On Advice Complexity of the k-server Problem under Sparse Metrics
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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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 | 0.003 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 0.001 |
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".