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

Sándor Kisfaludi-Bak*, Geert van Wordragen*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingsScientificpeer-review

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Abstract

We propose a data structure 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 O_{d,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 an elegant solution to the approximate nearest neighbour problem: given a point set P in d-dimensional hyperbolic space we build the data structure in O_{d,ε}(nlog n) time, using O_{d,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 O_{d,ε}(log n) time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time O_{d,ε}(log n). This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.
Original languageEnglish
Title of host publication40th International Symposium on Computational Geometry (SoCG 2024)
EditorsWolfgang Mulzer, Jeff M. Phillips
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
ISBN (Electronic)978-3-95977-316-4
DOIs
Publication statusPublished - Jun 2024
MoE publication typeA4 Conference publication
EventInternational Symposium on Computational Geometry - Athens, Greece
Duration: 11 Jun 202414 Jun 2024
Conference number: 40
https://socg24.athenarc.gr/socg.html

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume293
ISSN (Print)1868-8969

Conference

ConferenceInternational Symposium on Computational Geometry
Abbreviated titleSoCG
Country/TerritoryGreece
CityAthens
Period11/06/202414/06/2024
Internet address

Keywords

  • dynamic approximate nearest neighbours
  • hyperbolic geometry
  • Steiner spanner

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