Deterministic APSP, Orthogonal Vectors, and More
Why this work is in the frame
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Bibliographic record
Abstract
We show how to solve all-pairs shortest paths on n nodes in deterministic n 3> /2> Ω ( √ log n ) time, and how to count the pairs of orthogonal vectors among n 0−1 vectors in d = c log n dimensions in deterministic n 2−1/ O (log c ) time. These running times essentially match the best known randomized algorithms of Williams [46] and Abboud, Williams, and Yu [8], respectively, and the ability to count was open even for randomized algorithms. By reductions, these two results yield faster deterministic algorithms for many other problems. Our techniques can also be used to deterministically count k -satisfiability ( k -SAT) assignments on n variable formulas in 2 n - n / O ( k ) time, roughly matching the best known running times for detecting satisfiability and resolving an open problem of Santhanam [24]. A key to our constructions is an efficient way to deterministically simulate certain probabilistic polynomials critical to the algorithms of prior work, carefully applying small-biased sets and modulus-amplifying polynomials.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it