Fragile complexity of adaptive algorithms
Bibliographic record
Abstract
The fragile complexity of a comparison-based algorithm is $f(n)$ if each input<br/>element participates in $O(f(n))$ comparisons.<br/>In this paper, we explore the fragile complexity of algorithms adaptive to<br/>various restrictions on the input, i.e., algorithms with a fragile complexity<br/>parameterized by a quantity other than the input size~$n$. We show<br/>that searching for the predecessor in a sorted array has fragile complexity<br/>$\Theta(\log k)$, where $k$ is the rank of the query element, both<br/>in a randomized and a deterministic setting. For predecessor<br/>searches, we also show how to optimally reduce the amortized fragile complexity<br/>of the elements in the array. We also prove the following results:<br/>Selecting the $k$th smallest element has expected fragile complexity<br/>$O(\log\log k)$ for the element selected. Deterministically finding<br/>the minimum element has fragile complexity $\Theta(\log(\INV))$ and<br/>$\Theta(\log(\RUNS))$, where $\INV$ is the number of inversions in a<br/>sequence and $\RUNS$ is the number of increasing runs in a sequence.<br/>Deterministically finding the median has fragile complexity<br/>$O(\log(\RUNS) + \log\log n)$ and $\Theta(\log (\INV))$.<br/>Deterministic sorting has fragile complexity $\Theta(\log (\INV))$ but it has<br/>fragile complexity $\Theta(\log n)$ regardless of the number of runs.<br/>
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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 | 0.004 | 0.040 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.011 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.012 | 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".