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 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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.038 | 0.001 |
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 teacher head, 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".