Optimal search trees with 2-way comparisons?
Why this work is in the frame
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Bibliographic record
Abstract
Abstract. In 1971, Knuth gave an O(n2)-time algorithm for the clas-sic problem of finding an optimal binary search tree. Knuth’s algorithm works only for search trees based on 3-way comparisons, but most modern computers support only 2-way comparisons (<, ≤, =, ≥, and>). Un-til this paper, the problem of finding an optimal search tree using 2-way comparisons remained open — poly-time algorithms were known only for restricted variants. We solve the general case, giving (i) an O(n4)-time al-gorithm and (ii) an O(n logn)-time additive-3 approximation algorithm. For finding optimal binary split trees, we (iii) obtain a linear speedup and (iv) prove some previous work incorrect. 1 Background and statement of results In 1971, Knuth [10] gave an O(n2)-time dynamic-programming algorithm for a classic problem: given a set K of keys and a probability distribution on queries, find an optimal binary-search tree T. As shown in Fig. 1, a search in such a tree for a given value v compares v to the root key, then (i) recurses left if v is smaller, (ii) stops if v equals the key, or (iii) recurses right if v is larger, halting at a leaf. The comparisons made in the search must suffice to determine the relation of v to all keys in K. (Hence, T must have 2|K | + 1 leaves.) T is optimal if it has minimum cost, defined as the expected number of comparisons assuming the query v is chosen randomly from the specified probability distribution. Knuth assumed three-way comparisons at each node. With the rise of higher-level programming languages, most computers began supporting only two-way
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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 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it