A Priori System-Level
Interconnect Prediction
The Road to Future Computer Systems
Dirk Stroobandt
Ghent University
Electronics and Information Systems Department
Presentation at Northwestern University
May 11th, 2000
Outline
•
•
•
•
•
Why do we need a priori interconnect prediction?
Basic models
Rent’s rule with extensions and applications
A priori wirelength prediction
New evolutions:
• 3D and anisotropic systems
• System-level predictions
• Applications
• Conclusions
May 11th, 2000
Talk at NWU, Dirk Stroobandt
2
Outline
•
•
•
•
•
Why do we need a priori interconnect prediction?
Basic models
Rent’s rule with extensions and applications
A priori wirelength prediction
New evolutions:
• 3D and anisotropic systems
• System-level predictions
• Applications
• Conclusions
May 11th, 2000
Talk at NWU, Dirk Stroobandt
3
Why do we need
a priori interconnect prediction?
• Importance of wires increases (they do not scale as
components).
• For future designs, very little is known. Roadmapping
uses a priori estimation techniques.
• To improve CAD tools for design layout generation.
• CAD tools have to take into account: timing constraints,
area constraints, performance, power dissipation…
• All these constraints: wires should be as short as possible.
• Estimation at early stage aids the CAD tools in finding a
better solution through fewer design cycle iterations.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
4
Why do we need
a priori interconnect prediction?
To evaluate new computer architectures
• To adhere to the increasing performance demands, new
computer architectures are needed.
• Each of them must be evaluated thoroughly.
• A priori estimates immediately provide a ground for drawing
preliminary conclusions.
• Different architectures can be compared to each other.
• Applications for evaluating three-dimensional (optoelectronic) architectures, FPGA’s, MCM’s,...
May 11th, 2000
Talk at NWU, Dirk Stroobandt
5
Components of the
physical design step
circuit
architecture
Layout generation
layout
May 11th, 2000
Talk at NWU, Dirk Stroobandt
6
Circuit model
Logic block
Net
Internal
net
External net
Terminal / pin
Multi-terminal nets have
a net degree > 2
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Model for partitioning
8 nets cut
4 nets cut
Optimal partitioning:
minimal number of nets cut
May 11th, 2000
Talk at NWU, Dirk Stroobandt
8
Model for partitioning
New net
New terminal
Module
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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The three basic models
Circuit model
Model for the architecture
Cell
Pad
Channel
Manhattan grid
using Manhattan metric
d | x1  x2 |  | y1  y2 |
Placement and routing model
May 11th, 2000
Talk at NWU, Dirk Stroobandt
10
The three basic models
Optimal placement = placement with minimal total wire
length over all possible placements.
Optimal routing = routing through shortest path
• requires channels with sufficiently high density
• for multi-terminal nets: Steiner trees
This defines the net length for known endpoints
Placement and routing model
May 11th, 2000
Talk at NWU, Dirk Stroobandt
11
Outline
•
•
•
•
•
Why do we need a priori interconnect prediction?
Basic models
Rent’s rule with extensions and applications
A priori wirelength prediction
New evolutions:
• 3D and anisotropic systems
• System-level predictions
• Applications
• Conclusions
May 11th, 2000
Talk at NWU, Dirk Stroobandt
12
Rent’s rule
Rent’s rule was first described by [Landman and Russo, 1971]
For average number of terminals and blocks per module:
p
100
T=tB
T
p = Rent exponent
t = average # term./block
10
Measure for the complexity
of the interconnection topology
(simple) 0  p  1 (complex)
average
Rent’s rule
Normal values: 0.5  p  0.75
1
1
May 11th, 2000
10
B
100
1000
Talk at NWU, Dirk Stroobandt
13
Rent’s rule
If B cells are added, what is the increase T?
In the absence of any other information we guess
T 
T    B
B
T
B
B
T
Overestimate: many of T terminals connect to T
terminals and so do not contribute to the total.
We introduce a factor p (p <1) which indicates how
self connected the netlist is
Statistically homogenous
system
T 
T  p B
 B
Or, if B & T are small compared to B and T
dT
 dB 
p
 p
  T  tB
T
 B 
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Rent’s rule
T=tB
100
p
Rent’s rule is experimentally
validated for a lot of real circuits
and for different partitioning
methodologies.
T
10
average
Rent’s rule
1
1
10
100
B
Deviation for high B and T:
Rent’s region II (cfr. later).
May 11th, 2000
1000
Distinguish between:
• p* : intrinsic Rent exponent
• p : Rent exponent for
a given placement
• p’ : Rent exponent for
a given partitioning
Talk at NWU, Dirk Stroobandt
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Rent’s rule
Rent’s rule is a result of the self-similarity within circuits
Assumption: interconnection complexity is equal at all levels.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Extension: the local Rent exponent
Variations in Rent’s rule:
• global variations (e.g., lower complexity after Technology
mapping of the circuit, duplication);
• local variations.
Two kinds of local variations in Rent’s rule:
• hierarchical locality: some hierarchical levels are more
complex than others;
• spatial locality: some circuit parts are more complex than
others.
Both are deviations from Rent’s rule that can be
modelled well.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Hierarchical locality: Rent’s region II
100
T
10
average
Rent’s rule
1
1
10
100
B
May 11th, 2000
1000
Causes of region II:
- pin limitation problem;
- parallel to serial
(complexity is moved
from space to time,
number of pins is
lowered);
- coding (input and
output stream
compact).
Talk at NWU, Dirk Stroobandt
18
Hierarchical locality: region III
For some circuits: also deviation
at low end.
T
Mismatch between the available
(library) and the desired (design)
complexity of interconnect
topology.
Only for circuits with logic blocks
that have many inputs.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
19
Hierarchical locality: modelling
Use incremental Rent exponent (proportional to the
slope of Rent’s curve in a single point).
T  tB
p(B)
p2
 log( T )
p( B) 
 log( B)
T
p1
p3
B
May 11th, 2000
Talk at NWU, Dirk Stroobandt
20
Spatial locality in Rent’s rule
Inhomogeneous circuits: different parts have different
interconnection complexity.
For separate parts:
Ti  tBi
pi
p1  0.80
p2  0.35
Only one Rent exponent (heterogeneous) might not be realistic.
Clustering: simple parts will be absorbed by complex parts.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
21
Local Rent exponent
1
1
2
1
T
1
2
1
1
2
2
2
B
May 11th, 2000
2
Higher partitioning levels: Rent
exponents will merge.
Spreading of the values with
steep slope (decreasing) for
complex part and gentle slope
(increasing) for simple part.
Local Rent exponent
tangent slope of the line that
combines all partitions
containing the local block(s).
Talk at NWU, Dirk Stroobandt
22
Heterogeneous Rent’s rule
Suggested by (Zarkesh-Ha, Davis, Loh, and Meindl,’98)
Weighted arithmetic average of the logarithm of T:
B1 log T1  B2 log T2
log Teq 
B1  B2
Heterogeneous Rent’s rule (for 2 parts):
Teq  teq B
May 11th, 2000
peq
1

B B B B
 teq  (t1 1 t 2 2 ) 1 2

p1 B1  p2 B2
 peq 

B1  B2
Talk at NWU, Dirk Stroobandt
23
Use of Rent’s rule in CAD
Rent’s rule is very powerful as a measure of
interconnection complexity
Can aid in the partitioning process
Benchmark generators are based on Rent’s rule
Is basis for a priori estimates in CAD
May 11th, 2000
Talk at NWU, Dirk Stroobandt
24
Rent’s rule in partitioning
Actual goal: minimize the number of pins per module.
We should use a pin count criterion.
External multi-terminal
nets lead to only one
new pin instead of two
when cut.
Preferring external nets
to be cut will better keep
clusters together.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
25
Rent’s rule in partitioning
Solution: use a new ratio value (in ratiocut partitioning)
based on terminal count:
Tn
Rp 
| A || A' |
Better partitions are obtained because the total number
of pins for each module is taken into account by the
cost function.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
26
Rent’s rule in partitioning
Better (ratio cut) heuristic by using terminal count
prediction (Stroobandt, ISCAS‘99).
• Clustering property of the ratio cut: use Rent’s rule instead of
uniformly distributed random graph.
• New ratio:
Tn
Rp  p
p
p
B1  B2  ( B1  B2 )
Instead of old ratio:
May 11th, 2000
Rold
Tn

B1 B2
Talk at NWU, Dirk Stroobandt
27
Rent’s rule in partitioning
Important (especially in pin-limited designs): terminal
balancing (Stroobandt, Swiss CAD/CAM‘99).
• Minimizing the terminal count alone is not enough.
Additional cost function
for terminal balancing:
Terminal
May 11th, 2000
 TA TA ' 
Rb   p  p 
 B A BA' 
Talk at NWU, Dirk Stroobandt
2
28
Rent’s rule in benchmark generation
Generating benchmarks in a hierarchical way
• Rent’s rule is used for estimating the number of connections
• Other parameters have to be controlled as well:
– Classical parameters:
* total number of gates
* total number of nets
* total number of pins
– Gate terminal distribution
– Net degree distribution
• Other issues: gate functionality, redundancy, timing
constraints, ...
May 11th, 2000
Talk at NWU, Dirk Stroobandt
29
Outline
•
•
•
•
•
Why do we need a priori interconnect prediction?
Basic models
Rent’s rule with extensions and applications
A priori wirelength prediction
New evolutions:
• 3D and anisotropic systems
• System-level predictions
• Applications
• Conclusions
May 11th, 2000
Talk at NWU, Dirk Stroobandt
30
Donath’s hierarchical placement model
1. Partition the circuit into 4 modules of equal size such
that Rent’s rule applies (minimal number of pins).
2. Partition the Manhattan grid in 4 subgrids of equal
size in a symmetrical way.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
31
Donath’s hierarchical placement model
3. Each subcircuit (module) is mapped to a subgrid.
mapping
4. Repeat recursively until all logic blocks are assigned
to exactly one grid cell in the Manhattan grid.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
32
Donath’s length estimation model
At each level: Rent’s rule gives number of connections
• number of terminals per module directly from Rent’s rule
(partitioning based Rent exponent p’);
• every net not cut before (internal net): 2 new terminals;
• every net previously cut (external net): 1 new terminal;
• assumption: ratio f = (#internal nets)/(#nets cut) is constant
over all levels k (Stroobandt and Kurdahi, GLSVLSI’98);
• number of nets cut at level k (Nk) equals
N k  Tk
where =1/(1+f);  depends on the total number of nets in
the circuit and is bounded by 0.5 and 1.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
33
Donath’s length estimation model
Length of the connections at level k ?
Adjacent (A-)
combination
Diagonal (D-)
combination


Donath assumes: all connection source and destination
cells are uniformly distributed over the grid.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
34
Average interconnection length
p 1
k ( p 1)
Number of connections at level k: N k  tG(1  4 )4
4
1
l


Average length A-combination: k ,a 3 3 ,
Average length D-combination: lk ,d  2 ,
lk  149  92
Average length level k:
Total average length:
2K ( 2r x )  1
H ( K , r, x )  2 r  x
2
1
May 11th, 2000
L
14 H ( K , r,1)  2 H ( K , r,3)
9 H ( K , r,2)
with
and 2K = G = total number of gates
Talk at NWU, Dirk Stroobandt
35
Results Donath
Scaling of the average
length L as a function of
the number of logic
blocks G :
 G p 0.5 ( p  0.5)

L  log( G ) ( p  0.5)
 f ( p ) ( p  0.5)

30
25
p = 0.7
20
L 15
p = 0.5
10
p = 0.3
5
0
1
10 100 103 104 105 106 107
G
Similar to measurements on placed designs.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
36
Results Donath
L
8
7
6
5
theory
experiment
4
3
2
1
0
10
100
1000
10000
G
Theoretical average wire length too high by a factor 2
May 11th, 2000
Talk at NWU, Dirk Stroobandt
37
Including optimal placement model
• Keep wire length scaling by hierarchical placement.
• Improve on uniform probability for all connections at one
level (not a good model for an optimal placement).
Enumeration: site density function (only architecture
dependent). Occupying probability favours short
interconnections (for an optimal placement) (darker)
May 11th, 2000
Talk at NWU, Dirk Stroobandt
38
Including optimal placement model
Wirelength distributions contain two parts:
site density function and probability distribution
all possibilities
probability of occurrence
requires enumeration
shorter wires more probable
(use generating polynomials)
N (l )  K D(l ) q(l )
May 11th, 2000
Talk at NWU, Dirk Stroobandt
39
Wire length distribution
Local distributions at each level have similar
shapes (self-similarity)  peak values scale.
Integral of local distributions equals number of
connections.
Global distribution follows peaks.
From this we can deduct that
N (l )  l 2 p 3
For short lengths:
D(l )  l
N (l )
 q(l ) 
 l 2 p 4
D( L)
May 11th, 2000
Talk at NWU, Dirk Stroobandt
40
Occupying probability: results
Use probability on each hierarchical level (local distributions).
8
L
Occupying prob.
Donath
experiment
7
6
5
4
3
2
1
0
10
100
1000
10000
G
May 11th, 2000
Talk at NWU, Dirk Stroobandt
41
Occupying probability: results
Effect of the occupying probability: boosting the local wire
length distributions (per level) for short wire lengths
percent of wires
Donath
global trend
per level
total
Occupying prob.
100
global trend
per level
total
10
1
0,1
0,01
10-3
10-4
1
10
100
1000 10000
Wire length
May 11th, 2000
1
10
100
1000 10000
Wire length
Talk at NWU, Dirk Stroobandt
42
Occupying probability: results
Effect of the occupying probability on the total
distribution: more short wires = less long wires

average
wire length
is shorter
May 11th, 2000
percent wires
100
10
1
10-1
10-2
10-3
10-4
10-5
1
Donath
Occupying prob.
10
100
1000 10000
Wire length
Talk at NWU, Dirk Stroobandt
43
Occupying probability: results
Percent wires
60
-8%
50
-23%
40
Donath
Occupying prob.
global trend
30
+10%
+6%
20
10
1
May 11th, 2000
2
3
4 5 6 7
Wire length
8
9 10
Talk at NWU, Dirk Stroobandt
44
Occupying probability: results
Number of wires
1000
Donath
Occupying prob.
measurement
100
10
1
0,1
May 11th, 2000
1
10
Wire length
Talk at NWU, Dirk Stroobandt
100
45
Davis’ probability function
Introduced by Davis, De, and
Meindl (IEEE T El. Dev., ‘98).
Number of interconnections at
distance l is calculated for every
gate separately, using Rent’s
rule.
Three regions: gate under
investigation (A), target gates
(C), and gates in between (B).
Number of connections
between A and C is calculated.
May 11th, 2000
This approach alleviates the
discrete effects at the boundaries of
the hierarchical levels while
maintaining the scaling behaviour.
Talk at NWU, Dirk Stroobandt
46
Davis’ probability function
A
B
C
C
=
A
B
C
C
B
C
C
B
B
B
C
B
B
A
B
B
C
B
B
B
C
C
B
C
+
A
C
=
TAC
B
-
C
+
TAB
TBC
A
B
-
C
-
TB
A
B
C
-
TABC
C
Assumption: net cannot connect A,B, and C
C
TAB  t 1  BB 
TB  tBB

TBC  t BB  BC 
p
p
TABC  t 1  BB  BC 
p
N AC  TAC  t 1  BB   BB  BC   BBp  1  BB  BC 
May 11th, 2000
p
p
Talk at NWU, Dirk Stroobandt
p
p

47
Davis’ probability function
For cells placed in infinite 2D plane
C
C
C
B
C
C
B
B
B
C
B
B
A
B
B
C
B
B
B
C
C
B
C
BC  4l
C
l 1
BB   4l '  2l (l  1)
l '1
C

N AC  t 1  2l (l  1)  2l (l  1)  4l   2l (l  1)  1  2l (l  1)  4l 
p
q(l ) 
May 11th, 2000
p
p
N A C
 l 2 p4
4l
Talk at NWU, Dirk Stroobandt
48
p

28
Planar wirelength model A
L
Finite system, Btot=L2, no edges,
approximate form for q(l)
L
Da (l )  2lL2
q(l )  l
May 11th, 2000
2 p 4
N (l )  Ntot Da (l )q(l )

p
N tot  t Btot  Btot
Talk at NWU, Dirk Stroobandt

49
29
Planar wirelength model B (Davis)
L
Finite system, Btot=L2,
includes edge effects,
use q(l)
L
l l 2  1  6 L L  l  3
for 1  l  L

Db (l )  2 L  l  12 L  l 2 L  l  1 3 for L  l  2 L
0
else

N (l )  Ntot Db (l )q(l )

p
N tot  t Btot  Btot
May 11th, 2000

Talk at NWU, Dirk Stroobandt
50
30
10
Number of nets
10
10
4
3
2
Planar wirelength model comparison
10
1
0
10 0
10
1
10
Length
2
10
Btot = 1024
p = 0.66
Model A
Model A: Lav = 4.53
Model B: Lav = 2.27
Model B
May 11th, 2000
Talk at NWU, Dirk Stroobandt
51
2
x 10
Relationship between models from
Davis (planar model B) and Stroobandt
(hierarchical model C)
4
Number
1.5
1
0.5
0
0
10
20
40
30
Length
50
60
70
Db(l)
Dc(l,h)
H
Db (l )   4 H  h Dc (l , h)
h 1
same q’(l)
essentially identical!
May 11th, 2000
Talk at NWU, Dirk Stroobandt
52
Hierarchical wirelength model
comparison
10
Number
10
10
10
4
3
2
1
0
10 0
10
1
10
Length
2
10
Ctot = 1024
p = 0.66
Model D
Model C (Stroobandt):
Lav = 2.05
Model D (Donath):
Lav = 5.14
Model C: q(l) and hierarchy
Model D: only hierarchy (q(l)=1)
May 11th, 2000
Model C
Talk at NWU, Dirk Stroobandt
53
Number of nets
10
10
10
Planar and hierarchical model
comparison
4
3
10
2
10
Number
10
1
10 0
10
10
10
0
1
10
Length
2
10
4
3
2
1
0
10 0
10
1
10
Length
2
10
Model A
Model B
Model D
Model C
Models B (planar) and C (hierarchical ) are equivalent if the Rent exponent
used for the probability function (depends on placement) and the one used
for the number of nets per hierarchical level (based on partitioning) are the same
May 11th, 2000
Talk at NWU, Dirk Stroobandt
54
Outline
•
•
•
•
•
Why do we need a priori interconnect prediction?
Basic models
Rent’s rule with extensions and applications
A priori wirelength prediction
New evolutions:
• 3D and anisotropic systems
• System-level predictions
• Applications
• Conclusions
May 11th, 2000
Talk at NWU, Dirk Stroobandt
55
Extension to three-dimensional grids
May 11th, 2000
Talk at NWU, Dirk Stroobandt
56
Three-dimensional grids: basic results
Average length converges (for G) up to r = 2/3  1/2
Average wire length is lower than for 2D (no long wires)
May 11th, 2000
Talk at NWU, Dirk Stroobandt
57
Anisotropic systems
May 11th, 2000
Talk at NWU, Dirk Stroobandt
58
Anisotropic systems
Basic method: Donath’s method in 3D
Not all dimensions are equal (e.g., optical links in 3rd D)
• possibly larger latency of the optical link (compared to intrachip connection);
• influence of the spacing of the optical links across the area
(detours may have to be made);
• limitation of number of
optical layers
Introducing an optical cost
May 11th, 2000
Talk at NWU, Dirk Stroobandt
59
Anisotropic systems
If limited number of layers: use third dimension for
topmost hierarchical levels (fewest interconnections).
For lower levels: 2D method.
2D and 1D partitioning are sometimes used to get
closer to the (optimal in isotropic grids) cubic form.
Depending on the optical cost, it is advantageous either
to strive for getting to the electrical plains as soon as
possible (high optical cost, use at high levels only) or to
partition the electrical planes first (low optical cost).
May 11th, 2000
Talk at NWU, Dirk Stroobandt
60
External nets
Importance of good wire length estimates for external
nets during the placement process:
For highly pin-limited designs: placement will be in a
ring-shaped fashion (along the border of the chip).
May 11th, 2000
Talk at NWU, Dirk Stroobandt
61
Wire lengths at system level
At system level: many long wires (peak in distribution).
How to model these?
Davis and Meindl ‘98:
estimation based on
Rent’s rule with the
floorplanning blocks
as logic blocks.
IMPORTANT!
May 11th, 2000
Talk at NWU, Dirk Stroobandt
62
Improving CAD tools for design layout
Digital design is complex
Computer-aided
design (CAD)
More efficient layout
generation requires good wire
length estimates.
• Layer assignment in routing
• effects of vias, blockages
• congestion, ...
A priori estimates are rough
but can already provide us
with a lot of information.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Evaluating new computer architectures
Estimation for evaluating and comparing different
architectures
Circuit characterization
We need parameters to classify circuits in classes and
to optimize them.
Benchmark generation based on Rent’s rule.
May 11th, 2000
Talk at NWU, Dirk Stroobandt
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Conclusion
• Wire length estimates are becoming more and more
important.
• A priori estimates can provide a lot of information at
virtually no cost.
• Methods are based on Rent’s rule.
• Important for future research: how can we build a
priori estimates into CAD layout tools?
More information at http://www.elis.rug.ac.be/~dstr/dstr.html
May 11th, 2000
Talk at NWU, Dirk Stroobandt
65