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Dirk Stroobandt
Ghent UniversityElectronics and Information Systems Department
A Priori System-LevelInterconnect Prediction
The Road to Future Computer Systems
Presentation at Northwestern UniversityMay 11th, 2000
May 11th, 2000 Talk at NWU, Dirk Stroobandt 2
• 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
Outline
May 11th, 2000 Talk at NWU, Dirk Stroobandt 3
• 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
Outline
May 11th, 2000 Talk at NWU, Dirk Stroobandt 4
• 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.
Why do we needa priori interconnect prediction?
May 11th, 2000 Talk at NWU, Dirk Stroobandt 5
Why do we needa 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 (opto-
electronic) architectures, FPGA’s, MCM’s,...
May 11th, 2000 Talk at NWU, Dirk Stroobandt 6
Components of thephysical design step
layout
Layout generation
circuit architecture
May 11th, 2000 Talk at NWU, Dirk Stroobandt 7
Net
External net
Internal net
Logic block
Multi-terminal nets havea net degree > 2
Circuit model
Terminal / pin
May 11th, 2000 Talk at NWU, Dirk Stroobandt 8
8 nets cut
4 nets cut
Model for partitioning
Optimal partitioning:minimal number of nets cut
May 11th, 2000 Talk at NWU, Dirk Stroobandt 9
Model for partitioning
Module
New net
New terminal
May 11th, 2000 Talk at NWU, Dirk Stroobandt 10
The three basic models
Circuit model
Placement and routing model
Model for the architecture
Pad
Channel
Manhattan gridusing Manhattan metric
Cell
|||| 2121 yyxxd
May 11th, 2000 Talk at NWU, Dirk Stroobandt 11
The three basic models
Optimal routing = routing through shortest path• requires channels with sufficiently high density• for multi-terminal nets: Steiner treesThis defines the net length for known endpoints
Placement and routing model
Optimal placement = placement with minimal total wire length over all possible placements.
May 11th, 2000 Talk at NWU, Dirk Stroobandt 12
• 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
Outline
May 11th, 2000 Talk at NWU, Dirk Stroobandt 13
T = t B p
11
100010
10
100
100
T
B
Rent’s rule
averageRent’s rule
(simple) 0 p 1 (complex)
Normal values: 0.5 p 0.75
Measure for the complexityof the interconnection topology
p = Rent exponent
Rent’s rule was first described by [Landman and Russo, 1971]For average number of terminals and blocks per module:
t = average # term./block
May 11th, 2000 Talk at NWU, Dirk Stroobandt 14
BT B
B
TT
If B cells are added, what is the increase T?In the absence of any other information we guess
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
BB
TpT
ptBTB
dBp
T
dT
Or, if B & T are small compared to B and T
Statistically homogenous system
T
B
Rent’s rule
May 11th, 2000 Talk at NWU, Dirk Stroobandt 15
11
100010
10
100
100
T
B
Rent’s rule
averageRent’s rule
Rent’s rule is experimentallyvalidated for a lot of real circuitsand for different partitioningmethodologies.
T = t B p
Distinguish between:• p* : intrinsic Rent exponent• p : Rent exponent for a given placement• p’ : Rent exponent for a given partitioningDeviation for high B and T:
Rent’s region II (cfr. later).
May 11th, 2000 Talk at NWU, Dirk Stroobandt 16
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 17
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 18
Hierarchical locality: Rent’s region II
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).
11
100010
10
100
100
T
B
averageRent’s rule
May 11th, 2000 Talk at NWU, Dirk Stroobandt 19
Hierarchical locality: region III
For some circuits: also deviation at low end.
Mismatch between the available (library) and the desired (design) complexity of interconnect topology.
Only for circuits with logic blocks that have many inputs.
T
May 11th, 2000 Talk at NWU, Dirk Stroobandt 20
Hierarchical locality: modelling
Use incremental Rent exponent (proportional to the slope of Rent’s curve in a single point).
)(BptBT
)log(
)log()(
B
TBp
B
T p1
p2
p3
May 11th, 2000 Talk at NWU, Dirk Stroobandt 21
Spatial locality in Rent’s rule
Inhomogeneous circuits: different parts have different interconnection complexity.
For separate parts:
ipii tBT
35.0
80.0
2
1
p
p
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 22
Local Rent exponent
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).
T
B
11
11
11
2 22
2
2
2
May 11th, 2000 Talk at NWU, Dirk Stroobandt 23
Heterogeneous Rent’s rule
Suggested by (Zarkesh-Ha, Davis, Loh, and Meindl,’98)
Weighted arithmetic average of the logarithm of T:
Heterogeneous Rent’s rule (for 2 parts):
21
2211 logloglog
BB
TBTBTeq
eqpeqeq BtT
21
2211
1
212121 )(
BB
BpBpp
ttt
eq
BBBBeq
May 11th, 2000 Talk at NWU, Dirk Stroobandt 24
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 25
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 26
Rent’s rule in partitioning
Solution: use a new ratio value (in ratiocut partitioning) based on terminal count:
Better partitions are obtained because the total number of pins for each module is taken into account by the cost function.
|'||| AA
TR n
p
May 11th, 2000 Talk at NWU, Dirk Stroobandt 27
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:
Instead of old ratio:
pppn
p BBBB
TR
)( 2121
21BB
TR n
old
May 11th, 2000 Talk at NWU, Dirk Stroobandt 28
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.
2
'
'
pA
ApA
Ab B
T
B
TR
Additional cost functionfor terminal balancing:
Terminal
May 11th, 2000 Talk at NWU, Dirk Stroobandt 29
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 30
• 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
Outline
May 11th, 2000 Talk at NWU, Dirk Stroobandt 31
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.
Donath’s hierarchical placement model
May 11th, 2000 Talk at NWU, Dirk Stroobandt 32
3. Each subcircuit (module) is mapped to a subgrid.
4. Repeat recursively until all logic blocks are assigned to exactly one grid cell in the Manhattan grid.
Donath’s hierarchical placement model
mapping
May 11th, 2000 Talk at NWU, Dirk Stroobandt 33
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
where =1/(1+f); depends on the total number of nets in the circuit and is bounded by 0.5 and 1.
kk TN
May 11th, 2000 Talk at NWU, Dirk Stroobandt 34
Donath’s length estimation model
Length of the connections at level k ?
Donath assumes: all connection source and destination cells are uniformly distributed over the grid.
Adjacent (A-) combination
Diagonal (D-) combination
May 11th, 2000 Talk at NWU, Dirk Stroobandt 35
Number of connections at level k:
Average length A-combination:
Average length D-combination:
Average length level k:
Total average length: with
and 2K = G = total number of gates
Average interconnection length
92
914
,
31
34
,
,2
,
k
dk
ak
l
l
l
)2,,(9)3,,(2)1,,(14
rKHrKHrKH
L
1212
),,( 2
)2(
xr
xrK
xrKH
)1(1 4)41( pkpk tGN
May 11th, 2000 Talk at NWU, Dirk Stroobandt 36
Results Donath
Scaling of the average length L as a function of the number of logic blocks G :
L
G
0
5
10
15
20
25
30
1 10 100 103 106105104 107
p = 0.7
p = 0.5
p = 0.3
Similar to measurements on placed designs.
)5.0()(
)5.0()log(
)5.0(5.0
ppf
pG
pG
L
p
May 11th, 2000 Talk at NWU, Dirk Stroobandt 37
Results Donath
Theoretical average wire length too high by a factor 2
10000
L
G
1
23
4
6
5
7
10 100 1000
8
experimenttheory
0
May 11th, 2000 Talk at NWU, Dirk Stroobandt 38
Enumeration: site density function (only architecture dependent). Occupying probability favours short interconnections (for an optimal placement) (darker)
• 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).
Including optimal placement model
May 11th, 2000 Talk at NWU, Dirk Stroobandt 39
Including optimal placement modelWirelength distributions contain two parts:
site density function and probability distribution
all possibilities
requires enumeration
(use generating polynomials)
probability of occurrence
shorter wires more probable
)()()( lqlDKlN
May 11th, 2000 Talk at NWU, Dirk Stroobandt 40
Wire length distribution
From this we can deduct that
For short lengths:
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.
42
)(
)()( pl
LD
lNlq
32)( pllNllD )(
May 11th, 2000 Talk at NWU, Dirk Stroobandt 41
Occupying probability: results
8Occupying prob.
10000
L
G
1
23
4
6
5
7
10 100 10000
Donathexperiment
Use probability on each hierarchical level (local distributions).
May 11th, 2000 Talk at NWU, Dirk Stroobandt 42
Occupying probability: results
Effect of the occupying probability: boosting the local wire length distributions (per level) for short wire lengths
Occupying prob.100
Wire length
percent of wires
10
1
0,1
0,01
10-3
10-4
100001 10 100 1000
per leveltotal
global trend
Donath
1
Wire length
10 100 1000
per leveltotal
global trend
10000
May 11th, 2000 Talk at NWU, Dirk Stroobandt 43
Effect of the occupying probability on the total distribution: more short wires = less long wires
average wire length is shorter
Occupying probability: results
100
10
1
10-1
10-2
10-3
10-4
10-5
1 10 100 1000 10000Wire length
percent wires
Occupying prob.Donath
May 11th, 2000 Talk at NWU, Dirk Stroobandt 44
Occupying probability: results
10
20
30
40
50
60
1 10Wire length
Percent wires
Occupying prob.Donath
2 3 4 5 6 7 8 9
global trend-23%
-8%
+10%
+6%
May 11th, 2000 Talk at NWU, Dirk Stroobandt 45
Occupying probability: results
0,1
1
10
100
1000
1 100Wire length
Number of wires
Occupying prob.Donath
measurement
10
May 11th, 2000 Talk at NWU, Dirk Stroobandt 46
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.
This approach alleviates the discrete effects at the boundaries of the hierarchical levels while maintaining the scaling behaviour.
May 11th, 2000 Talk at NWU, Dirk Stroobandt 47
Davis’ probability function
B
B
BB
B B
B
A B B
BB
B
C
C
C
C
C
C
C
C
C
C
C
C
C
BA
C
BA
C
BA
C
BA
C
BA= + - -
TAC TAB TBC TB TABC= + - -
Assumption: net cannot connect A,B, and C
pBAB BtT 1 pCBBC BBtT
pBB tBT pCBABC BBtT 1
pCBpB
pCB
pBCACA BBBBBBtTN 11
May 11th, 2000 Talk at NWU, Dirk Stroobandt 48
Davis’ probability function
B
B
BB
B B
B
A B B
BB
B
C
C
C
C
C
C
C
C
C
C
C
C
For cells placed in infinite 2D plane
lBC 4
)1(2'41
1'
lllBl
lB
ppppCA lllllllllltN 4)1(21)1(24)1(2)1(21
42
4)( pCA l
l
Nlq
May 11th, 2000 Talk at NWU, Dirk Stroobandt 49
Planar wirelength model AL
L
Finite system, Btot=L2, no edges, approximate form for q(l)
)()()( lqlDNlN atot
ptottottot BBtN
22)( lLlDa
28
42)( pllq
May 11th, 2000 Talk at NWU, Dirk Stroobandt 50
Planar wirelength model B (Davis)L
L
else0
2for 312212
1for 361
)(
2
LlLlLlLlL
LllLLll
lDb
Finite system, Btot=L2, includes edge effects,use q(l)
29
)()()( lqlDNlN btot
ptottottot BBtN
May 11th, 2000 Talk at NWU, Dirk Stroobandt 51
Planar wirelength model comparison
100
101
102
100
101
102
103
104
Length
Num
ber
of n
ets
Model A
Model B
Btot = 1024p = 0.66
Model A: Lav = 4.53Model B: Lav = 2.27
30
May 11th, 2000 Talk at NWU, Dirk Stroobandt 52
Relationship between models from Davis (planar model B) and Stroobandt
(hierarchical model C)
0 10 20 30 40 50 60 700
0.5
1
1.5
2x 10
4
Length
Num
ber
H
hc
hHb hlDlD
1
),(4)(
Dc(l,h)
Db(l)
same q’(l)essentially identical!
May 11th, 2000 Talk at NWU, Dirk Stroobandt 53
Hierarchical wirelength model comparison
100
101
102
100
101
102
103
104
Length
Num
ber
Model D
Model C
Ctot = 1024p = 0.66
Model C (Stroobandt):Lav = 2.05Model D (Donath):Lav = 5.14
Model C: q(l) and hierarchyModel D: only hierarchy (q(l)=1)
May 11th, 2000 Talk at NWU, Dirk Stroobandt 54
Planar and hierarchical model comparison
100
101
102
100
101
102
103
104
Length
Num
ber
Model D
Model C
100
101
102
100
101
102
103
104
Length
Num
ber
of n
ets
Model A
Model B
Models B (planar) and C (hierarchical ) are equivalent if the Rent exponentused for the probability function (depends on placement) and the one usedfor the number of nets per hierarchical level (based on partitioning) are the same
May 11th, 2000 Talk at NWU, Dirk Stroobandt 55
• 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
Outline
May 11th, 2000 Talk at NWU, Dirk Stroobandt 56
Extension to three-dimensional grids
May 11th, 2000 Talk at NWU, Dirk Stroobandt 57
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 58
Anisotropic systems
May 11th, 2000 Talk at NWU, Dirk Stroobandt 59
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 intra-
chip 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 60
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 61
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 62
Wire lengths at system level
At system level: many long wires (peak in distribution).
How to model these?Davis and Meindl ‘98:estimation based onRent’s rule with thefloorplanning blocksas logic blocks.IMPORTANT!
May 11th, 2000 Talk at NWU, Dirk Stroobandt 63
Improving CAD tools for design layout
Digital design is complex
Computer-aideddesign (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 64
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 65
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