A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space

We propose a data structure for point sets in d-dimensional hyperbolic space
that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. Using these quadtrees and the L-order we build geometric spanners. Near-linear size (1 + ε)-spanners do not exist in hyperbolic spaces, but we create a Steiner spanner that achieves a spanning ratio of 1 + ε with Od,ε(n) edges, using a simple construction that can be maintained dynamically. As a corollary, we also get a (2 + ε)-spanner (in the classical
sense) of the same size, where the spanning ratio 2 + ε is almost optimal among spanners of subquadratic size. Finally, we show that our Steiner spanner directly provides a solution to the approximate nearest neighbour problem: given a point set P in d-dimensional hyperbolic space we build the data structure in Od,ε(n log n) time, using Od,ε(n) space. Then for any query
point q we can find a point p ∈ P that is at most 1 + ε times farther from q than its nearest neighbour in P in Od,ε(log n) time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time Od,ε(log n). This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.