The exact channel density and compound design for generic universal switch blocks
Bibliographic record
Abstract
A switch block of k sides W terminals on each side is said to be universal (a ( k , W )-USB) if it is routable for every set of 2-pin nets of channel density at most W . The generic optimum universal switch block design problem is to design a ( k , W )-USB with the minimum number of switches for every pair of ( k , W ). This problem was first proposed and solved for k =4 in Chang et al. [1996], and then solved for even W or for k ≤6 in Shuy et al. [2000] and Fan et al. [2002b]. No optimum ( k , W )-USB is known for k ≥7 and odd W ≥3. But it is already known that when W is a large odd number, a near-optimum ( k , W )-USB can be obtained by a disjoint union of ( W − f 2 ( k ))/2 copies of the optimum ( k , 2)-USB and a noncompound ( k , f 2 ( k ))-USB, where the value of f 2 ( k ) is unknown for k ≥8. In this article, we show that f 2 ( k ) = k +3− i /3, where 1≤ i ≤6 and i ≡ k (mod 6), and present an explicit design for the noncompound ( k , f 2 ( k ))-USB. Combining these two results we obtain the exact designs of ( k , W )-USBs for all k ≥7 and odd W ≥3. The new ( k , W )-USB designs also yield an efficient detailed routing algorithm.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".