Quasi-Universal Switch Matrices for FPD Design

An FPD switch module $M$ with $w$ terminals on each side is said to be universal if every set of nets satisfying the dimension constraint (i.e., the number of nets on each side of $M$ is at most $w$) is simultaneously routable through $M$ [8]. Chang et al. have identified a class of universal switch blocks in [8]. In this paper, we consider the design and routing problems for another popular model of switch modules called switch matrices. Unlike switch blocks, we prove that there exist no universal switch matrices. Nevertheless, we present quasi-universal switch matrices which have the maximum possible routing capacities among all switch matrices of the same size and show that their routing capacities converge to those of universal switch blocks. Each of the quasi-universal switch matrices of size $w$ has a total of only $14w-20$ ($14w-21$) switches if $w$ is even (odd), $w>1$, compared to a fully populated one which has $3w^2-2w$ switches. We prove that no switch matrix with less than $14w-20$ ($14w-21$) switches can be quasi-universal. Experimental results demonstrate that the quasi-universal switch matrices improve routabilty at the chip level.