Maximum-Weight Planar Boxes in O(n 2 ) Time (and Better).
Why this work is in the frame
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Bibliographic record
Abstract
Given a set P of n points in Rd, where each point p of P is associated with a weight w(p) (positive or negative), the Maximum-Weight Box problem consists in finding an axis-aligned box B maximizing ∑ p∈B∩P w(p). We describe algorithms for this problem in two dimensions that run in the worst case in O(n2) time, and much less on more specific classes of instances. In particular, these results imply similar ones for the Maximum Bichromatic Discrepancy Box problem. These improve by a factor of Θ(log n) on the best worst-case complexity previously known for these problems, O(n 2 lg n) [Cortés et al., J. Alg., 2009; Dobkin et al., J. Comput. Syst. Sci., 1996]. Although the O(n 2) result can be deduced from new results on the Klee’s Measure problem [Chan, 2013], it is a more direct and simplified (nontrivial) solution, which further provides smaller running times on specific classes on instances. 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) | 0.008 | 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 it