On Optimum Switch Box Designs for 2-D FPGAs

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Unformatted text preview: ¥ ¤ ¥ ¦¥ §¤ ¨¦  ¥ E IPPPPP P ROOF. If , then and any -global routing can be decomposed into a union of global routings and a -global routing. (sometimes, can be decomposed into -global routings and a -design. But this does not always happen.) Now it is easy to see that contains a detailed routing of as is a disjoint union of ’s and an , and and are hyper-universal design and -design, respectively by Lemma 4. Let where . Since the densities of minimal -way global routings are or , any -global routing can be decomposed into -global routings and a global routing for . Since is a disjoint union  ¥ 4 4 4 4 4 8 4 4 4 ¥ ¥£ ¦¥ ! R2  , if  FF  FF  FF if if if if if if ’s and a ’s and a ’s and a ’s and a ’s and a 2 XSS 3 WWX "# WXS 33 8 WXS 3 4 WXSS 3 3 © C C ¥ ¥ ¤   †R  ¡ ‚§   2  2 ¥ HR  £ ¥ ¤¥ ¡ I P ©  ¥ $$8  $  $8    8     %%4 #  &%# "  &%!"  &0$4 !"  &$ %4 !# '# "8 $$   $  $  ...
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This document was uploaded on 12/26/2013.

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