Space-Efficient Data Structures for Polyominoes and Bar Graphs
Bibliographic record
Abstract
We provide a compact data structure for representing polyominoes that supports neighborhood and visibility queries. Neighborhood queries concern reporting adjacent cells to a given cell, and visibility queries determine whether a straight line can be drawn within the polyomino that connects two specified cells. For an arbitrary small ϵ > 0, our data structure can encode a polyomino with n cells in (3 + ϵ)n + o(n) bits while supporting all queries in constant time. The space complexity can be improved to 3n + o(n), while supporting neighborhood queries in $\mathcal{O}(1)$ and visibility queries in $\mathcal{O}(t(n))$ for any arbitrary t(n) ∈ ω(1). Previous attempts at enumerating polyominoes have indicated that at least 2.00091n−o(n) bits are required to differentiate between distinct polyominoes, which shows our data structure is compact.In addition, we introduce a succinct data structure tailored for bar graphs, a specific subclass of polyominoes resembling histograms. We show that a bar graph comprising n cells can be encoded using n + o(n) bits, enabling constant-time query processing. Meanwhile, n − 1 bits are necessary to represent any bar graph, proving our data structure is succinct.
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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 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.003 |
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".