Comparison-based time-space lower bounds for selection
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Bibliographic record
Abstract
We establish the first nontrivial lower bounds on time-space tradeoffs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(n log logS n) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the read-only input array. This bound is tight for all S À logn, and remains true even if the array is given in a random order. Our result thus answers a 16-year-old question of Munro and Raman, and also complements recent lower bounds that are restricted to sequential access, as in the multi-pass streaming model [Chakrabarti et al., SODA 2008]. We also prove that any comparison-based, determinis-tic, multi-pass streaming algorithm for finding the median requires Ω(n log∗(n/s) + n logs n) worst-case time (in scan-ning plus comparisons), if we have s cells of space. This bound is also tight for all s À log2 n. We get deterministic lower bounds for I/O-efficient algorithms as well. All proofs in this paper involve “elementary ” techniques only. 1
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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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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