Succinct indexable dictionaries with applications to encoding <i>k</i> -ary trees, prefix sums and multisets
Bibliographic record
Abstract
We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,…, m − 1} for some integer m while supporting the operations of rank( x ), which returns the number of elements in S that are less than x if x ∈ S , and −1 otherwise; and select( i ), which returns the i th smallest element in S . We give a data structure that supports both operations in O (1) time on the RAM model and requires B( n, m ) + o ( n ) + O (lg lg m ) bits to store a set of size n , where B( n, m ) = ⌊lg ( m / n )⌋ is the minimum number of bits required to store any n -element subset from a universe of size m . Previous dictionaries taking this space only supported (yes/no) membership queries in O (1) time. In the cell probe model we can remove the O (lg lg m ) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001]. We present extensions and applications of our indexable dictionary data structure, including: —an information-theoretically optimal representation of a k -ary cardinal tree that supports standard operations in constant time; —a representation of a multiset of size n from {0,…, m − 1} in B( n, m + n ) + o ( n ) bits that supports (appropriate generalizations of) rank and select operations in constant time; and + O (lg lg m ) —a representation of a sequence of n nonnegative integers summing up to m in B( n, m + n ) + o ( n ) bits that supports prefix sum queries in constant time.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".