Implicit dictionaries with O(1) modifications per update and fast search
Bibliographic record
Abstract
The implicit dictionary problem is that of maintaining a dynamic ordered set, S, under the operations search, insert and delete, so that the elements of S are stored in the first |S| locations of an array. No operations are permitted on the data other than comparisons (≤) and interchanges. The only auxiliary memory permitted is a constant number of O(log |S|) bit integers. The organization will, then, rely heavily on the permutations of the relative order of the values in which the data is stored. While such a structure can be maintained in O(log |S|) time, the most interesting lower bound on the topic is that of Borodin, Fich, Meyer auf der Heide, Upfal and Wigderson [3]. They proved a tradeoff between search and update time in implicit dictionaries: if the update cost (comparisons and exchanges) is O(1), then the search cost must be Ω(|S|e), for some constant e > 0. The authors left open the question of whether such a tradeoff would hold if only the modifications performed during an update were considered. They conjectured that any implicit dictionary performing only O(1) exchanges per update should very quickly become disorganized, and so require Ω(|S|e) comparisons per search. We answer this long-standing open question by disproving the conjecture.
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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 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.011 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".