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We consider the optimal design problem for arbitrary-shaped switch box, (r_1, \ldots, r_k){\hbox{-}}{\rm SB}, in which r_i terminals are located on side i for i = 1, \ldots, k and programmable switches are joining pairs of terminals from different sides. Previous investigations on switch box designs mainly focused on regular switch boxes in which all sides have the same number of terminals. By allowing different numbers of terminals on different sides, irregular switch boxes are more general and flexible for applications such as customized FPGAs and reconfigurable interconnection networks. The optimal switch box design problem is to design a switch box satisfying the given shape and routing capacity specifications with the minimum number of switches. We present a decomposition design method for a wide range of irregular switch boxes. The main idea of our method is to model a routing requirement as a nonnegative integer vector satisfying a system of linear equations and then derive a decomposition theory of routing requirements based on the theory of systems of linear Diophantine equations. The decomposition theory makes it possible to construct a large irregular switch box by combining small switch boxes of fixed sizes. Specifically, we can design a family of hyperuniversal (universal) (w {\bf{d}}+{\bf{c}}){\hbox{-}}{\rm SBs} with \Theta(w) switches, where {\bf{d}} and {\bf{c}} are constant vectors and w is a scalar. We illustrate the design method by designing a class of optimal hyperuniversal irregular 3-sided switch boxes and a class of optimal rectangular universal switch boxes. Experimental results on the rectangular universal switch boxes with the VPR router show that the optimal design of irregular switch boxes does pay off.
FPGA, reconfigurable interconnection network, switch box, switch block, universal, hyperuniversal.

Y. Wu, R. C. Cheung, J. Liu and H. Fan, "Decomposition Design Theory and Methodology for Arbitrary-Shaped Switch Boxes," in IEEE Transactions on Computers, vol. 55, no. , pp. 373-384, 2006.
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