Finding the shortest bottleneck edge in a parametric minimum spanning tree
Bibliographic record
Abstract
The result. Parametric optimization pwblems that concern graphs with continuously changing edge weights have been explored by numerous researctmrs, with moti-vation ranging fl'om sensitivity analysis to mobile-data applications. For instance, Dey [5] has shown that for an undirected graph with n vertices and m edges where the edge weights are linear functions in one parameter ("time"), the minimum spanning tree (MST) can un-dergo at most O(mn 1/3) changes (edge swaps). Agarwal et al. [1] have given data structures to maintain the MST over time, with a cost of O(n 2/3 polylog n) per change. Fernandez-Baca et al. [7] have given an algorithm to compute all changes to the MST in O(mnlog n) total time. In this note, we focus on a problem studied by Katoh and Tokuyama [8]: Given a parametric graph with edge weights changing linearly in time, find the time value when the weight of the largest MST edge (the so-called bottleneck edge) is minimized. The bottleneck edge weight is of particular interest, because it represents a threshold for connectivity: it is equal to the smallest value r such that the subgraph of edges with weight < r stays connected. For this problem, Katoh and ~Ibkuyama [8] have given an O((ms/7nU7 + mn U3) polylogn) algorithm, which is faster than the current methods for computing all MSTs over time. Katoh and Tokuyama's method uses advanced ata structures for range searching and is therefore difficult to implement. Here, we give a much faster and simpler randomized algorithm that runs in O(n(m/n)Slogn + m) expected time for any fixed ¢> 0. This time bound is at least as good as 1 O(mlogn) and O(nlog + ~ n + m) for any fixed ~ '> 0, and almost matches an fi(n log n+m) lower bound. The new result is obtained by an interesting combination of techniques from computational geometry and graph data structures. The geometric aspects are similar to those used for the problem of 2-d feasible linear programming with violations fi'om a previous paper [4].
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".