Space Complexity of Minimum Cut Problems in Single-Pass Streams
Bibliographic record
Abstract
We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an Õ(n/ε) space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the Ω(n/ε²) space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of Õ(n/ε) for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is Õ(1), provided that the number of edges in the input graph is at least (n/ε²)^{1+o(1)}. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the exact minimum cut on a simple, unweighted graph using Õ(n) space. Finally, we give an Ω(n/ε²) space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.
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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 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 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".