Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive
orderings (LSO) for Euclidean space. A (τ,ρ)-LSO is a collection
Σ of orderings such that for every x,y∈Rd there is an
ordering σ∈Σ, where all the points between x and y w.r.t.
σ are in the ρ-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,
ℓp, 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