"Fast estimation of the partitioning Rent characteristic"
Fast estimation of the Fast estimation of the partitioning Rent partitioning Rent
characteristic characteristic using a recursive partitioning using a recursive partitioning
modelmodelJJ. Dambre, D. Stroobandt and J. Van . Dambre, D. Stroobandt and J. Van
CampenhoutCampenhoutSLIP’03 - April 5-6, 2003SLIP’03 - April 5-6, 2003
"Fast estimation of the partitioning Rent characteristic"
Presentation outlinePresentation outline
• Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic• Recursive (bi-)partitioning equations Recursive (bi-)partitioning equations • Cut probability modelsCut probability models• Validation and application for fast Validation and application for fast
estimation of the Rent characteristicestimation of the Rent characteristic• Conclusions & future workConclusions & future work
"Fast estimation of the partitioning Rent characteristic"
• NNcc connections between connections between gates (gates (internal netsinternal nets))
• TTextext Connections to circuit’s Connections to circuit’s exterior (exterior (external nets external nets oror pinspins))
The circuit graphThe circuit graph
Circuit Circuit netlistnetlist, consists of:, consists of:
• GGcc gates gates
Only two-terminal connections Only two-terminal connections considered !considered !
"Fast estimation of the partitioning Rent characteristic"
Donath-based prediction of Donath-based prediction of placement wire lengthsplacement wire lengths
circuitcircuit architecturearchitecture
Perform recursive partitioning of circuit and Perform recursive partitioning of circuit and architecturearchitecture
"Fast estimation of the partitioning Rent characteristic"
Donath-based prediction of Donath-based prediction of placement wire lengthsplacement wire lengths
circuitcircuit
At each level:At each level:calculate number of calculate number of cut nets ...cut nets ...
"Fast estimation of the partitioning Rent characteristic"
Donath-based prediction of Donath-based prediction of placement wire lengthsplacement wire lengths
architecturearchitecture
... and their length ... and their length distribution from the distribution from the architecture site architecture site functionsfunctions
"Fast estimation of the partitioning Rent characteristic"
Recursive circuit partitioning ...Recursive circuit partitioning ...modeled by Rent’s rule??modeled by Rent’s rule??
For a particular circuit netlist:For a particular circuit netlist:
ptG=Tp: Rent exponentp: Rent exponentt: Rent coefficientt: Rent coefficient
Regi
on II
I
Regi
on II
Region I
• Perform complete recursive circuit partitioningPerform complete recursive circuit partitioning• Find average data points Find average data points
for T vs. G for T vs. G (=Rent characteristic)(=Rent characteristic)
• Fit power law to region I:Fit power law to region I:
"Fast estimation of the partitioning Rent characteristic"
Recursive circuit partitioning ...Recursive circuit partitioning ...modeled by Rent’s rule??modeled by Rent’s rule??
Benchmark ibm171
10
100
1000
10000
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06
Module size G
Mod
ule te
rmina
ls T
T = 51.58 G0.45
?
T = 7.45 G0.78
?
Which Rent parameters to Which Rent parameters to use ??use ??
"Fast estimation of the partitioning Rent characteristic"
Rent’s rule vs. the Rent Rent’s rule vs. the Rent characteristiccharacteristic
Rent’s rule is not Rent’s rule is not a very accurate a very accurate model, so ... model, so ...
.... use the Rent .... use the Rent characteristic characteristic instead of Rent’s instead of Rent’s rule!rule!
ibm171
10
100
1000
10000
1E+00 1E+02 1E+04 1E+06Module size G
Mod
ule te
rmina
ls T
"Fast estimation of the partitioning Rent characteristic"
0
5
10
15
20
25
ibm0
1
ibm0
2
ibm0
3
ibm0
4
ibm0
5
ibm0
6
ibm0
7
ibm0
8
ibm0
9
Aver
age
wire
leng
th
predictionPlatoQplaceSA
Last year’s results ...Last year’s results ...
Prediction of average Prediction of average wire lengths, based on wire lengths, based on Donath’s technique, Donath’s technique, but ...but ...
using the Rent using the Rent characteristic instead of characteristic instead of Rent’s ruleRent’s rule
Overestimation, but Overestimation, but very good correlation !very good correlation !
"Fast estimation of the partitioning Rent characteristic"
ibm171
10
100
1000
10000
1E+00 1E+02 1E+04 1E+06Module size G
Mod
ule te
rmina
ls T
ibm171
10
100
1000
10000
1E+00 1E+02 1E+04 1E+06Module size G
Mod
ule te
rmina
ls T
... becomes very ... becomes very inaccurate if too few inaccurate if too few levelslevels... doesn’t help much if ... doesn’t help much if many levelsmany levels
5 levels of 5 levels of partitioningpartitioning
Rent’s rule vs. the Rent Rent’s rule vs. the Rent characteristiccharacteristic
Problem: full recursive partitioning takes a lot of Problem: full recursive partitioning takes a lot of timetimeSolution I*:Solution I*:
perform only limited perform only limited number of partitioning number of partitioning levels and levels and interpolate ...interpolate ...
* suggested in, e.g. Hagen et al., “* suggested in, e.g. Hagen et al., “On the intrinsic Rent On the intrinsic Rent parameter and spectra-based partitioning methodologiesparameter and spectra-based partitioning methodologies”, ”, 19941994
"Fast estimation of the partitioning Rent characteristic"
Rent’s rule vs. the Rent Rent’s rule vs. the Rent characteristiccharacteristic
Problem: recursive partitioning takes a lot of timeProblem: recursive partitioning takes a lot of time
Solution II:Solution II:find a better model find a better model than Rent’s rule ??than Rent’s rule ??
ibm171
10
100
1000
10000
1E+00 1E+02 1E+04 1E+06Module size G
Mod
ule te
rmina
ls T
"Fast estimation of the partitioning Rent characteristic"
Presentation outlinePresentation outline
• Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic• Recursive (bi-)partitioning equations Recursive (bi-)partitioning equations • Cut probability modelsCut probability models• Validation and application for fast Validation and application for fast
estimation of the Rent characteristicestimation of the Rent characteristic• Conclusions & future workConclusions & future work
"Fast estimation of the partitioning Rent characteristic"
Verplaetse’s recursive Verplaetse’s recursive (bi-)partitioning equations(bi-)partitioning equations
Level k-1
Gk-1 = 16
Tk-1 = 6
Nk-1 = 30
Hierarchy levelsk = 0 ...H, with:level 0 = entire circuit,level H = single gate
"Fast estimation of the partitioning Rent characteristic"
Verplaetse’s recursive Verplaetse’s recursive (bi-)partitioning equations(bi-)partitioning equations
Level k Level k
Partition in two equal parts (balanced)Partition in two equal parts (balanced)
"Fast estimation of the partitioning Rent characteristic"
Define:k-1 : fraction of Nk-1 that is cut (=cut probability)
Verplaetse’s recursive Verplaetse’s recursive bipartitioning modelbipartitioning model
Level k
21 k
kGG
111
2 kk
kk NTT
11
21
kk
k NN
Recursive partitioning equations:
"Fast estimation of the partitioning Rent characteristic"
Verplaetse’s recursive Verplaetse’s recursive bipartitioning modelbipartitioning model
Level k
New problem: find expression for k-1 instead of expression for Tk-1 ??
Define:k-1 : fraction of Nk-1 that is cut (=cut probability)
"Fast estimation of the partitioning Rent characteristic"
Presentation outlinePresentation outline
• Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic• Recursive (bi-)partitioning equations Recursive (bi-)partitioning equations • Cut probability modelsCut probability models• Validation and application for fast Validation and application for fast
estimation of the Rent characteristicestimation of the Rent characteristic• Conclusions & future workConclusions & future work
"Fast estimation of the partitioning Rent characteristic"
module 1 module 2
Cut probabilities for random Cut probabilities for random partitioningpartitioning
Random bipartitioning of a Random bipartitioning of a two-terminal net = randomly two-terminal net = randomly assigning both connected assigning both connected gates to two modulesgates to two modules
Pcut = (1/2) Gk/(Gk-1) = k,
rand
4 possible assignments, net 4 possible assignments, net is cut for two of them:is cut for two of them:
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for random Cut probabilities for random partitioningpartitioning
0.1
1
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05Module size Gk
Cut p
roba
bilit
y
k
0.5
k, rand = (1/2) Gk/(Gk-1)
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
Measured values of Measured values of kk ?? ??Must perform some manipulations on Must perform some manipulations on Rent characteristic because:Rent characteristic because:• circuit size not equal to 2circuit size not equal to 2HH • partitionings not perfectly balancedpartitionings not perfectly balanced
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
0.001
0.01
0.1
1
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06
Module size Gk
Cut p
roba
bilit
y
k randomrandom
ibm17ibm17
Always = 1(boundary condition)
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
Verplaetse’s model for Verplaetse’s model for kk : :
1
2/12
k
k G
With With and and parameters to be fitted parameters to be fitted
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
0.001
0.01
0.1
1
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06Module size Gk
Cut p
roba
bilit
y
k randomrandom
ibm17ibm17
VerplaetseVerplaetse
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
New model for New model for kk : based on observed : based on observed relationship between relationship between kk and and k,randk,rand
0.001
0.01
0.1
1
1.0E+00 1.0E+02 1.0E+04 1.0E+06Module size Gk
k/
k ,
rand
12/ 1
k
kk G
G
randkkk G ,~ 11 H+
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
0.001
0.01
0.1
1
1 10 100 1000 10000 100000 1E+06
Module size Gk
Cut p
roba
bilit
y
k randomrandom
ibm17ibm17
VerplaetseVerplaetse
New modelNew model
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
1
10
100
1000
10000
1 10 100 1000 10000 100000 1E+06Module size Gk
Mod
ule
term
inal
s k
ibm17VerplaetseNew model
"Fast estimation of the partitioning Rent characteristic"
Presentation outlinePresentation outline
• Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic• Recursive (bi-)partitioning equations Recursive (bi-)partitioning equations • Cut probability modelsCut probability models• Validation and application for fast Validation and application for fast
estimation of the Rent characteristicestimation of the Rent characteristic• Conclusions & future workConclusions & future work
"Fast estimation of the partitioning Rent characteristic"
Cut probabilities for optimal Cut probabilities for optimal circuit partitioningcircuit partitioning
0
0.05
0.1
0.15
0.2
0.25ib
m01
ibm0
2ib
m03
ibm0
4ib
m05
ibm0
6ib
m07
ibm0
8ib
m09
ibm1
0ib
m11
ibm1
2ib
m13
ibm1
4ib
m15
ibm1
6ib
m17
ibm1
8Av
erag
e
Erro
r on fi
tted
k -
value
s
VerplaetseNew model
Both models have comparable fitting qualityBoth models have comparable fitting qualityNew model only has a single fitting parameterNew model only has a single fitting parameter
"Fast estimation of the partitioning Rent characteristic"
5
10
15
20
25
ibm01
ibm02
ibm03
ibm04
ibm05
ibm06
ibm07
ibm08
ibm09
ibm10
ibm11
ibm12
ibm13
ibm14
ibm15
ibm16
ibm17
ibm18
Aver
age
wire
leng
th
Rent characteristicVerplaetse (using all levels)New model (using all levels)
Correlation coefficients:Correlation coefficients:• Verplaetse vs. Rc: 0.988Verplaetse vs. Rc: 0.988• New model vs. Rc: 0.995New model vs. Rc: 0.995
Update of last year’s results ...Update of last year’s results ...
Prediction of average Prediction of average wire lengths, based on wire lengths, based on Donath’s technique, Donath’s technique, but ...but ...
using the new using the new partitioning model partitioning model instead of the Rent instead of the Rent characteristic (Rc)characteristic (Rc)
"Fast estimation of the partitioning Rent characteristic"
Can models for Can models for kk be applied to obtain be applied to obtain accurate estimations of the entire Rent accurate estimations of the entire Rent characteristic based on a few partitioning characteristic based on a few partitioning levels only??levels only??
Fast estimation of the Rent Fast estimation of the Rent characteristiccharacteristic
"Fast estimation of the partitioning Rent characteristic"
Fast estimation of the Rent Fast estimation of the Rent characteristiccharacteristic
0.01
0.1
1
10
0 2 4 6 8 10 12Number of partitioning levels
Aver
age e
rror
(a
cros
s ibm
01-ib
m18)
I nterpolationVerplaetseNew model
all levels
New model converges faster: New model converges faster: for most benchmarks, two levels is for most benchmarks, two levels is enoughenoughuse 3 or 4 levels to be sureuse 3 or 4 levels to be sure
"Fast estimation of the partitioning Rent characteristic"
5
10
15
20
25
ibm01
ibm02
ibm03
ibm04
ibm05
ibm06
ibm07
ibm08
ibm09
ibm10
ibm11
ibm12
ibm13
ibm14
ibm15
ibm16
ibm17
ibm18
Aver
age
wire
leng
th
Rent characteristicVerplaetse (fi t f or 3 levels)New model (fi t f or 3 levels)
Update of last year’s results ...Update of last year’s results ...
Prediction of average Prediction of average wire lengths, based on wire lengths, based on Donath’s technique, Donath’s technique, but ...but ...
using the new using the new partitioning model, fit partitioning model, fit for 3 levels of for 3 levels of partitioningpartitioningCorrelation coefficients:Correlation coefficients:• Verplaetse vs. Rc: 0.902Verplaetse vs. Rc: 0.902• New model vs. Rc: 0.824New model vs. Rc: 0.824
"Fast estimation of the partitioning Rent characteristic"
Presentation outlinePresentation outline
• Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic• Recursive (bi-)partitioning equations Recursive (bi-)partitioning equations • Cut probability modelsCut probability models• Validation and application for fast Validation and application for fast
estimation of the Rent characteristicestimation of the Rent characteristic• Conclusions & future workConclusions & future work
"Fast estimation of the partitioning Rent characteristic"
ConclusionsConclusions
• Adaptation of Verplaetse’s recursive Adaptation of Verplaetse’s recursive partitioning equations and cut partitioning equations and cut probability modelprobability model
• Application of this model for fast Application of this model for fast estimation of entire partitioning Rent estimation of entire partitioning Rent characteristiccharacteristic
"Fast estimation of the partitioning Rent characteristic"
Future workFuture work
• Extension to multi-terminal netsExtension to multi-terminal nets• Try to find theoretical foundations for Try to find theoretical foundations for
cut probability modelcut probability model• Is it possible to estimate cut Is it possible to estimate cut
probabilities from circuit graph probabilities from circuit graph withoutwithout performing partitioning??performing partitioning??