Faster Path Queries in Colored Trees via Sparse Matrix Multiplication and Min-Plus Product
Bibliographic record
Abstract
Let T be an ordinal tree on n nodes in which each node is assigned a color. We consider the batched colored path counting problem and the batched path mode/least frequent element query problem, in which given n query paths, each identified by a pair of nodes in T, one is asked to answer queries of the following forms: How many distinct colors are there on each query path (i.e. the colored path counting problem); what is the color on each query path that occurs at least/most as frequently as any other colors (i.e. the path mode/least frequent element query problem). By reducing the batched colored path counting problem to sparse matrix multiplication, we design a solution that answers n colored path counting queries in Õ(n^{2ω/(ω+1)}) = O(n^1.40704) time in total, while we reduce batched path mode/least frequent element query to the min-plus-query-witness problem so that we can answer a batch of n queries in Õ(n^{{24+2ω}/{17+ω}}) = O(n^1.483814) time. Previously, both problems could only be solved in Õ(n^1.5) time. Based on similar techniques, we design a dynamic colored path counting structure supporting both queries and updates in Õ(n^{{ω+1}/{ω+3}}) = O(n^0.627759) time, while our dynamic path mode/least frequent element query structures support each operation in Õ(n^{{16+ω(1,2,1)}/{26+ω(1,2,1)}}) = O(n^0.658139) time, where ω(1, 2, 1) denotes the minimum value such that the product of an n × n² matrix and an n² × n matrix can be computed in O(n^{ω(1, 2, 1)+ε}) time for any constant ε > 0. We also solve batched range mode/least frequent element query problems over arrays in Õ(n^{{18+2ω}/{13+ω}}) = O(n^1.479603) time. Both problems can be viewed as special cases of these batched path queries, and previously, the fastest algorithm for batched range mode queries and batched range least frequent element queries use O(n^1.4805) and Õ(n^1.5) time, respectively.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".