Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings

TL;DR

This paper introduces new locality-sensitive orderings for Euclidean, ℓ_p, and doubling spaces that enable efficient construction of spanners and solve the labeled nearest neighbor search problem by assigning labels to metric points.

Contribution

It develops novel LSO's with a trade-off between stretch and the number of orderings, and applies them to create efficient spanners and labeled NNS data structures.

Findings

New LSO's for Euclidean, ℓ_p, and doubling spaces with fewer orderings.

Construction of path reporting, fault tolerant, reliable, and light spanners.

Efficient labeled NNS data structures for metric spaces with labels.

Abstract

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.