Dynamic Maintenance of Low-Stretch Probabilistic Tree Embeddings with Applications
Bibliographic record
Abstract
We give the first non-trivial fully dynamic probabilistic tree embedding algorithm for a weighted, undirected graph G with n nodes and at most m edges undergoing edge insertions and deletions. The goal in this problem is to maintain a tree containing all nodes of G with a randomized algorithm such that for every edge (u, v) of G the expected length of the path from u to v in the tree exceeds the weight of the edge (u, v) only by a small multiplicative factor, called the stretch of the embedding. In this paper, we obtain a trade-off between amortized update time and expected stretch against an oblivious adversary. At the two extremes of this trade-off, we can maintain a tree of expected stretch O(log4 n) with update time m1/2+o(1) or a tree of expected stretch no(1) with update time no(1) (for edge weights polynomial in n). A guarantee of the latter type has so far only been known for maintaining tree embeddings with average (instead of expected) stretch [Chechik/Zhang, SODA '20]. Our main result has direct implications to fully dynamic approximate distance oracles and fully dynamic buy-at-bulk network design as our trade-off from above carries over to these two problems with minor overheads. For dynamic distance oracles, our result is the first to break the update-time barrier. For buy-at-bulk network design, a problem which also in the static setting heavily relies on probabilistic tree embeddings, we give the first non-trivial dynamic algorithm. As probabilistic tree embeddings are an important tool in static approximation algorithms, we expect our result to have further applications in dynamic approximation algorithms. From a technical perspective, we obtain our main result by first designing a decremental (i.e., deletionsonly) algorithm for probabilistic low-diameter decompositions via a careful combination of Bartal's ball-growing approach [FOCS ‘96] with the pruning framework of Chechik and Zhang [SODA ‘20]. Such a low-diameter decomposition is the heart of Bartal's seminal tree embedding construction and we show how to adapt it to the decremental setting. We then extend this to a fully dynamic algorithm by significantly enriching a well-known “decremental to fully dynamic” reduction with a new bootstrapping idea to recursively employ a fully dynamic algorithm instead of a static one in this reduction. By additionally exploiting certain properties of our tree embedding, this bootstrapping scheme can be made highly efficient.
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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 | 0.002 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.010 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.002 | 0.003 |
| 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".