Bibliographic record
Abstract
We introduce the lazy search tree data structure. The lazy search tree is a comparison-based data structure on the pointer machine that supports order-based operations such as rank, select, membership, predecessor, successor, minimum, and maximum while providing dynamic operations insert, delete, change-key, split, and merge. We analyze the performance of our data structure based on a partition of current elements into a set of gaps $\{\Delta_i\}$ based on rank. A query falls into a particular gap and splits the gap into two new gaps at a rank $r$ associated with the query operation. If we define $B = \sum_i |\Delta_i| \log_2(n/|\Delta_i|)$, our performance over a sequence of $n$ insertions and $q$ distinct queries is $O(B + \min(n \log \log n, n \log q))$. We show $B$ is a lower bound. Effectively, we reduce the insertion time of binary search trees from $\Theta(\log n)$ to $O(\min(\log(n/|\Delta_i|) + \log \log |\Delta_i|, \; \log q))$, where $\Delta_i$ is the gap in which the inserted element falls. Over a sequence of $n$ insertions and $q$ queries, a time bound of $O(n \log q + q \log n)$ holds; better bounds are possible when queries are non-uniformly distributed. As an extreme case of non-uniformity, if all queries are for the minimum element, the lazy search tree performs as a priority queue with $O(\log \log n)$ time insert and decrease-key operations. The same data structure supports queries for any rank, interpolating between binary search trees and efficient priority queues. Lazy search trees can be implemented to operate mostly on arrays, requiring only $O(\min(q, n))$ pointers. Via direct reduction, our data structure also supports the efficient access theorems of the splay tree, providing a powerful data structure for non-uniform element access, both when the number of accesses is small and large.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.012 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.006 |
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".