Succinct geometric indexes supporting point location queries
Bibliographic record
Abstract
We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time. We also design three variants of this index. The first supports point location using lg n + 2 √lg n + O(lg1/4n) point-line comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H + 1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg2n) time.
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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 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.011 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".