Unified Quadratic Programming Approach for Mixed Mode Placement

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Unified Quadratic Programming Approach for Mixed
Mode Placement

• Bo Yao, Hongyu Chen, Chung-Kuan Cheng, Nan-Chi
  Chou, Lung-Tien Liu, Peter Suaris
• CSE Department
• University of California, San Diego
• Mentor Graphics Corporation

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Outline

• Introduction to the mixed mode placement
• Unified cost function
• DCT based cell density cost
• Experimental results
• Conclusions

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Mixed Mode Placement

• Common design needs
• Mixed signal designs (analog and RF parts are
  macros)
• Hierarchical design style
• IP blocks
• Memory blocks
• Challenges for placement
• Huge amount of components
• Heterogeneous module sizes/shapes

Analog
IP
Memory
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Previous Works on Mixed Mode Placement

• Combined floorplanning and std. cell placement
• Capo (Markov, ISPD 02, ICCAD 2003)
• Multi-level annealing placement
• mPG-MS (Cong, ASPDAC 2003)
• Partitioning based approaches
• Feng Shui (Madden, ISPD 04)
• Force-directed / analytical approaches
• Kraftwork (Eisenmann and Johannas, DAC 98 )
• FastPlace (Chu, ISPD 04)
• APlace (Kahng, ISPD 04, ICCAD 04)

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UPlace Optimization Flow
Analytical Placement
Discrete Optimization
Detailed Placement
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Unified Cost Function

• Combined object function for global placement
• DP Penalties for un-even cell densities
• WL Wire length cost function
• Quadratic analytical placement
• WL 1/2xTQxpx 1/2yTQypy
• Bounding box wire length for discrete optimization

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Cell Density

• Common strategy
• Partition the placement area into N by N rooms
• Cell density matrix D dij
• dij is the total cell area in room (i,j)

A
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DCT Cell Density in Frequency Domain

• 2-D Discrete Cosine Transform (DCT)
• Cell density matrix D gt Frequency matrix F
  fij
• where fij is the weight of density pattern (i,j)

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Properties of Frequency Matrix

• Each fuv is the weight of frequency (u,v)
• Inverse DCT recovers the cell density

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Frequency Matrix An Example

• Density matrix D and frequency matrix F

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Properties of DCT

• Cell density energy ?dij2 ?fij2
• Cell perturbation and frequency matrix
• Uniform density ? fij 0

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Density Cost of a Placement

• Weighted sum of fij2
• Higher weight for lower frequency

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Approximation of the Density Cost

• Approximate the density cost with a quadratic
  function
• DP ½aixi2 bixici
• Make DP convex
• ai gt 0 to ensure
• Matrix form
• DP ½xTAxBx
• A diag(a1, a2, , an),
• B(b1, b2, , bn)T

DP
ai gt 0
xi
x-? x x?
DP
ai 0
xi
x-? x x?
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UPlace Minimize Combined Objective Function

• Combine quadratic objectives
• WL ?DP
• WL ½ xTQxpx
• DP ½xTAxBx
• Solve linear equation for each minimization
• (Q ?A)x -p - ?B
• Use Lagrange relaxation to reduce cell congestion
• ?(k1) ?(k) ?(k) DP
• ?0 0, ?0 Const
• ?(k1) ?(k) ?, 0lt ? ? 1

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Discrete Optimization

• Try ?-distance moves in four directions. Pick the
  best position.
• Sweep all the cells in each iteration

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Legalization/Detailed Placement Zone Refinement

• One cell a time, ceiling -gt floor
• Two directional alternation

A
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Experimental Results Wire length
Normalized Wire Length
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Experimental results CPU time
CPU (Min)
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UPlace Placement Results

• IBM-02

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Conclusions

• We propose a unified cost function for global
  optimization, which provides good trade-offs
  between wire length minimization and cell
  spreading.
• We introduce a DCT based cell density calculation
  method, and a quadratic approximation.
• The unified placement approach generates
  promising results on mixed mode designs.

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Thank You !