Approximating the minimum closest pair distance and nearest neighbor distances of linearly moving points
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Bibliographic record
Abstract
Given a set of n moving points in R d , where each point moves along a linear trajectory at arbitrary but constant velocity, we present an O ź ( n 5 / 3 ) -time algorithm1 to compute a ( 1 + ź ) -factor approximation to the minimum closest pair distance over time, for any constant ź 0 and any constant dimension d. This addresses an open problem posed by Gupta, Janardan, and Smid 1.More generally, we consider a data structure version of the problem: for any linearly moving query point q, we want a ( 1 + ź ) -factor approximation to the minimum nearest neighbor distance to q over time. We present a data structure that requires O ź ( n 5 / 3 ) space and O ź ( n 2 / 3 ) query time, O ź ( n 5 ) space and polylogarithmic query time, or O ź ( n ) space and O ź ( n 4 / 5 ) query time, for any constant ź 0 and any constant dimension d. 1The notation O ź is used to hide polylogarithmic factors. That is, O ź ( f ( n ) ) = O ( f ( n ) log c ź n ) , where c is a constant.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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