Tight Bounds for Monotone Minimal Perfect Hashing
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Bibliographic record
Abstract
The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set \(S=\{s_{1},\ldots,s_{n}\}\) of \(n\) distinct keys from a universe \(U\) of size \(u\) , create a data structure \(\mathbf{D}\) that answers the following query: \(\rm{{R\small{ANK}}}(q)=\begin{cases}\text{rank of }q\text{ in }S&q\in S \\ \text{arbitrary answer}&\text{otherwise.}\end{cases}\) Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes \(\mathbf{D}\) in \(O(n\log\log\log u)\) bits and performs queries in \(O(\log u)\) time. It has been an open problem to either improve the space upper bound or to show that this somewhat odd looking bound is tight. In this article, we show the latter: any data structure (deterministic or randomized) for monotone minimal perfect hashing of any collection of \(n\) elements from a universe of size \(u\) requires \(\Omega(n\cdot\log\log\log{u})\) expected bits to answer every query correctly. We achieve our lower bound by defining a graph \(\mathbf{G}\) where the nodes are the possible \({u\choose n}\) inputs and where two nodes are adjacent if they cannot share the same \(\mathbf{D}\) . The size of \(\mathbf{D}\) is then lower bounded by the log of the chromatic number of \(\mathbf{G}\) . Finally, we show that the fractional chromatic number (and hence the chromatic number) of \(\mathbf{G}\) is lower bounded by \(2^{\Omega(n\log\log\log u)}\) .
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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 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 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