The exact channel density and compound design for generic universal switch blocks

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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.