Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima
Bibliographic record
Abstract
We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of $2n+o(n)$ bits, but the space drops to $1.736n + o(n)$ bits on average for random permutations of $n$ elements, and $2\lg\binom nr + o(n)$ for arrays with $r$ increasing runs, respectively. Both results are optimal; the former answers an open problem of Davoodi et al. (2014) and Golin et al. (2016). Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that they simultaneously achieve the optimal space usage to within lower order terms for a wide range of distributions over tree shapes, including: binary search trees (BSTs) generated by insertions in random order / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees.
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.006 | 0.011 |
| Research integrity | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
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".