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On Optimum Switch Box Designs for 2-D FPGAs

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Unformatted text preview: fixed, each a unique , we may have more than one as varies but there are finitely many such ’s for all ). , ¥ ¦¥  £   PR   I P I P I. Find and all -way minimal global routings. The existence of and the finiteness of the number of minimal -way global routings are guaranteed by Lemma 1. , ¤ IP ¥ V H£ E  U¥ HP ¥ E ‰£ I  ‰I PP ¥ ¨¤ ¡ be a hyper-universal L EMMA 2. Let -design. Then . restricting on any two parts gives a hyper-universal -design, and restricting on any three parts gives a hyper-universal -design. The optimum -design is a perfect matching, and an optimum -design is a Hamilton cycle. Moreover, the optimum -design must be a Hamiltonian cycle. In [8], we have developed a general reduction technique for designing S-boxes. , 8 L EMMA 1. For any integer with , there exists an integer such that any -way global routing could be decomposed into minimal -way subglobal routings with densities at most . Moreover, for . , ¥ U¥  £  For Step III, [8] gave a hyper-universal -design with less than switches. Our goal in this paper is to further investigate Step III to obtain better -designs for and and hence obtain a better -design than the -design constructed in [8]. The following result was proved in [8] which will be used in this paper. ¥£ U¥ ! Our approach depends on a very nice decomposition property of global routings. Let be a -global routing and be a sub-collection of . If is a -global routing with , is called a sub-global routing o...
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