"Fast estimation of the partitioning Rent characteristic" Fast estimation of the partitioning Rent...

"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


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


• 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 !


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



circuitcircuit

At each level:At each level:calculate number of calculate number of cut nets ...cut nets ...



architecturearchitecture

... and their length ... and their length distribution from the distribution from the architecture site architecture site functionsfunctions


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:


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 ??


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


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 !


ibm171

10

100

1000

10000


Mod

ule te

rmina

ls T

ibm171

10

100

1000

10000


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


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



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


Mod

ule te

rmina

ls T


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


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)


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:


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)


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:


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)


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



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)



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



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



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+



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



1

10

100

1000

10000

1 10 100 1000 10000 100000 1E+06Module size Gk

Mod

ule

term

inal

s k

ibm17VerplaetseNew model



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


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 ...


using the new using the new partitioning model partitioning model instead of the Rent instead of the Rent characteristic (Rc)characteristic (Rc)


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 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


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 ...


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


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


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??

"Fast estimation of the partitioning Rent characteristic" Fast estimation of the partitioning Rent...

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