Convex Hull of the Union of Convex Objects in the Plane: an Adaptive Analysis.
Bibliographic record
Abstract
We prove a tight asymptotic bound of Θ(δ log(n/δ)) on the worst case computational complexity of the convex hull of the union of two convex objects of sizes summing to n requiring δ orientation tests to certify the answer. For more convex objects, we prove a (non optimal) asymptotic bound of O(δ ∑ k i=1 log(ni/δ)) on the worst case computational complexity of the convex hull of the union of k convex objects of respective sizes (n1,..., nk) requiring δ orientation tests to certify the answer. Our algorithms are deterministic, they use portions of the convex hull of input objects to describe the final convex hull, and take advantage of easy instances, such as those where large parts of two objects are horizontally or vertically separated. 1
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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 | 0.004 | 0.035 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".