新的LSO，允许以更少的排序数量为代价换取更大的拉伸比。随后，我们利用新LSO（以及之前的LSO）为不同度量空间构建了路径报告低跳伸缩子、容错伸缩子、可靠伸缩子以及轻量伸缩子。尽管许多最近邻搜索数据结构是针对具有隐式距离表示（例如，通过名称可计算两点距离，如欧几里得空间）的度量空间构建的，但对于其他空间几乎一无所知。本文开创性地研究了标记最近邻搜索问题，在该问题中，允许人为地将标签（短名称）分配给度量点。我们利用LSO在此模型下构建了高效的标记最近邻搜索数据结构。