Time-space tradeoffs for all-nearest-larger-neighbors problems
Bibliographic record
Abstract
This paper addresses two versions of a fundamental problem, referred to as the All-Nearest-Larger-Neighbors (ANLN) problem, defined as follows: given a one-dimensional array A of n real-valued keys, find, for each array element A[i], the index of a nearest array element, if one exists, whose key is strictly larger than A[i]. We develop algorithms for one- and two-sided versions of the ANLN problem that run in O(n logbn) time, using Θ(b) work-space, for all b=O(n), exhibiting a full time-space tradeoff that subsumes all known (memory-restricted) special cases. In addition, a non-trivial lower bound is developed for the time complexity of solving both versions on a pointer machine with limited work-space. This lower bound matches the time complexity of our algorithms, when restricted to constant space. The fundamental nature of ANLN problems make them intrinsically interesting to study. They also capture the essence of a variety of other familiar problems, such as determining the forest structure associated with a given string of nested parentheses, and triangulating monotone polygons. For both of these, we describe reductions to versions of the ANLN problem, achieving the same time-space tradeoffs.
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How this classification was reachedexpand
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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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 teacher head, 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".