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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Wirelength Estimation based on Rent Wirelength Estimation based on Rent Exponents of Partitioning and PlacementExponents of Partitioning and Placement

Xiaojian Yang

Elaheh Bozorgzadeh

Majid Sarrafzadeh

Embedded and Reconfigurable System Lab

Computer Science Department, UCLA

Xiaojian Yang

Elaheh Bozorgzadeh

Majid Sarrafzadeh

Embedded and Reconfigurable System Lab

Computer Science Department, UCLA

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

OutlineOutlineOutlineOutline

Introduction Motivation Rent Exponents of Partitioning and Placement Wirelength Estimation based on Rent’s rule Rent Exponent and Placement Quality Conclusion

Introduction Motivation Rent Exponents of Partitioning and Placement Wirelength Estimation based on Rent’s rule Rent Exponent and Placement Quality Conclusion

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

IntroductionIntroductionIntroductionIntroduction

Rent’s rule and its application P = TB r

Introduced by Landman and Russo, 1971 Used for Wirelength estimation

Rent Exponent Key role in Rent’s rule applications Extracted from partitioning-based method “Intrinsic Rent exponent”, Hagen, et.al 1994

Rent’s rule and its application P = TB r

Introduced by Landman and Russo, 1971 Used for Wirelength estimation

Rent Exponent Key role in Rent’s rule applications Extracted from partitioning-based method “Intrinsic Rent exponent”, Hagen, et.al 1994

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Introduction (cont’d)Introduction (cont’d)Introduction (cont’d)Introduction (cont’d)

Two Rent Exponents Topological and Geometrical (Christie, SLIP2000) Partitioning and Placement

Questions: Same or different? Which one is appropriate for Rent’s rule applications? Relationship?

Two Rent Exponents Topological and Geometrical (Christie, SLIP2000) Partitioning and Placement

Questions: Same or different? Which one is appropriate for Rent’s rule applications? Relationship?

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Partitioning Rent ExponentPartitioning Rent ExponentPartitioning Rent ExponentPartitioning Rent Exponent

log P

log B B – Number of cells

P – Number of external nets

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Partitioning Rent ExponentPartitioning Rent ExponentPartitioning Rent ExponentPartitioning Rent Exponent

slope = r

log P

log B B – Number of cells

P – Number of external nets

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Placement Rent exponentPlacement Rent exponentPlacement Rent exponentPlacement Rent exponent

log P

log B

slope = r’

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Difference between two exponentsDifference between two exponentsDifference between two exponentsDifference between two exponents

Partitioning objective: Minimizing cut-size Embed partitions into two-dimensional plane Cut-size increases in placement compared to

partitioning

Partitioning objective: Minimizing cut-size Embed partitions into two-dimensional plane Cut-size increases in placement compared to

partitioning log P

log B

Placement

Partitioning

Placement r’ > Partitioning rPlacement r’ > Partitioning r

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Relation between two Rent exponentsRelation between two Rent exponentsRelation between two Rent exponentsRelation between two Rent exponents

Based on min-cut placement approaches (recursively bipartitioning)

Different partitioning instances Partitioning tree approach: Pure Partitioning Partitioning in Placement: terminal propagation

Based on min-cut placement approaches (recursively bipartitioning)

Different partitioning instances Partitioning tree approach: Pure Partitioning Partitioning in Placement: terminal propagation

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Pure PartitioningPure PartitioningPure PartitioningPure Partitioning

Cut-size = CCut-size = C Cut-size = CCut-size = C

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Terminal PropagationTerminal PropagationTerminal PropagationTerminal Propagation

Cut-size = C’ > CCut-size = C’ > C Cut-size = C’ > CCut-size = C’ > C

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Cut size increasesCut size increasesCut size increasesCut size increases

cut-size : C cut-size : C C’ C’ cut-size : C cut-size : C C’ C’

PP PP

PP11 PP11 PP22 PP22

CCCC BB22 BB22

BB11 BB11

uu uu

PP11+C = TB+C = TB11r r = P= P PP11+C = TB+C = TB11r r = P= P

PP11+P+P22 = T(B = T(B11+B+B22))rr PP11+P+P22 = T(B = T(B11+B+B22))rr PP11= 2= 2r-1 r-1 PP PP11= 2= 2r-1 r-1 PP

--- effect of external net--- effect of external net --- effect of external net--- effect of external net

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

RelationshipRelationshipRelationshipRelationship

Brr

r

log

)21log( 1

r --- Partitioning Rent exponentr --- Partitioning Rent exponentr’ --- Placement Rent exponentr’ --- Placement Rent exponentB --- number of cellsB --- number of cells --- 0 --- 0 1, effect of external 1, effect of external netnet

r --- Partitioning Rent exponentr --- Partitioning Rent exponentr’ --- Placement Rent exponentr’ --- Placement Rent exponentB --- number of cellsB --- number of cells --- 0 --- 0 1, effect of external 1, effect of external netnet

• Limited RangeLimited Range• Rough Estimation from r to r’Rough Estimation from r to r’• Limited RangeLimited Range• Rough Estimation from r to r’Rough Estimation from r to r’

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Experiment BackgroundExperiment BackgroundExperiment BackgroundExperiment Background

Benchmark: MCNC+IBM IBM: Derived from ISPD98 partitioning benchmark Size from 20k cells --- 220k cells

Partitioning: hMetis Placement: wirelength-driven

Capo, Feng Shui, Dragon Rent exponent extraction

Linear regression Each point corresponds to one level in partitioning or

placement

Benchmark: MCNC+IBM IBM: Derived from ISPD98 partitioning benchmark Size from 20k cells --- 220k cells

Partitioning: hMetis Placement: wirelength-driven

Capo, Feng Shui, Dragon Rent exponent extraction

Linear regression Each point corresponds to one level in partitioning or

placement

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Experimental Observation (1)Experimental Observation (1)Experimental Observation (1)Experimental Observation (1)

Example: ibm11, 68k cells Example: ibm11, 68k cells

0.608

0.6930.682

0.6670.680

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

Partitioning rPartitioning rPartitioning rPartitioning r Placement r’Placement r’Placement r’Placement r’ Estimated Placement r’Estimated Placement r’Estimated Placement r’Estimated Placement r’

CapoCapoCapoCapo Feng ShuiFeng ShuiFeng ShuiFeng Shui DragonDragonDragonDragon

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Wirelength Estimation based on Wirelength Estimation based on Rent’s ruleRent’s ruleWirelength Estimation based on Wirelength Estimation based on Rent’s ruleRent’s rule Classical problem

Donath 1979 Stroobandt et.al 1994 Davis et.al 1998

Needs geometrical (placement) Rent exponent Comparison

Estimated WL using Partitioning Rent exponent Estimated WL using Placement Rent exponent Total Wirelength after global routing (maze-based)

Classical problem Donath 1979 Stroobandt et.al 1994 Davis et.al 1998

Needs geometrical (placement) Rent exponent Comparison

Estimated WL using Partitioning Rent exponent Estimated WL using Placement Rent exponent Total Wirelength after global routing (maze-based)

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Experimental Observation (2)Experimental Observation (2)Experimental Observation (2)Experimental Observation (2)

Example: ibm13, 81k cells

Overall: Estimation based on Partitioning Rent exponent under-estimate total wirelength 19% --- 32%

Example: ibm13, 81k cells

Overall: Estimation based on Partitioning Rent exponent under-estimate total wirelength 19% --- 32%

PartitioningPartitioningr = 0.600r = 0.600PartitioningPartitioningr = 0.600r = 0.600

Actual WLActual WLCapo FS DragonCapo FS Dragon

Actual WLActual WLCapo FS DragonCapo FS Dragon

Placement Rent r’Placement Rent r’Capo FS DragonCapo FS DragonPlacement Rent r’Placement Rent r’Capo FS DragonCapo FS Dragon

500

0

676716

605

0

675

627586

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Estimation based on r’Estimation based on r’Estimation based on r’Estimation based on r’

Recursively bipartitioning

Derivation of Placement Rent exponentcircuitcircuitcircuitcircuit

Wirelength Estimation

Estimated Estimated total wirelengthtotal wirelength

Estimated Estimated total wirelengthtotal wirelength

r r (partition r)

r r (partition r)

r’ r’ (place r)r’ r’ (place r)

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Estimation based on r’Estimation based on r’Estimation based on r’Estimation based on r’

Estimation results: -12% --- +14%

Total wirelength estimation is hard Rent exponent Placement approach Routing approach Congestion --- unevenly distributed wires

Estimation results: -12% --- +14%

Total wirelength estimation is hard Rent exponent Placement approach Routing approach Congestion --- unevenly distributed wires

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Rent exponent, a placement metric?Rent exponent, a placement metric?Rent exponent, a placement metric?Rent exponent, a placement metric?

Hagen et.al Rent exponent is a measurement of partitioning

approach Ratio-cut gives the smallest Rent exponent

Similar case in Placement? Ordinary placement measurement

Total bounding box wirelength or routed wirelength Correlation between wirelength and Rent exponent?

Hagen et.al Rent exponent is a measurement of partitioning

approach Ratio-cut gives the smallest Rent exponent

Similar case in Placement? Ordinary placement measurement

Total bounding box wirelength or routed wirelength Correlation between wirelength and Rent exponent?

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

Experimental ObservationExperimental ObservationExperimental ObservationExperimental Observation

Rent exponentRent exponentRent exponentRent exponent Bounding box Bounding box wirelengthwirelength

Bounding box Bounding box wirelengthwirelength

Routed Routed wirelengthwirelengthRouted Routed

wirelengthwirelength

• Weak correlation: most shorter wirelengths Weak correlation: most shorter wirelengths correspond to lower Rent exponents correspond to lower Rent exponents • Open questionOpen question

• Weak correlation: most shorter wirelengths Weak correlation: most shorter wirelengths correspond to lower Rent exponents correspond to lower Rent exponents • Open questionOpen question

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International Workshop on System-Level Interconnection Prediction, Sonoma County, CA March 2001

ERER UCLAUCLA

ConclusionConclusionConclusionConclusion

Topological (partitioning) Rent exponent and Geometrical (placement) Rent exponent are different.

Relationship between two Rent exponents. Wirelength Estimation should use Geometrical

Rent exponent. Open question: Is Rent exponent a metric of

placement quality?

Topological (partitioning) Rent exponent and Geometrical (placement) Rent exponent are different.

Relationship between two Rent exponents. Wirelength Estimation should use Geometrical

Rent exponent. Open question: Is Rent exponent a metric of

placement quality?