Fast Algorithms for Slew Constrained Minimum Cost Buffering

Fast Algorithms for Slew Constrained Minimum Cost Buffering

S Hu C Alpert J Hu S Karandikar Z Li W Shi and C SzeS Hu C Alpert J Hu S Karandikar Z Li W Shi and C Sze

Dept of ECE Texas AampM UniversityDept of ECE Texas AampM University

IBM Austin Research LabIBM Austin Research Lab

2

OutlineOutline

MotivationMotivation Slew ModelSlew Model AlgorithmsAlgorithms

ndash Discrete slew buffering with fixed input slewDiscrete slew buffering with fixed input slewndash Discrete slew buffering with non-fixed input Discrete slew buffering with non-fixed input

slewslewndash Continuous slew bufferingContinuous slew buffering

Experimental ResultsExperimental Results ConclusionConclusion

3

MotivationMotivation

Buffer insertion is prevalent in VLSI designsBuffer insertion is prevalent in VLSI designsndash eg for timing optimization and improving signal eg for timing optimization and improving signal

integrityintegrity ProblemsProblems

ndash Most nets (90-95) are NOT timing critical so donrsquot Most nets (90-95) are NOT timing critical so donrsquot need a timing-driven formulationneed a timing-driven formulation

ndash Takes a long time as millions of nets need bufferingTakes a long time as millions of nets need bufferingndash Uses a ton of area so area minimization is criticalUses a ton of area so area minimization is critical

Our solutionOur solutionndash Replace timing-driven formulation with slew-driven Replace timing-driven formulation with slew-driven

formulation - formulation - Slew BufferingSlew Bufferingndash Good enough for Good enough for most netsmost nets as they are not critical as they are not criticalndash gt100xgt100x faster than timing-driven buffering faster than timing-driven bufferingndash Still saving area compared to timing bufferingStill saving area compared to timing buffering

4

A New Flow for Buffering 1M NetsA New Flow for Buffering 1M Nets

First buffering 1 million nets using slew First buffering 1 million nets using slew buffering algorithm buffering algorithm ndash Can be done efficiently as our algorithm can Can be done efficiently as our algorithm can

buffer buffer 10001000 industrial nets ( industrial nets (4848 buffer types) in buffer types) in 55 seconds seconds

Timing analysis finds that most nets except Timing analysis finds that most nets except say 50K critical nets satisfy the timing say 50K critical nets satisfy the timing constraintconstraint

Rip up these 50K critical nets for rebuffering Rip up these 50K critical nets for rebuffering by timing buffering algorithmby timing buffering algorithm

BenefitsBenefitsndash Much fasterMuch fasterndash More area savingMore area saving

5

Slew DefinitionSlew Definition

Delay

Slew (Transition Time)

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

2

OutlineOutline

MotivationMotivation Slew ModelSlew Model AlgorithmsAlgorithms

ndash Discrete slew buffering with fixed input slewDiscrete slew buffering with fixed input slewndash Discrete slew buffering with non-fixed input Discrete slew buffering with non-fixed input

slewslewndash Continuous slew bufferingContinuous slew buffering

Experimental ResultsExperimental Results ConclusionConclusion

3

MotivationMotivation

Buffer insertion is prevalent in VLSI designsBuffer insertion is prevalent in VLSI designsndash eg for timing optimization and improving signal eg for timing optimization and improving signal

integrityintegrity ProblemsProblems

ndash Most nets (90-95) are NOT timing critical so donrsquot Most nets (90-95) are NOT timing critical so donrsquot need a timing-driven formulationneed a timing-driven formulation

ndash Takes a long time as millions of nets need bufferingTakes a long time as millions of nets need bufferingndash Uses a ton of area so area minimization is criticalUses a ton of area so area minimization is critical

Our solutionOur solutionndash Replace timing-driven formulation with slew-driven Replace timing-driven formulation with slew-driven

formulation - formulation - Slew BufferingSlew Bufferingndash Good enough for Good enough for most netsmost nets as they are not critical as they are not criticalndash gt100xgt100x faster than timing-driven buffering faster than timing-driven bufferingndash Still saving area compared to timing bufferingStill saving area compared to timing buffering

4

A New Flow for Buffering 1M NetsA New Flow for Buffering 1M Nets

First buffering 1 million nets using slew First buffering 1 million nets using slew buffering algorithm buffering algorithm ndash Can be done efficiently as our algorithm can Can be done efficiently as our algorithm can

buffer buffer 10001000 industrial nets ( industrial nets (4848 buffer types) in buffer types) in 55 seconds seconds

Timing analysis finds that most nets except Timing analysis finds that most nets except say 50K critical nets satisfy the timing say 50K critical nets satisfy the timing constraintconstraint

Rip up these 50K critical nets for rebuffering Rip up these 50K critical nets for rebuffering by timing buffering algorithmby timing buffering algorithm

BenefitsBenefitsndash Much fasterMuch fasterndash More area savingMore area saving

5

Slew DefinitionSlew Definition

Delay

Slew (Transition Time)

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

3

MotivationMotivation

Buffer insertion is prevalent in VLSI designsBuffer insertion is prevalent in VLSI designsndash eg for timing optimization and improving signal eg for timing optimization and improving signal

integrityintegrity ProblemsProblems

ndash Most nets (90-95) are NOT timing critical so donrsquot Most nets (90-95) are NOT timing critical so donrsquot need a timing-driven formulationneed a timing-driven formulation

ndash Takes a long time as millions of nets need bufferingTakes a long time as millions of nets need bufferingndash Uses a ton of area so area minimization is criticalUses a ton of area so area minimization is critical

Our solutionOur solutionndash Replace timing-driven formulation with slew-driven Replace timing-driven formulation with slew-driven

formulation - formulation - Slew BufferingSlew Bufferingndash Good enough for Good enough for most netsmost nets as they are not critical as they are not criticalndash gt100xgt100x faster than timing-driven buffering faster than timing-driven bufferingndash Still saving area compared to timing bufferingStill saving area compared to timing buffering

4

A New Flow for Buffering 1M NetsA New Flow for Buffering 1M Nets

First buffering 1 million nets using slew First buffering 1 million nets using slew buffering algorithm buffering algorithm ndash Can be done efficiently as our algorithm can Can be done efficiently as our algorithm can

buffer buffer 10001000 industrial nets ( industrial nets (4848 buffer types) in buffer types) in 55 seconds seconds

Timing analysis finds that most nets except Timing analysis finds that most nets except say 50K critical nets satisfy the timing say 50K critical nets satisfy the timing constraintconstraint

Rip up these 50K critical nets for rebuffering Rip up these 50K critical nets for rebuffering by timing buffering algorithmby timing buffering algorithm

BenefitsBenefitsndash Much fasterMuch fasterndash More area savingMore area saving

5

Slew DefinitionSlew Definition

Delay

Slew (Transition Time)

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

4

A New Flow for Buffering 1M NetsA New Flow for Buffering 1M Nets

First buffering 1 million nets using slew First buffering 1 million nets using slew buffering algorithm buffering algorithm ndash Can be done efficiently as our algorithm can Can be done efficiently as our algorithm can

buffer buffer 10001000 industrial nets ( industrial nets (4848 buffer types) in buffer types) in 55 seconds seconds

Timing analysis finds that most nets except Timing analysis finds that most nets except say 50K critical nets satisfy the timing say 50K critical nets satisfy the timing constraintconstraint

Rip up these 50K critical nets for rebuffering Rip up these 50K critical nets for rebuffering by timing buffering algorithmby timing buffering algorithm

BenefitsBenefitsndash Much fasterMuch fasterndash More area savingMore area saving

5

Slew DefinitionSlew Definition

Delay

Slew (Transition Time)

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

5

Slew DefinitionSlew Definition

Delay

Slew (Transition Time)

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

6

Slew ModelSlew Model

( ) ( ) ln 9w i j i js v v ElmoreDelay v v

2 2( ) ( ) ( )j b out i w i js v s v s v v

Upstream Downstream

Sbout(vi) Slew degradation on wire Sw(vivj)

S(vj)

vi vj

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

7

BufferDriver Input Slew Assumption BufferDriver Input Slew Assumption

Output slew of a buffer depends on its input Output slew of a buffer depends on its input slewslewbottom-up dynamic programming bottom-up dynamic programming inapplicableinapplicable

AssumptionAssumption the input slew of each buffer is the input slew of each buffer is conservatively assumed to be a fixed valueconservatively assumed to be a fixed value

Then the output slew of a buffer isThen the output slew of a buffer is

where Rwhere Rbb and K and Kbb are called are called slew resistanceslew resistance and and intrinsic slewintrinsic slew

( ) ( )b out i b i bs v R C v K

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

8

Slew Resistance of An InverterSlew Resistance of An Inverter

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

9

Problem FormulationProblem Formulation

GivenGivenndash A Steiner treeA Steiner treendash Maximum input slew rate α Maximum input slew rate α

at each buffersink (at each buffersink (slew constraintslew constraint))ndash A buffer libraryA buffer libraryndash RC parametersRC parametersndash Candidate buffer locationsCandidate buffer locations

Find a Find a minimal areaminimal area buffer insertion solution buffer insertion solution such that the slew constraint is satisfiedsuch that the slew constraint is satisfied

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

10

NP-Complete ProofNP-Complete Proof

hellip

nN

1nN

2N

nN

1

1

There is a solution with slew constraint and cost n

n i

i

N M N N

BufferBuffer Slew Slew ResRes

Input Input CapCap

CostCost

BB1111 XX11 XX22+N+Nnn

BB2211 XX22 XX11+N+Nnn

BB33NN XX33 XX44+N+Nn-1n-1

BB44NN XX44 XX33+N+Nn-1n-1

helliphellip helliphellip helliphellip helliphellip

BB2n-12n-1 NNn-1n-1 XX2n-12n-1 XX2n2n+N+N

BB2n2n NNn-1n-1 xx2n2n XX2n-12n-1+N+N

2

1 2 21

2 -1 2

2-1 partition problem given 2 positive integers and 2

is there an index set which contains exactly one of and for each 1 such that

n

n ii

i i ii I

n x x x x N

I x x i n x N

1

1 1

n n

n n i ii i

i I i I i i

N x N x N N N

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

11

Fixed-Input Slew Buffering Candidate Solution CharacteristicsFixed-Input Slew Buffering Candidate Solution Characteristics

Each candidate Each candidate solution is associated solution is associated withwithndash vvii a node a nodendash ccii downstream downstream

capacitancecapacitancendash ssii cumulative slew cumulative slew

degradation along degradation along wirewire

ndash wwii cumulative buffer cumulative buffer areaarea

vi is a sinkci is sink capacitance

v is an internal node

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

12

Dynamic ProgrammingDynamic Programming

Candidate solutions are propagated toward the source

Start from sinks Candidate

solutions are generated

Three operationsndash Add Wirendash Insert Bufferndash Merge

Solution Pruning

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

13

Solution Propagation Add WireSolution Propagation Add Wire

cc22 = c = c11 + cx + cx ss22 = s = s11 + (rcx + (rcx222 + rxc2 + rxc11)ln9)ln9 s slew degradation along wiress slew degradation along wires r wire slew resistance per unit lengthr wire slew resistance per unit length c wire capacitance per unit lengthc wire capacitance per unit length

(v1 c1 w1 s1)

(v2 c2 w2 s2)

x

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

14

Solution Propagation Insert BufferSolution Propagation Insert Buffer

c1b = Cb s1b = 0

w1b = w1+w(b)Cb buffer input capacitancePruned if the following slew constraint is violated

Rb buffer output slew resistanceKb buffer intrinsic slew

(v1 c1 w1 s1)(v1 c1b w1b s1b)

221 1( )b bR c K s

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

15

Solution Propagation MergeSolution Propagation Merge

ccmerge merge = c= cl l + c+ crr

wwmerge merge = w= wl l + w+ wrr

ssmergemerge = max(s = max(sl l s srr))

(v cl wl sl) (v cr wlrsr)

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

16

Solution PruningSolution Pruning

Two candidate solutionsTwo candidate solutionsndash Solution 1 (v c1 w1 s1)Solution 1 (v c1 w1 s1)ndash Solution 2 (v c2 w2 s2)Solution 2 (v c2 w2 s2)

Solution 1 is Solution 1 is inferiorinferior if if ndash c1 gt c2 larger loadc1 gt c2 larger loadndash and w1 gt w2 larger buffer areaand w1 gt w2 larger buffer areandash and s1 gt s2 worse cumulative slew and s1 gt s2 worse cumulative slew

degradationdegradationon wireon wire

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

17

Timing vs Slew Buffering (I)Timing vs Slew Buffering (I)

A buffer insertion S=0 C=C(b) A buffer insertion S=0 C=C(b) Inserting one buffer Inserting one buffer one new one new

solutionsolution (the one with the smallest (the one with the smallest cost) cost)

In min-cost timing buffering a buffer In min-cost timing buffering a buffer insertion brings many non-inferior insertion brings many non-inferior (CWQ) with the same C where Q is (CWQ) with the same C where Q is the required arrival time (RAT)the required arrival time (RAT)

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

18

Timing vs Slew Buffering (II)Timing vs Slew Buffering (II)

Slew constraint is close to length Slew constraint is close to length constraint constraint

An extreme case An extreme case ndash in min-cost timing buffering solutions with no buffer in min-cost timing buffering solutions with no buffer

inserted live till driver inserted live till driver ndash Soon become infeasible in slew bufferingSoon become infeasible in slew buffering

A A linear timelinear time optimal algorithm for slew optimal algorithm for slew buffering with a single buffer type buffering with a single buffer type

No polynomial timeNo polynomial time min-cost timing min-cost timing buffering algorithm in the same case buffering algorithm in the same case

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

19

Non-Fixed Input SlewNon-Fixed Input Slew

Input slew to each buffer can vary Input slew to each buffer can vary Our idea discretize the possible input slew Our idea discretize the possible input slew

values into values into input slew binsinput slew bins For each input slew bin carry out the above For each input slew bin carry out the above

procedure (for the fixed input slew case) to procedure (for the fixed input slew case) to propagate solutions propagate solutions

Some detailsSome detailsndash Input slew bins can be merged for speedupInput slew bins can be merged for speedupndash Inferiority also depends on the slew binInferiority also depends on the slew binndash A maximum bipartite matching algorithm is A maximum bipartite matching algorithm is

used for pruningused for pruning

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

20

Continuous Slew BufferingContinuous Slew Buffering

Buffers are allowed to be inserted Buffers are allowed to be inserted anywhereanywhere

Single buffer type a linear greedy optimal Single buffer type a linear greedy optimal algorithm ndash add buffer as upstream as algorithm ndash add buffer as upstream as possiblepossible

Start from sinks

Greedy algorithm toward the source

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

21

Multiple Buffer TypesMultiple Buffer Types

Multiple buffer types greedily inserts Multiple buffer types greedily inserts buffers for every possibilitybuffers for every possibilityndash SlowSlow

Approximation via Approximation via adaptive buffer adaptive buffer selectionselection ndash Buffer library is shrunkenBuffer library is shrunkenndash Prefer buffer types with small slew Prefer buffer types with small slew

resistanceresistance Tight slew constraint choose top few buffer Tight slew constraint choose top few buffer

typestypes Loose constraint choose more buffer typesLoose constraint choose more buffer types

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

22

ExperimentsExperiments

Experiment SetupExperiment Setupndash 1000 industrial netlists1000 industrial netlistsndash 48 buffer types including non-inverting 48 buffer types including non-inverting

buffers and inverting buffersbuffers and inverting buffersndash A Pentium 4 machine with a 32GHz CPU A Pentium 4 machine with a 32GHz CPU

1G memory1G memory

Compared to slew constrained min-Compared to slew constrained min-cost timing buffering cost timing buffering ndash Pruning based on (QCW) S is maintainedPruning based on (QCW) S is maintainedndash S is only responsible for checking whether S is only responsible for checking whether

the solution violates the slew constraintthe solution violates the slew constraint

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

23

Slew Constraint vs Buffer AreaSlew Constraint vs Buffer Area

05000

100001500020000250003000035000400004500050000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

Are

a

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

24

Slew Constraint vs CPU Time (s)Slew Constraint vs CPU Time (s)

0100200300400500600700800900

1000

03 05 1 15 2

Fixed SBTiming BufNon-fixed SBContinuous

Slew Constraint

CPU

Tim

e (s)

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

25

Slew Constraint vs SlackSlew Constraint vs Slack

7600

7800

8000

8200

8400

8600

8800

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sla

ck

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

26

Slew Con vs Solutions at DriverSlew Con vs Solutions at Driver

0

50

100

150

200

250

300

350

03 05 1 15 2

Fixed SBTiming Buf

Slew Constraint

Sol a

t Driv

er

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

27

ObservationsObservations

Discrete Fixed-Input Slew BufferingDiscrete Fixed-Input Slew Bufferingndash Loose slew constraint smaller areaLoose slew constraint smaller areandash gt100x faster than timing bufferinggt100x faster than timing bufferingndash Saves 6 area over timing bufferingSaves 6 area over timing bufferingndash Small slack sacrificeSmall slack sacrifice

Non-fixed input slew buffering Non-fixed input slew buffering ndash Save up to 40 area over fixed input slew bufferingSave up to 40 area over fixed input slew buffering

Continuous slew bufferingContinuous slew bufferingndash Tight slew constraint causes many buffer insertions Tight slew constraint causes many buffer insertions

Not-well set candidate buffer positions significant Not-well set candidate buffer positions significant buffer waste buffer waste

ndash Continuous slew buffering reduces the wasteContinuous slew buffering reduces the wastendash Fast due to adaptive buffer selection strategyFast due to adaptive buffer selection strategy

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion

28

ConclusionConclusion

Propose three slew buffering algorithms Propose three slew buffering algorithms ndash Discrete fixed-input slew bufferingDiscrete fixed-input slew bufferingndash Discrete non-fixed input slew bufferingDiscrete non-fixed input slew bufferingndash Continuous slew bufferingContinuous slew buffering

gt100x faster while still saving area gt100x faster while still saving area compared to timing bufferingcompared to timing buffering

gt90 nets are not timing critical in reality gt90 nets are not timing critical in reality and thus can be buffered by our and thus can be buffered by our algorithmalgorithm

• Fast Algorithms for Slew Constrained Minimum Cost Buffering
• Outline
• Motivation
• A New Flow for Buffering 1M Nets
• Slew Definition
• Slew Model
• BufferDriver Input Slew Assumption
• Slew Resistance of An Inverter
• Problem Formulation
• NP-Complete Proof
• Fixed-Input Slew Buffering Candidate Solution Characteristics
• Dynamic Programming
• Solution Propagation Add Wire
• Solution Propagation Insert Buffer
• Solution Propagation Merge
• Solution Pruning
• Timing vs Slew Buffering (I)
• Timing vs Slew Buffering (II)
• Non-Fixed Input Slew
• Continuous Slew Buffering
• Multiple Buffer Types
• Experiments
• Slew Constraint vs Buffer Area
• Slew Constraint vs CPU Time (s)
• Slew Constraint vs Slack
• Slew Con vs Solutions at Driver
• Observations
• Conclusion