Kein Folientitel

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Analog Placement with Common Centroid Constraints
Qiang Ma, Evangeline F. Y. Young and K. P. Pun
The Chinese University of Hong Kong Hong
Kong
8-Nov-2007
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Outline

• Introduction
• Problem formulation
• Methodology
• Experimental results
• An Alternative Grid-based Approach
• Conclusion

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Introduction

• In todays SoC design, a chip may consist of both
  digital and analog parts
• Need a compatible CAD tool for both the digital
  and analog parts
• In analog circuits, component mismatch has
  significant adverse effects on the performance
• Placement with symmetry and common centroid
  constraints help to reduce mismatch errors greatly

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Previous Work

• 1-D symmetry
• 1 KOAN/ANAGRAMII New Tools for Device-Level
  Analog Layout (Cohn et al. )
• 2 A Performance-driven Placement Tool for
  Analog Integrated Circuits. (Lampaert et al.)
• 3 Automation of IC Layout with Analog
  Constraints
• (Malavasi et al.)
• 4 Symmetry within the Sequence-Pair
  Representation in the Constext of Placement for
  Analog Design. (Balasa et al.)
• 5 Block Placement with Symmetry Constraints
  based on the O-tree Nonslicing Representation.(Pan
  g et al.)
• 6 On the Exploration of the Solution Space in
  Analog Placement with Symmetry Constraints.
  (Balasa et al.)
• 7 Analog Placement with Symmetry and Other
  Placement Constraints (Tam et al.)
• 8 Analog Placement Based on Novel
  Symmetry-Island Formulation (Lin et al.)

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Common Centroid Placement

• Difference between Common Centroid Placement and
  2-D Symmetry Placement
• In 2-D symmetry placement, blocks can only be
  placed along the horizontal and vertical axes of
  the symmetry group
• Thus 2-D symmetry placement is a small subset of
  common centroid placement

2-D symmetry
common centroid
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Problem Formulation

• Analog placement with common centroid constraints
• A set of blocks without constraints
• A set of nets
• A set of p common centroid groups G1,G2, , Gp
• Each group Gi consists of a number of si
  devices
• The si devices are with areas gi1, gi2, , gisi,
  respectively
• The si devices are sliceable and required to be
  placed in a common centroid geometry
• Cost function
• Cost Area ?Wirelength

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Methodology (C-CBL)

• Slice each device gij of Gi into two blocks aij
  and bij with exactly the same dimension
• Gi contains si pairs of blocks (ai1,bi1),(ai2,bi2
  ),,(aisi,bisi)
• Let the si pairs share a common centroid
• Center-based Corner Block List (C-CBL)
• Represent the placement of a common centroid
  group
• Extended from the basic CBL representation
• Complete and non-redundant in representing any
  common centroid mosaic packings with pairs of
  devices to be matched
• Global Sequence Pair
• Each common centroid group is regarded as a
  super-block
• Describe global topology between the super-blocks
  and other ordinary blocks
• Simulated Annealing is used as the basic
  searching engine

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A Common Centroid Placement Example

• Two common centroid groups
• G1 (a11,b11), (a12,b12), c
• G2 (a21,b21), (a22,b22), (a23,b23)

b12
a11
c
b11
a12
b22
a23
a21
b23
b21
a22
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Review of Corner Block List

• Corner Block List
• 3-tuple S,L,T
• A representation for mosaic floorplan
• Corner Block
• The block located at the upper right corner

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Review of Corner Block List (Cont)

• Corner Block List
• 3-tuple S, L, T
• S is a sequence of block names
• L is a sequence of block orientations
• T is a sequence of T-junction information
• Obtained by iteratively deleting Corner Block

a2
S b2 L 0 T 1
S b1,b2 L 1,0 T 1,1
S c,b1,b2 L 1,1,0 T 0,1,1
S a1,c,b1,b2 L 0,1,1,0 T 0,0,1,1
S a2,a1,c,b1,b2 L 0,1,1,0 T 0,0,1,1
CBL S (a2,a1,c,b1,b2), L (0,1,1,0), T
(0,0,1,1)
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C-CBL

• Center-based Corner Block List (C-CBL)
• Two corner blocks
• Upper-Right (UR) Corner Block
• Lower-Left (LL) Corner Block

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C-CBL (cont)

• Definition
• Corner Block Pair
• UR Corner Block and LL Corner Block form a Corner
  Block Pair iff they have the same orientation and
  T-junction information

UR Corner Block
b2
a1
b1
a2
LL Corner Block
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C-CBL (cont)

• C-CBL is a 4-tuple C, S, L, T
• C is a center block
• C can be a real block located at the center or a
  dummy block with 0 width and height
• S is a list of Corner Block Pairs
• L is a list of block orientations
• T is a list of T-junction information

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Derive C-CBL from packing
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Theorem 1

• A unique C-CBL can be obtained from a mosaic
• packing with n pairs of blocks (or n pairs
  plus one
• block) satisfying the common centroid
  constraint.

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Construct a packing from C-CBL
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Theorem 2

• A packing constructed from a C-CBL satisfies the
• common centroid constraint.

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Obtain exact coordinates

• Given a C-CBL representation C, S, L, T
• Construct HCG and VCG
• Longest Path algorithm
• Bottom-left compacted packing
• Not necessarily a common centroid placement

b2
b1
c
a1
h(Gi)
a2
w(Gi)
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Coordinates Adjustment

• Fix the coordinates of the LL corner blocks
• Recalculate the coordinates of the UR corner
  blocks
• bij.x w(Gi)-aij.x-width(bij),
• bij.y h(Gi)-aij.y-height(bij)

b2
b1
c
a1
h(Gi)
a2
Fix LL corner blocks
w(Gi)
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Global Sequence Pair

• Globally, use sequence pair as representation
• The common centroid groups are regarded as
  super-blocks
• Global sequence pair consists of super-blocks and
  other ordinary blocks
• Easy to deal with other placement constraints
• Just inserting additional constraint edges into
  constraint graphs

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Simulated Annealing

• Random Moves
• SP perturbations
• Change aspect ratio
• Swap two blocks in one sequence
• Swap two blocks in both sequences
• C-CBL perturbations
• Change aspect ratio of symmetry blocks
• Perturb S
• Perturb L
• Perturb T

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Comparison with extension of 7
7 Analog Placement with Symmetry and Other
Placement Constraints (Tam et al. ICCAD06)
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Experimental Results with C-CBL

• A packing contains two common centroid groups

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Experimental Results with C-CBL (cont)

• A common centroid group with a 1-D symmetry group
• 1-D symmetry group is handled with the approach
  in 7
• Our approach is compatible with other methods to
  solve mixed constraints

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Experimental Results with C-CBL

• A resultant packing with 300 non-symmetry blocks
  and six common centroid groups

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An Alternative Grid-based Approach

• In C-CBL approach, each device in a common
  centroid group is sliced into two blocks
• We want to distribute the devices in a group more
  uniformly
• Average out the influence of parasitic errors
• More stable
• Less sensitive to noise
• Split each device into a set of basic unit blocks

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Grid-based Approach

• Split each device in a group into a number of
  basic unit blocks according to the area
• Solution generation for each group
• Match the basic unit blocks in each device into
  singles, pairs and triples
• Generate all possible packing solutions by
  assigning those singles, pairs and triples into a
  grid in a common centroid way
• Note that if all the matched blocks share a
  common centroid, all devices have a common
  centroid
• At the end, prunings are done to remove redundant
  solutions
• Global SP
• Common centroid groups are regarded as
  super-blocks
• With a set of aspect ratios available to choose

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Example of Assigning into Grid

• Suppose a group contains 4 devices g1,g2,g3,g4
  with areas 1, 3, 3 and 2 respectively.
• After matching
• g1 has a single (a1)
• g2 has a triple (a2,b2,c2)
• g3 has a triple (a3,b3,c3)
• g4 has a pair (a4,b4)
• These matched basic unit blocks will be assigned
  into a grid in a common centroid way in the order
  of singles, triples and then pairs.

b2
b4
c3
a1
a2
a3
a4
c2
b3
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Solution Pruning

• After assignment, every group will have a number
  of grid packing solutions with different column
  numbers.
• The redundant ones are then pruned.
• Example
• Consider all the packing solutions of a group
  with 2 pairs

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Simulated Annealing

• Random moves
• Change packing solution of a group
• Rotate a group
• Perturb global SP
• Change aspect ratio
• Swap two blocks in one sequence
• Swap two blocks in both sequences

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Experimental Results

• A Digital-Analog Converter Circuit
• Containing a capacitor array of 9 devices
• 1,1,2,4,8,16,32,64,128

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Experimental Results

• A data set with a common centroid group
• 1,1,2,3,4,5,6,7,8,9,10

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Conclusion

• Placement with common centroid constraints has
  been studied
• A representation C-CBL is developed
• Represent common centroid placement with pairs of
  blocks to be matched
• An alternative Grid-based approach
• Distribute the devices more uniformly
• Deal with the case that devices are with integer
  area ratios
• First work in literature to handle common
  centroid constraints topologically

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• Thank You!