Weighted Region Problem in Arrangement of Lines
Bibliographic record
Abstract
In this paper, a geometric shortest path problem in weighted regions is discussed. An arrangement of lines A, a source s, and a target t are given. The objective is to nd a weighted shortest path, st, from s to t. Existing approximation algorithms for weighted shortest paths work within bounded regions (typically triangulated). To apply these algorithms to unbounded regions, such as arrangements of lines, there is a need to bound the regions. Here, we present a minimal region that contains st, called SP-Hull of A. It is a closed polygonal region that only depends on the geometry of the arrangement A and is independent of the weights. It is minimal in the sense that for any arrangement of lines A, it is possible to assign weights to the faces of A and choose s and t such that st is arbitrary close to the boundary of SP-Hull of A. We show that SP-Hull can be constructed in O(n logn) time, where n is the number of lines in the arrangement. As a direct consequence we obtain a shortest path algorithm for weighted arrangements of lines.
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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 | 0.004 |
| 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 | 0.001 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".