Efficient algorithms for centers and medians in interval and circular‐arc graphs
Bibliographic record
Abstract
Abstract Thep‐center problem is to locatepfacilities on a network so as to minimize the largest distance from a demand point to its nearest facility. Thep‐median problem is to locatepfacilities on a network so as to minimize the average distance from a demand point to its closest facility. We consider these problems when the network can be modeled by an interval or circular‐arc graph whose edges have unit lengths. We provide, given the interval model of annvertex interval graph, anO(n) time algorithm for the 1‐median problem on the interval graph. We also show how to solve thep‐median problem, for arbitraryp, on an interval graph inO(pnlogn) time and on a circular‐arc graph inO(pn2logn) time. We introduce a spring representation of the objective function and show how to solve thep‐center problem on a circular‐arc graph inO(pn) time, assuming that the arc endpoints are sorted. © 2002 Wiley Periodicals, Inc.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| 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".