Optimal search trees with 2-way comparisons?
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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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.017 | 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".