The Trie Measure, Revisited
Bibliographic record
Abstract
In this paper, we study the following problem: given $n$ subsets $S_1, \dots, S_n$ of an integer universe $U = \{0,\dots, u-1\}$, having total cardinality $N = \sum_{i=1}^n |S_i|$, find a prefix-free encoding $enc : U \rightarrow \{0,1\}^+$ minimizing the so-called trie measure, i.e., the total number of edges in the $n$ binary tries $\mathcal T_1, \dots, \mathcal T_n$, where $\mathcal T_i$ is the trie packing the encoded integers $\{enc(x):x\in S_i\}$. We first observe that this problem is equivalent to that of merging $u$ sets with the cheapest sequence of binary unions, a problem which in [Ghosh et al., ICDCS 2015] is shown to be NP-hard. Motivated by the hardness of the general problem, we focus on particular families of prefix-free encodings. We start by studying the fixed-length shifted encoding of [Gupta et al., Theoretical Computer Science 2007]. Given a parameter $0\le a < u$, this encoding sends each $x \in U$ to $(x + a) \mod u$, interpreted as a bit-string of $\log u$ bits. We develop the first efficient algorithms that find the value of $a$ minimizing the trie measure when this encoding is used. Our two algorithms run in $O(u + N\log u)$ and $O(N\log^2 u)$ time, respectively. We proceed by studying ordered encodings (a.k.a. monotone or alphabetic), and describe an algorithm finding the optimal such encoding in $O(N+u^3)$ time. Within the same running time, we show how to compute the best shifted ordered encoding, provably no worse than both the optimal shifted and optimal ordered encodings. We provide implementations of our algorithms and discuss how these encodings perform in practice.
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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 | 0.002 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.014 | 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".