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Interconnect Implications of Growth-Based Structural Models for VLSI Circuits*
Chung-Kuan Cheng, Andrew B. Kahng and Bao Liu
UC San Diego CSE Dept.
e-mail: {kuan,abk,bliu}@cs.ucsd.edu
*Supported by a grant from Cadence Design Systems, Inc. and by the MARCO Gigascale Silicon Research Center.
Presentation Outline Introduction and Motivation Random Growth Models Experiments Conclusion and Future Work
VLSI circuits:degree d == # adjacent gates
P(d) == # gates with degree d f == # gates being drivenN(f) == # nets with fanout fG == # gatesT == # terminalsE == # crossing edges
(connections between two gates on different sides of a partition)
Definitions
g3
g1
g2
G = 3E = 6T = 4
g3
D(g3) = 5
P(3) += 1
P(5) += 1
P(2) += 1
f = 3N(1) += 2
VLSI Power-Law Phenomena Rent’s rule
pkGT Crossing edge scaling
epeGkE
T == # terminal, G: # gate,
p == Rent exponentE == # connections between two gates on
different sides of the partition
VLSI Power-Law Phenomena (cont.) Vertex degree
dpddkP(d)
Net fanoutfp
f fkN(f) P(d) == # vertices with degree d
d == vertex degree
N(f) == # nets with fanout f
f == net fanout
Power-Law Phenomena in other Contexts Zief’s law
English word frequency with rank i is proportional to i-
Lotka’s law (Yule’s law)# authors (# papers)-2
Power-law vertex degree distribution WWW (in-degree exponent 2.1, out-degree 2.45) actor connectivity (exponent 2.3)paper citation (exponent 3)power grid (exponent 4)
Rent’s Rule Based VLSI Models Claims that Rent’s rule implies fanout distribution
Zarkesh-Ha:Stroobandt-Kurdahi: logistic equations
Are they really correlated?Rent p depends on partitioning method, fanout distribution
does notFamilies of topologies with different p and identical N(f)
1-D mesh: p = 0, N(1) = # nets, N(f 1) = 02-D mesh: p = 0.5, N(1) = # nets, N(f 1) = 03-D mesh: p = 0.667, N(1) = # nets, N(f 1) = 0
Our experiments fail to confirm the p-3 fanout exponent
3-pp cfN(f) ,kGT
Our Motivation Open problems
what are the reasons behind all these power-law scaling phenomena?
what are the relations between these power-law scaling phenomena? Are they correlated?
Our aim to better understand scaling phenomena and structural
properties in VLSI circuits eventually, to better estimate VLSI interconnect parameters
Presentation Outline Introduction and Motivation Random Growth Models Experiments Conclusion and Future Work
Random Growth Model (Framework)
Random growth in time n0 primary vertices at timestep 0
1 new vertex with m edges to existing vertices, added at each time step
Preferential attachment (Barabasi, Kumar, Pref, Temp 1, Temp 2...) Interpretation as hypergraph
Each vertex has m input (backward) edges and 1 output (forward) hyperedge
Barabasi Model
Given:Random growthPreferential attachment
Result:
1
0
)(
)()(t
jj
ii
td
tdm
t
td
3)( cddPVertex degree
Kumar Model
Given:Random growth of verticesRandom link to other vertices with probability Copy links from a random vertex with probability 1-
Results:Power-law vertex degree distribution
New Pref Model
tq
qmtd
qmtd
qmtdm
t
td it
jj
ii
)2(
)(
))((
)()(1
0
0 ,)1
(
0 ,)1()(
2
1
0
2
1
itqn
mqm
ii
tqmqm
tdq
q
i
Preferential attachment
After integration, vertex degree
New Pref Model
32
0
)())(2())((
)(
qqi qmdqmmq
nt
t
d
dtdPdP
320 )())(2()()()( qq qmmfqmmqNmfPnNfN
i qmd
qmm
nt
t
qmd
qmmtiPdtdP
2
0
2
1)())((Vertex degree probability
Probability density
d = f + m, so fanout
New Pref Model
2
1
2
1
2
1
02
1
1
)1()2()1(
2)()0()(
qqqq
n
ii
GNqmmGqNmqnmN
mnNdEGE
GnNqmMinGT
q
qq ,))(1()(
2
2
1
2
1
Terminal
Crossing edge
New Temporal Models Temporal attachment:
1
0
)(t
j
s
si
j
im
t
td
Temp 1 (s = 1): attachments that prefer temporal locality Temp 2 (s = 0): random equiprobable attachment to all
previous verticesTemp 3 (s = ): extreme temporal locality (a vertex
connects only to its temporally immediate neighbors)
Summary of Models
Barabasi Pref
1
0
)(
)()(t
ji
ii
td
tdm
t
td
3)( cddP3)()( mfcfN
GcGccGE 35.0
21)(
},)({)( 25.021 GGccMinGT
1
0
)(t
ji
ii
qd
qdm
t
td
321 )()( qcdcdP
321 )()( qcfcfN
2
1
21)( qGcGcGE
},)({)( 22
1
21 GGccMinGT qq
t
m
t
tdi
)(
dcedP )(
fcefN )(
321 log)( cGGccGE
},log{)( 21 GGGcGcMinGT
j
i
j
im
t
td )(
322
1)( cGcGcGE
},)1
log({)( 121 G
G
GccMinGT
Temp 1 Temp 2
Presentation Outline Introduction and Motivation Random Growth Models Experiments Conclusion and Future Work
Experimental Setting 21 industry standard-cell test cases with between 4K and
283K cells Fanout and vertex degree obtained by scanning netlist files E and T from UCLA Capo placer
remove Rent region II data average blocks with same gate number
Best-fitted exponents by linear regression Minimum standard deviation fit from non-linear
regression (Levenberg-Marquardt variant)
Experimental Observations Pref model provides most reasonable fanout
distribution and vertex degree distribution prediction Barabasi model gives best E prediction Temp 2 model gives best T prediction
Case19 183k 181k 1.1e6 1.1e6 5.3e6 1.6e6 1.4e6 4.7e4 6.0e4 4.8e5 4.8e4
Case #cells #nets standard deviation of E standard deviation of T
Test Total Total best-fit Bara. Pref Temp 1 Temp 2 best-fit Bara. Temp 1 Temp 2
Case18 182k 181k 1.3e6 1.3e6 4.9e6 1.5e6 1.4e6 3.2e4 4.5e4 2.5e5 3.3e4
Case17 118k 125k 1.4e6 1.5e6 5.4e6 3.1e6 1.4e6 2.4e3 6.5e3 1.8e4 3.8e3
Case16 86k 87k 5.4e5 5.4e5 1.4e6 8.3e5 6.4e5 4.8e3 5.2e3 5.1e4 5.9e3
Case21 283k 285k 1.5e8 1.5e8 1.5e9 1.3e7 4.0e7 2.0e4 3.0e3 9.8e4 2.0e4
Case20 210k 200k 2.2e6 2.2e6 7.3e6 2.4e6 2.3e6 6.6e4 8.0e4 2.7e5 6.7e4
Experimental Observations
ZH does not fit data very well
Case21 -1.201 -2.495 5.7e7 6.4e8
Case20 -3.303 -2.405 2.3e8 2.4e8
Case19 -3.983 -2.351 2.5e8 2.8e8
Case18 -4.099 -2.405 2.9e8 3.2e8
Case17 -2.122 -2.644 8.0e7 8.5e7
Case exp. exp. std.dev. std.dev.Test fitted ZH fitted ZH
Case16 -2.053 -2.448 2.8e6 3.1e7
N(f) = c1 (f+c2)q-3
N(f) = c fp-3
N(f) = c (f+m)-3
N(f) = c e-f
N(f)
Experimental Observations
Correlation between T and E Correlation between T and N(f)
T and E correlated, T and N(f) not correlated
Experimental Observations
Correlation between T and P(d) Correlation between P(d) and N(f)
T and P(d), P(d) and N(f) not correlated
Presentation Outline Introduction and Motivation Random Growth Models Experiments Conclusion and Future Work
Conclusion Have explored possibility of non-Rent based scaling
phenomena in VLSI circuits Proposed new random growth models and studied
their implications for VLSI interconnect structure Empirically studied relationships between various
interconnect structural characteristics T, E, N(f), P(d)
Current Work and Open Questions Calculation methodology for confirmation of scaling laws Generation of random netlists that observe multiple scaling
laws simultaneously Analytical models with more than one scaling parameter
Are these power-law scaling phenomena correlated to each other?
Evolution models with copying (“reuse”)Can we have closed-form results?Do evolution models converge or diverge?
What are root causes of these scaling phenomena?Design hierarchy?Reuse?
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