SIAM Journal on Computing, Volume 49, Issue 3, Page 583-600, January 2020.
For any constant $d$ and parameter $\varepsilon \in (0,1/2]$, we show the existence of (roughly) $1/\varepsilon^d$ orderings on the unit cube $[0,1)^d$ such that for any two points $p, q\in [0,1)^d$ close together under the Euclidean metric, there is a linear ordering in which all points between $p$ and $q$ in the ordering are “close” to $p$ or $q$. More precisely, the only points that could lie between $p$ and $q$ in the ordering are points with Euclidean distance at most $\varepsilon\left\| {p} - {q} \right\|$ from either $p$ or $q$. These orderings are extensions of the Z-order, and they can be efficiently computed. Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

中文翻译：

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局部敏感排序及其应用

SIAM计算学报，第49卷，第3期，第583-600页，2020年1月。
对于任何常数$ d $和参数$ \ varepsilon \ in（0,1 / 2] $，我们显示在单位立方体$ [0,1）^ d上存在（大约）$ 1 / \ varepsilon ^ d $排序。 $，使得对于在欧几里得度量下在[0,1）^ d $中靠拢的任何两个点$ p，q \，都有一个线性排序，其中该排序中$ p $和$ q $之间的所有点都是“接近”到$ p $或$ q $。更准确地说，在排序中可能位于$ p $和$ q $之间的唯一点是欧几里德距离最多为$ \ varepsilon \ left \ |的点​​。{p}-{q} \ right \ | $来自$ p $或$ q $。这些顺序是Z顺序的扩展，可以有效地计算它们。从功能上讲，可以将顺序视为四叉树和相关结构的替代（例如，分隔良好的对分解）。