Embeddings of complete binary trees into star graphs with congestion 1
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Bibliographic record
Abstract
Gives a construction of embeddings of vertex-congestion 1 and dilation 4 of complete binary trees into star graphs. The height of the trees embedded into the n-dimensional star graph S/sub n/ is (n+1)[log/sub 2/ n]-2/sup [log n]+1/+1, which improves the previous result from Bouabdallah and Heydemann (1993, 1994) by more than n/2-1. We then construct embeddings of vertex-congestion 1, dilation at most (5n/4)+2, of complete binary trees into the n-dimensional star graph, whose height differs from the theoretical upper bound of log/sub 2/n! by less than 3[log/sub 2/n]. Our results show that the star networks can efficiently simulate algorithms that are intended for a binary tree architecture.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
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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 | 0.000 |
| 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 it