Expected complexity of routing in $\Theta_6$ and half-$\Theta_6$ graphs
Bibliographic record
Abstract
We study online routing algorithms on the $\Theta_6$-graph and the half-$\Theta_6$-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex $s$ and a target vertex $t$ in the $\Theta_6$-graph (resp. half-$\Theta_6$-graph), there exists a deterministic online routing algorithm that finds a path from $s$ to $t$ whose length is at most $2\|st\|$ (resp. $2.89\|st\|$) which is optimal in the worst case [Bose et al, SIAM J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $\Theta_6$-graph and half-$\Theta_6$-graph defined on a Poisson point process. For the $\Theta_6$-graph, our online routing algorithm has an expected routing factor of $1.161$ when $s$ and $t$ are random. The routing factor is the length of the route between $s$ and $t$ produced by our algorithm divided by the Euclidean distance between $s$ and $t$. Moreover, our routing algorithm has a maximum expected routing factor of $1.22$, where the maximum is for fixed $s$ and $t$ and all other points are random. This is much better than the worst-case routing ratio of $2$. The routing ratio is the maximum routing factor among all pairs of points. For the half-$\Theta_6$-graph, our memoryless online routing algorithm has an expected routing factor of $1.43$ and a maximum expected routing factor of $1.58$. Our online routing algorithm that uses a constant amount of additional memory has an expected routing factor of $1.34$ and a maximum expected routing factor of $1.40$. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing factor that is much better than their worst-case routing ratio of $2.89$.
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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 | 0.002 | 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 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".