Two-center of the Convex Hull of a Point Set: Dynamic Model, and Restricted Streaming Model
Bibliographic record
Abstract
In this paper, we consider the dynamic version of covering the convex hull of a point set P in ℝ 2 by two congruent disks of minimum size. Here, the points can be added or deleted in the set P, and the objective is to maintain a data structure that, at any instant of time, can efficiently report two disks of minimum size whose union completely covers the boundary of the convex hull of the point set P. We show that maintaining a linear size data structure, we can report a radius r satisfying r ≤ 2 r opt at any query time, where r opt is the optimum solution at that instant of time. For each insertion or deletion of a point in P, the update time of our data structure is O(log n). Our algorithm can be tailored to work in the restricted streaming model where only insertions are allowed, using constant work-space. The problem studied in this paper has novelty in two ways: (i) it computes the covering of the convex hull of a point set P, which has lot of surveillance related applications, but not studied in the literature, and (ii) it also considers the dynamic version of the problem. In the dynamic setup, the extent measure problems are studied very little, and in particular, the k-center problem is not at all studied for any k ≥ 2.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".