Dynamic graph connectivity in polylogarithmic worst case time
Bibliographic record
Abstract
The dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes which is undergoing a sequence of edge insertions and deletions, answer queries of the form q(a, b): Is there a path between nodes a and b? While data structures for this problem with polylogarithmic amortized time per operation have been known since the mid-1990's, these data structures have Θ(n) worst case time. In fact, no previously known solution has worst case time per operation which is o(√n).We present a solution with worst case times O(log4n) per edge insertion, O(log5n) per edge deletion, and O(log n/log log n) per query. The answer to each query is correct if the answer is yes and is correct with high probability if the answer is no. The data structure is based on a simple novel idea which can be used to quickly identify an edge in a cutset.Our technique can be used to simplify and significantly speed up the preprocessing time for the emergency planning problem while matching previous bounds for an update, and to approximate the sizes of cutsets of dynamic graphs in time O(min{|S|, |V\S|}) for an oblivious adversary.
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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 | 0.004 | 0.030 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.006 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.008 | 0.017 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.004 | 0.005 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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".