Interconnect Implications of Growth-Based Structural Models for VLSI Circuits*

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26 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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 driven


N(f) == # nets with fanout f


G == # gates


T == # terminals


E == # crossing edges
(connections between two
gates on different sides of a
partition)

Definitions

g3

g1

g2

G = 3

E = 6

T = 4

g3

D(g3) = 5

P(3) += 1

P(5) += 1

P(2) += 1

f = 3

N(1) += 2

VLSI Power
-
Law Phenomena


Rent’s rule

p
kG
T


Crossing edge scaling

e
p
e
G
k
E

T == # terminal, G: # gate,

p == Rent exponent

E == # connections between two gates
on different sides of the partition

VLSI Power
-
Law Phenomena (cont.)


Vertex degree

d
p
d
d
k
P(d)


Net fanout

f
p
f
f
k
N(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
-
a


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 not


Families of topologies with different p and identical N(f)


1
-
D mesh: p
=
0, N(1) = # nets, N(f


1) = 0


2
-
D mesh: p

=
0.5, N(1) = # nets, N(f


1) = 0


3
-
D mesh: p

=
0.667, N(1) = # nets, N(f


1) = 0


Our experiments fail to confirm the p
-
3 fanout exponent

3
-
p
p
cf
N(f)

,
kG
T


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


n
0

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 growth


Preferential attachment


Result:







1
0
)
(
)
(
)
(
t
j
j
i
i
t
d
t
d
m
t
t
d
3
)
(


cd
d
P
Vertex degree

Kumar Model


Given:


Random growth of vertices


Random link to other vertices with probability
a


Copy links from a random vertex with probability 1
-
a


Results:


Power
-
law vertex degree distribution

New Pref Model

t
q
qm
t
d
qm
t
d
qm
t
d
m
t
t
d
i
t
j
j
i
i
)
2
(
)
(
)
)
(
(
)
(
)
(
1
0

































0

,
)
1
(
0

,
)
1
(
)
(
2
1
0
2
1
i
t
q
n
m
qm
i
i
t
q
m
qm
t
d
q
q
i
Preferential attachment

After integration, vertex degree

New Pref Model

3
2
0
)
(
)
)(
2
(
)
)
(
(
)
(











q
q
i
qm
d
qm
m
q
n
t
t
d
d
t
d
P
d
P
3
2
0
)
(
)
)(
2
(
)
(
)
(
)
(










q
q
qm
m
f
qm
m
q
N
m
f
P
n
N
f
N
q
q
i
qm
d
qm
m
n
t
t
qm
d
qm
m
t
i
P
d
t
d
P




























2
0
2
1
)
(
)
)
(
(
Vertex degree probability

Probability density

d = f + m, so fanout

New Pref Model

2
1
2
1
2
1
0
2
1
1
)
1
(
)
2
(
)
1
(
2
)
(
)
0
(
)
(
















q
q
q
q
n
i
i
G
N
q
m
mG
q
N
mqn
mN
mn
N
d
E
G
E
























G
n
N
q
m
Min
G
T
q
q
q
,
)
)(
1
(
)
(
2
2
1
2
1
Terminal

Crossing edge

New Temporal Models


Temporal attachment:







1
0
)
(
t
j
s
s
i
j
i
m
t
t
d

Temp 1 (s = 1): attachments that prefer temporal locality


Temp 2 (s = 0): random equiprobable attachment to all
previous vertices


Temp 3 (s =

): extreme temporal locality (a vertex
connects only to its temporally immediate neighbors)

Summary of Models

Barabasi

Pref







1
0
)
(
)
(
)
(
t
j
i
i
i
t
d
t
d
m
t
t
d
3
)
(


cd
d
P
3
)
(
)
(



m
f
c
f
N
G
c
G
c
c
G
E
3
5
.
0
2
1
)
(



}
,
)
(
{
)
(
2
5
.
0
2
1
G
G
c
c
Min
G
T










1
0
)
(
t
j
i
i
i
q
d
q
d
m
t
t
d
3
2
1
)
(
)
(



q
c
d
c
d
P
3
2
1
)
(
)
(



q
c
f
c
f
N
2
1
2
1
)
(



q
G
c
G
c
G
E
}
,
)
(
{
)
(
2
2
1
2
1
G
G
c
c
Min
G
T
q
q




t
m
t
t
d
i



)
(
d
ce
d
P


)
(
f
ce
f
N


)
(
3
2
1
log
)
(
c
G
G
c
c
G
E



}
,
log
{
)
(
2
1
G
G
G
c
G
c
Min
G
T






j
i
j
i
m
t
t
d
)
(
3
2
2
1
)
(
c
G
c
G
c
G
E



}
,
)
1
log
(
{
)
(
1
2
1
G
G
G
c
c
Min
G
T




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) = c
1

(f+c
2
)
q
-
3


N(f) = c f
p
-
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?