Bibliographic record
Abstract
Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any d-dimensional n-point Euclidean space, there exists a (1+ε) -spanner with O(nε^-d+1) edges and lightness (normalized weight) O(ε^-2d)^1. Surprisingly, the fundamental question of whether or not these dependencies on ε and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus ε is tiny). In the most extreme case ε is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n. The state-of-the-art bounds O(nε^-d+1) and O(ε^-2d) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is “near-optimal''; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is two-fold. 1) We resolve these longstanding questions by nailing down the exact dependencies on ε and d and showing that the greedy spanner is truly optimal. Specifically, for any d= O(1), ε = Ω(n^-1/d-1): • We show that any (1+ε) -spanner must have Ω(nε^-d+1) edges, implying that the greedy (and other) spanners achieve the optimal size. • We show that any (1+ε) -spanner must have lightness Ω(ε^-d), and then improve the upper bound on the lightness of the greedy spanner from O(ε^-2d) to Õ_ε (ε^-d). 2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners is tight as well (up to lower-order terms).
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.003 |
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".