Degree four plane spanners: Simpler and better
Bibliographic record
Abstract
Let $\mathcal{P}$ be a set of $n$ points embedded in the plane, and let $\mathcal{C}$ be the complete Euclidean graph whose point-set is $\mathcal{P}$. Each edge in $\mathcal{C}$ between two points $p$, $q$ is realized as the line segment $[pq]$ and is assigned a weight equal to the Euclidean distance $|pq|$. In this paper, we show how to construct in $O(n \lg n)$ time a plane spanner of $\mathcal{C}$ of maximum degree at most $4$ and of stretch factor at most $20$. This improves a long sequence of results on the construction of bounded degree plane spanners of $\mathcal{C}$. Our result matches the smallest known upper bound of $4$ by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from $156.82$ to $20$. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner and reveals useful structural properties of Delaunay triangulations defined with respect to the equilateral-triangle distance. The structure of the constructed spanner implies that when $\mathcal{P}$ is in convex position, the maximum degree of the spanner is at most $3$. Combining the above degree upper bound with the fact that $3$ is a lower bound on the maximum degree of any plane spanner of $\mathcal{C}$ when the point-set $\mathcal{P}$ is in convex position, the results in this paper give a tight bound of $3$ on the maximum degree of plane spanners of $\mathcal{C}$ for point-sets in convex position.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 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".