Criticality Computation in Parameterized Statistical Timing
Jinjun Xiong
Vladimir Zolotov
Natesan Venkateswaran Chandu Visweswariah
Univ. of California
IBM Research
IBM Sys. & Tech. Group
IBM Research
Los
Angeles,
CA
Yorktown
Heights,
NY
East
Fishkill,
NY
Yorktown
Heights, NY
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ABSTRACT
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C
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Complement slack computation
E
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Level 0
max(EFGH)
EDGE CRITICALITY COMPUTATION
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CONCLUSIONS AND DISCUSSION
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Edge
Edge
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7.! ý" çREFERENCES
û#é ò$í6ò$ñ1õé ñè&%(' ü í&ñ1öè&%öñíòPñ1ø|ù*) ñ1é + é ø,ó&è|é ê|ò% -.0/2143506878789
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! "TS ûUûVú|öè|ñ1ú|ò%WLû8#íø ñ1öí6ò$ñ1õñ1øX%çû#é ò?$íò$ñ1õé ñèX%ñ1ø)ù
#LûY÷+ ÷öA÷[Z\%('V4ñöAí³òAé ìé ø],Lú{òé ø],Lé ø)óõí^í6øöñ+ê|ñ1õñí&öAíõé ìí6ù
òöñ1öAé òöé óñ+öé é ø],ñ1ø|ñ+ _òAé ò% -R`?a0a0ab` ABFc; / A H Fc> 1 A H]d K 1 A[M?; / ;[A 3 ; 1 A
K 18Gfe E]Fc; /^g CT>ih8;jhk:<;4=j> @lA %ê|êû ýþ N ýþ O%W÷4Z1í^Qm"í6õn þþ1ÿ ûë{ñø
P÷1òAí]%çkïû
! " ïLûïf,ñõ$ñ+%o`ûçPè)÷1ê|õAñp% ü ûðq+ ññ1ú8$%ñø{ùr#LûY'÷+ ÷öA÷4Z\%('AçYé õó6ú|é ö
÷1ê|öé wé ìñöAé ÷øOú)òAé ø,òöñ1öé òöé ó6ñ+öAé é ø,rñø)ñ+ _òé ò% -s.0/143450687878t
:<;=?> @BADCEBF 18GIH FJ> 1 ALK 1 AM?; / ;4A 3 ; %ê)êû ý N %Pú|ø)íQ þ1þ1ÿ û
ïø|ñ1è|í6é R%çPïû
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P÷1òAí]%çkïû
! " çû#é ò$í6ò$ñ1õé ñè&%orû8Ukñ[Zé ø)ù{õñø&%o`ûo³ñ+ ñ) ñ+ ñO%ëûVOû}-ñ+ íõ%ñø{ù
ëûWñõñ4_1ñ1øw%('?~é õAòö:÷õù|íõPé ø|ó6õívwíøöñ+(mB+ ÷ó m{ñòíùOòöAñöé òöAé ó6ñ+
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