On Locality-Sensitive Orderings and their Applications

TL;DR

This paper introduces a set of orderings on the unit cube that efficiently approximate Euclidean proximity, enabling simpler algorithms for various low-dimensional geometric problems.

Contribution

It presents a new family of orderings extending the $ ext{Z}$-order, which approximate Euclidean closeness and can replace complex data structures in geometric algorithms.

Findings

Orderings are roughly $1/ extvarepsilon^d$ in number for dimension $d$.

Orderings can be computed efficiently.

Enable simpler algorithms for geometric problems like spanners and nearest neighbor search.

Abstract

For any constant $d$ and parameter $\varepsilon > 0$, we show the existence of (roughly) $1/\varepsilon^d$ orderings on the unit cube $[0,1)^d$, such that any two points $p,q\in [0,1)^d$ that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between $p$ and $q$ in the ordering are points with Euclidean distance at most $\varepsilon\| p - q \|$ from $p$ or $q$. These orderings are extensions of the $\mathcal{Z}$-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.