Statistical analysis and optimization in the presence of gate and interconnect delay variations.

Statistical Analysis and Optimization in the Presence of
Gate and Interconnect Delay Variations
[Extended Abstract]
Chandu Visweswariah
IBM Thomas J. Watson Research Center
Yorktown Heights, NY, USA
[email protected]
Categories and Subject Descriptors
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General Terms
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Keywords
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1.
ABSTRACT
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SLIP’06 March 4–5, 2006, Munich, Germany
Copyright 2006 ACM 1-59593-255-0/06/0003 ... ’ 5.00.
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2. ACKNOWLEDGMENTS
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3.
§ ¨Œ©}ªc«^REFERENCES
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