A quadtree, a Steiner spanner, and approximate nearest neighbours in hyperbolic space
Bibliographic record
Abstract
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 $(1+\varepsilon)$-spanners do not exist in hyperbolic spaces, but we create a Steiner spanner that achieves a spanning ratio of $1+\varepsilon$ with $\mathcal O_{d,\varepsilon}(n)$ edges, using a simple construction that can be maintained dynamically. As a corollary, we also get a $(2+\varepsilon)$-spanner (in the classical sense) of the same size, where the spanning ratio $2+\varepsilon$ 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 $\mathcal O_{d,\varepsilon}(n\log n)$ time, using $\mathcal O_{d,\varepsilon}(n)$ space. Then for any query point $q$ we can find a point $p\in P$ that is at most $1+\varepsilon$ times farther from $q$ than its nearest neighbour in $P$ in $\mathcal O_{d,\varepsilon}(\log n)$ time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time $\mathcal O_{d,\varepsilon}(\log n)$. This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".