On Advice Complexity of the k-server Problem under Sparse Metrics
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Bibliographic record
Abstract
We consider the k-server problem under the advice model of computation when the underlying metric space is sparse. On one side, we show that an advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of size n, even for the 2-server problem on a path metric of size N >= 5. Through another lower bound argument, we show that at least (n/2)(log α - 1.22) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 <= α < 2k. On the other side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs, which require advice of (almost) linear size. Namely, we show that for graphs of size N and treewidth α, there is an online algorithm which receives $O(n (log α + log log N))$ bits of advice and optimally serves a sequence of length n. With a different argument, we show that if a graph admits a system of μ collective tree (q,r)-spanners, then there is a (q+r)-competitive algorithm which receives O(n (log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, provided with O(n log log N) bits of advice.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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