Upper and Lower Bounds for Competitive Online Routing on Delaunay\n Triangulations
Bibliographic record
Abstract
Consider a weighted graph G where vertices are points in the plane and edges\nare line segments. The weight of each edge is the Euclidean distance between\nits two endpoints. A routing algorithm on G has a competitive ratio of c if the\nlength of the path produced by the algorithm from any vertex s to any vertex t\nis at most c times the length of the shortest path from s to t in G. If the\nlength of the path is at most c times the Euclidean distance from s to t, we\nsay that the routing algorithm on G has a routing ratio of c.We present an\nonline routing algorithm on the Delaunay triangulation with competitive and\nrouting ratios of 5.90. This improves upon the best known algorithm that has\ncompetitive and routing ratio 15.48. The algorithm is a generalization of the\ndeterministic 1-local routing algorithm by Chew on the L1-Delaunay\ntriangulation. When a message follows the routing path produced by our\nalgorithm, its header need only contain the coordinates of s and t. This is an\nimprovement over the currently known competitive routing algorithms on the\nDelaunay triangulation, for which the header of a message must additionally\ncontain partial sums of distances along the routing path.We also show that the\nrouting ratio of any deterministic k-local algorithm is at least 1.70 for the\nDelaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case\nof the L1-Delaunay triangulation, this implies that even though there exists a\npath between two points x and y whose length is at most 2.61|[xy]| (where\n|[xy]| denotes the length of the line segment [xy]), it is not always possible\nto route a message along a path of length less than 2.70|[xy]|. From these\nbounds on the routing ratio, we derive lower bounds on the competitive ratio of\n1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations.\n
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How this classification was reachedexpand
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 | 0.000 |
| Open science | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".