Parks and Recreation: Color Fault-Tolerant Spanners Made Local
Bibliographic record
Abstract
We provide new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to $f$ failures of entire color classes. The failure of even a single color may cause a linear number of individual edge/vertex faults. In a recent work, Petruschka, Sapir and Tzalik [ITCS `24] gave tight bounds for the (worst-case) size $s$ of such spanners, where $s=Θ(f n^{1+1/k})$ or $s=Θ(f^{1-1/k} n^{1+1/k})$ for spanners with stretch $(2k-1)$ that are resilient to at most $f$ edge- or vertex-color faults, respectively. Additionally, they showed an algorithm for computing spanners of size $\tilde{O}(s)$, running in $\tilde{O}(msf)$ sequential time, based on the (FT) greedy spanner algorithm. The problem of providing faster and/or distributed algorithms was left open therein. We address this problem and provide a novel variant of the classical Baswana-Sen algorithm [RSA `07] in the spirit of Parter's algorithm for vertex fault-tolerant spanners [STOC `22]. In a nutshell, our algorithms produce color fault-tolerant spanners of size $\tilde{O}_k (s)$ (hence near-optimal for any fixed $k$), have optimal locality $O(k)$ (i.e., take $O(k)$ rounds in the LOCAL model), can be implemented in $O_k (f^{k-1})$ rounds in CONGEST, and take $\tilde{O}_k (m + sf^{k-1})$ sequential time. To handle the considerably more difficult setting of color faults, our approach differs from [BS07, Par22] by taking a novel edge-centric perspective, instead of (FT)-clustering of vertices; in fact, we demonstrate that this point of view simplifies their algorithms. Another key technical contribution is in constructing and using collections of short paths that are "colorful at all scales", which we call "parks". These are intimately connected with the notion of spread set-systems that found use in recent breakthroughs regarding the famous Sunflower Conjecture.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".