Optimal halfspace range reporting in three dimensions
Why this work is in the frame
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Bibliographic record
Abstract
We give the first optimal solution to a standard problem in computational geometry: three-dimensional halfspace range reporting. We show that n points in 3-d can be stored in a linear-space data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data structure can be built in O(n log n) expected time. The previous methods with optimal query time required superlinear (O(n log log n)) space. We also mention consequences, for example, to higher dimensions and to external-memory data structures. As an aside, we partially answer another open question concerning the crossing number in Matouˇsek’s shallow partition theorem in the 3-d case (a tool used in many known halfspace range reporting methods). 1
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it