EE382V Fall 2006 VLSI Physical Design Automation

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EE382V Fall 2006VLSI Physical Design Automation
Lecture 5. Circuit Partitioning (III) Spectral
and Flow

• Prof. David Pan
• dpan_at_ece.utexas.edu
• Office ACES 5.434

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Recap of what we have learned

• KL Algorithm
• FM algorithm
• Variation and Extension
• Multilevel partition (hMetis)
• Simulated Annealing

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Partitioning Algorithms

• Two elegant partition algorithms
• although not the fastest
• Learn how to formulate the problem!
• gt Key to VLSI CAD
• (1) Spectral based partitioning algorithms
• (2) Max-flow based partition algorithm

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Spectral Based Partitioning Algorithms
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Eigenvalues and Eigenvectors
If Ax?x then ? is an eigenvalue of A x
is an eignevector of A w.r.t. ? (note that Kx is
also a eigenvector, for any constant K).
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A Basic Property
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Basic Idea of Laplacian Spectrum Based Graph
Partitioning

• Given a bisection ( X, X ), define a
  partitioning vector
• clearly, x ? 1, x ? 0 ( 1 (1, 1, , 1 ), 0
  (0, 0, , 0))
• For a partition vector x
• Let S x x ? 1 and x ? 0. Finding best
  partition vector x such that the total edge cut
  C(X,X) is minimum is relaxed to finding x in S
  such that is minimum
• Linear placement interpretation
• minimize total squared wirelength

Î
-
ì

1
X
i



.
s.t

)
,
,
,
(
x
x
x
x
x
í
L
2
1
i
n
Î
'

1
X
i
î
4 C(X, X)
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Property of Laplacian Matrix
å

T
( 1 )
x
x
Q
Qx
x
i
ij
j
-

T
T
Ax
x
Dx
x
å
å
-

2
x
x
A
x
d
j
i
ij
i
i
å
å
-

2
x
aij x
x
d
2
j
i
i
i
Î
E
j
i
)
,
(
å
-

2
x
x
)
(
squared wirelength
j
i
Î
E
j
i
)
,
(
If x is a partition vector
4 C(X, X)
Therefore, we want to minimize
T
Qx
x
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Property of Laplacian Matrix (Contd)
( 2 ) Q is symmetric and semi-definite, i.e. (i)
(ii) all eigenvalues of Q are ?0 ( 3 ) The
smallest eigenvalue of Q is 0
corresponding eigenvector of Q is x0 (1, 1, , 1
) (not interesting, all modules overlap
and x0?S ) ( 4 ) According to Courant-Fischer
minimax principle the 2nd smallest
eigenvalue satisfies
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Results on Spectral Based Graph Partitioning

• Min bisection cost c(x, x) ? n??/4
• Min ratio-cut cost c(x,x)/x?x ? ?/n
• The second smallest eigenvalue gives the best
  linear placement
• Compute the best bisection or ratio-cut
• based on the second smallest eigenvector

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Computing the Eigenvector

• Only need one eigenvector
• (the second smallest one)
• Q is symmetric and sparse
• Use block Lanczos Algorithm

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Interpreting the Eigenvector
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Experiments on Special Graphs

• GBUI(2n, d, b) Bui-Chaudhurl-Leighton 1987

2n nodes d-regular min-bisection?b
n
n
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Some Applications of Laplacian Spectrum

• Placement and floorplan
• Hall 1970
• Otten 1982
• Frankle-Karp 1986
• Tsay-Kuh 1986
• Bisection lower bound and computation
• Donath-Hoffman 1973
• Barnes 1982
• Boppana 1987
• Ratio-cut lower bound and computation
• Hagen-Kahng 1991
• Cong-Hagen-Kahng 1992