Succinct indexable dictionaries with applications to encoding <i>k</i> -ary trees, prefix sums and multisets
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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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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 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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