Multidimensional Included and Excluded Sums
Bibliographic record
Abstract
This paper presents algorithms for the included-sums and excluded-sums problems used by scientific computing applications such as the fast multipole method. These problems are defined in terms of a d-dimensional array of N elements and a binary associative operator ⊕ on the elements. The included-sum problem requires that the elements within overlapping boxes cornered at each element within the array be reduced using ⊕. The excluded-sum problem reduces the elements outside each box. The weak versions of these problems assume that the operator ⊕ has an inverse ⊖, whereas the strong versions do not require this assumption. In addition to studying existing algorithms to solve these problems, we introduce three new algorithms. The bidirectional box-sum (BDBS) algorithm solves the strong included-sums problem in Θ(dN) time, asymptotically beating the classical summed-area table (SAT) algorithm, which runs in Θ(2dN) and which only solves the weak version of the problem. Empirically, the BDBS algorithm outperforms the SAT algorithm in higher dimensions by up to 17.1×. The box-complement algorithm solves the strong excluded-sums problem in Θ(dN) time, asymptotically beating the state-of-the-art corners algorithm by Demaine et al., which runs in Ω(2dN) time. The box-complement algorithm empirically outperforms the corners algorithm by about 1.4× given similar amounts of space in three dimensions. If the assumptions for the weak excluded-sums problem can be satisfied, the bidirectional box-sum complement (BDBSC) algorithm, which is a trivial extension of the BDBS algorithm, can beat box-complement by up to a factor of 4.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.020 | 0.004 |
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".