Adaptive and Approximate Orthogonal Range Counting
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Bibliographic record
Abstract
We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model. It is well known that there are linear-space data structures for 2-D orthogonal range counting with worst-case optimal query time O(logw n). We give an O(n log log n)-space adaptive data structure that improves the query time to O (log log n + logw k), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan, Larsen, and Pătraşcu, SoCG 2011]. We give an O(n log log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in O (log log n) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem. Lastly, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [SODA 2011] presented a linear-space adaptive data structure with query time O (log log n + logw k). We give a new linear-space structure that improves the query time to O(1 + logw k), exactly matching the lower bound proved by Jørgensen and Larsen.
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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 | 0.001 |
| 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