Closing the Smoothness and Uniformity Gap in Area Fill Synthesis

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Closing the Smoothness and Uniformity Gap in Area
Fill Synthesis
Supported by Cadence Design Systems, Inc., NSF,
the Packard Foundation, and State of Georgias
Yamacraw Initiative
Y. Chen, A. B. Kahng, G. Robins, A. Zelikovsky
(UCLA, UCSD, UVA and GSU) http//vlsicad.ucsd.ed
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Fixed-Dissection Fill Problem

• Use dummy features to improve layout uniformity
  for CMP process

• Fixed-Dissection Fill Problem
• given
• rule-correct layout in n ? n region
• upper bound U on density

• partition layout into nr/w ? nr/w fixed
  dissections

• monitor only fixed set of windows consisting of
  r2 tiles
• fill layout subject to given constraints w.r.t.
  Min-Var or Min-Fill objective

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The Smoothness Gap

• Fixed-dissection analysis ? floating window
  analysis

• Fill result will not satisfy the given bounds
• Despite this gap (observed in 1998), all
  published filling methods fail to consider this
  smoothness gap

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Accurate Layout Density Analysis

• Optimal extremal-density analysis with complexity
• inefficient

• Multi-level density analysis algorithm

• any arbitrary window contains some shrunk on-grid
  window

• any arbitrary window is contained in some bloated
  on-grid window

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Smoothness Gap in Existing Methods

• Window density variation and violation of max
  window density in fixed-dissection filling are
  underestimated

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Local Density Variations

• Type I max density variation of every r
  neighboring windows in each row of the
  fixed-dissection

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Linear Programming Formulations

• Lipschitz Types

with

• Combined Objective
• linear summation of Min-Var, Lip-I and Lip-II
  objectives with specific coefficients
• add Lip-I and Lip-II constraints as well as

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Computational Experience

• Solutions with best Min-Var objective value do
  not always have the best value in terms of local
  smoothness
• LP with combined objective achieves best
  comprehensive solutions

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Thank you!