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

Ηλεκτρονική - Συσκευές

26 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

130 εμφανίσεις

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

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?