Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM
Bibliographic record
Abstract
Abstract. We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(n logn + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((n logn+ k) log logn) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Pătraşcu [SoCG’11] that attains the same time complexity but in expectation. The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d ≥ 4. A key tool behind Chan et al.’s previous randomized algorithm is shal-low cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA’14] obtained a deterministic O(n logn)-time algorithm to con-struct such cuttings. We first present an improved deterministic con-struction algorithm that runs in O(n log logn) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cut-tings and an algorithm for an offline tree point location problem. 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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".