Post on 15-Jan-2016
Directions in Low-Power CADDirections in Low-Power CAD
Dennis Sylvester
University of Michigan
dennis@eecs.umich.edu
http://vlsida.eecs.umich.edu
With acknowledgements to: Prof. David Blaauw, Dr. Sarvesh Kulkarni, Saumil Shah, Kavi Chopra
TopicsTopics
A new dual-Vth assignment formulation
Dual-Vdd power distribution
Approaches to parametric yield optimization: statistical leakage + delay
MotivationMotivation We require high-performance yet low-power circuits
Leakage power contributes significantly to total power
All High- Vth implementation too slow
All Low-Vth implementation too leaky
Dual- Vth processes popular
Problem Definition Minimize
Total Circuit Power Subject to
Circuit Delay Constraint Sizing Constraints
Optimization Variables Gate Sizes Gate Threshold Voltages
SwitchingSubthreshold
leakage
S. Narendra et al [ICCAD ’03]
Gate Sizing + Vth Assignment Problem Prior WorkGate Sizing + Vth Assignment Problem Prior Work
Traditionally a discrete problem
Previous approaches Separate Sizing and Vth Assignment Mixed Integer Non-Linear Programming Sensitivity-based methods (DUET, etc) Continuous formulation [Chen, ASP-DAC ‘05]
Very reliant on discretization heuristic
Proposed Approach – Self-snapping formulationProposed Approach – Self-snapping formulation
Continuous formulation – Use of large variety of algorithms/powerful non-linear optimizers possible
Solution has almost all gates assigned to one of the two available threshold voltages
Small fraction of gates with intermediate Vth’s, can be handled heuristically
Discretization algorithm has negligible power impact and can be very simple
Proposed Approach – Mixed- Vth GatesProposed Approach – Mixed- Vth Gates Consider each gate to be a parallel combination of high and low
Vth gates
RC Delay ModelD R Ceff l
D=R Ceff l
Cl
R Rl h=R W +R Wl h h l
C =C +K (W +W )l Load SL l h
HVt Gate
LVt Gate
Mixed Gate Linear Power Model
W +P Wl l h h=PLVt HVtP=P +P
HVt LVtl h
l h
/ W / Wl h Cl/ W / Wl h
R R=
R +R
Complete Dual- Vth Problem FormulationComplete Dual- Vth Problem Formulation Similar to single-Vth gate sizing problem, with simple gate delays
replaced with High Vth/Low Vth parallel combinations
Minimize
Subject to: , ,, ,W P Wh i h iPl i l ii G
0a Aj
a D aj i i ({1,..., } { })i n inputs
D ai i { }i inputs
{ ( )}j input i
,0 Wl i 1, ..., .i n
,0 Wh i 1, ..., .i n
, ,i UiW Wl i h iL 1, ..., .i n
Proof of Discretized SolutionProof of Discretized Solution
Conceptually separate optimization process into two distinct phases: D-Phase : Fix delays of all gates W-Phase : Find the minimum-power sizing solution
that satisfies the chosen D vector
Hypothetical separation for proof – Not implemented in actual optimization procedure
W-PhaseW-Phase
Proof of discrete optimal solution under arbitrary D-vector sufficient
W-Phase formulation
Minimize
Subject to:
, , , ,P W P Wl i l i h i h ii G
( ( ( )) ( )), , , , ,( ), ,
, , , ,
C W W K W Winp j l j h j SL l i h ij fanout i
i
R Rl i h iR W R Wl i h i h i l i D
0,Wl i
0,Wh i
1, ..., .i n
1, ..., .i n
1, ..., .i n
W-PhaseW-Phase
Linear programming problem
n basic variables, n non-basic variables
Therefore, only n non-zero variables
Every gate snapped to either high-Vth or low-Vth
Addition of upper and lower bounds on total size leads to some non-snapped gates
Number extremely small – simple heuristic achieves good results
Practical Constraint – Fixed-Width Input DriversPractical Constraint – Fixed-Width Input Drivers Sequential elements driving the combinational circuit
Delay of these elements affected by primary input widths
Modeled as fixed-width drivers
Extension of Discretization Analysis Extension of Discretization Analysis m+n constraints in the optimization problem
n+m basic variables, n-m non-basic variables
Therefore, n+m positive variables
Total number of non-snapped gates bounded by number of inputs Once again, small in number; can be handled
heuristically In practice, number of non-snapped gates found to be
much less than the number of inputs
Discretization HeuristicsDiscretization Heuristics
Iterative snapping Round gates to closer Vth and re-optimize until non-
snapped solution achieved
Single-pass Vth assignment Fix all gates to closer Vth and re-optimize only for gate
sizes
Second heuristic faster with negligible power impact
ResultsResults Snapping properties of some circuits
# of non-snapped gates is very small
Dominated by gates at upper and lower size bounds
Approach is easily extendable to multi-Vth AND multi-Lgate
c2670 c3540 c5315 c6288 c7552 i8 i9 i10 --0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.0016
0.0018
0.0020
% o
f tot
al n
on-s
napp
ed g
ates
due
to in
put d
rive
rs
% o
f tot
al n
on-s
napp
ed g
ates
Circuit
ResultsResults Power and runtime comparisons between proposed
approach and sensitivity-based algorithm at 2% timing backoff (results shown for larger circuits only)
Average: 31% leakage reduction vs. previous approaches
Ckt
SBA Continuous Formulation
% Improvement Runtime(s)
Static Dyn. Static Dyn. Total Static Total SBA Cont
C3540 0.26 0.74 0.16 0.78 0.94 38.14 6.46 28 51
C5315 0.22 0.78 0.15 0.80 0.95 30.53 5.11 52 133
C6288 0.35 0.65 0.26 0.65 0.91 24.69 9.04 136 443
C7552 0.31 0.69 0.24 0.68 0.91 23.93 8.87 94 171
i8 0.24 0.76 0.19 0.75 0.94 21.57 5.87 24 35
i9 0.20 0.80 0.16 0.77 0.94 17.65 6.47 9 21
i10 0.31 0.69 0.23 0.69 0.92 24.8 7.69 287 373
TopicsTopics
A new dual-Vth assignment formulation
Dual-Vdd power distribution
Approaches to parametric yield optimization: statistical leakage + delay
Multiple supply designMultiple supply design
Relies on applying a lower supply (VDDL) to gates along non-critical paths thus reducing power while meeting timing
A flexible fine-grained VDD assignment scheme promises best power reduction Gate-level Extended Clustered Voltage Scaling
However, physical design and power delivery are complicated
VDDH
IN
VDDL
VDDL Swing
Need for Level Conversion
DC Current
FF
FF
FF
FF
FF
Implications of using multiple suppliesImplications of using multiple supplies
Coupled
issuesAlgorithmsVDD assignment
CircuitsLevel shifting
Physical designVDD Granularity
Power deliveryDistributionGeneration
Critical Non-critical
CVS ECVS
Fine-grained
Islanding
IN
OUT
Power delivery for dual-VDD circuitsPower delivery for dual-VDD circuits
Power grid integrity vital for circuit performance
Dual-VDD circuits require two supply voltages for operation
Fine-grained dual-VDD can place VDDL/VDDH gates arbitrarily on the die
Implications at the board, package and die level Fixed resources need to be split between VDDL and VDDH
However, load on each supply is lower than on original single supply:
Power supply current demanded by a dual-VDD circuit is significantly lower than the corresponding single-VDD circuit, allowing robust power delivery within available resources (decap, C4, wiring)
Reduced current load on VDDL/VDDHReduced current load on VDDL/VDDH
Gate level comparison Avg. 54% (33%) for VDDL = 0.8V (0.6V)
Circuit level comparison Avg. 49% (51%) and 28% (14%) for VDDH and VDDL for 0.8V (0.6V)
Single VDDVDD VDDH VDDL VDDH VDDL
c880 9.7 5.6 2.2 5.9 1.3c2670 23.6 11.9 6.5 10.1 3.0c5315 36.7 20.9 7.2 20.9 3.6c7552 47.9 13.9 19.4 20.4 8.5
AVERAGE % 100.0 48.5 27.7 50.7 13.5
Dual VDD: VDDL=0.8V Dual VDD: VDDL=0.6V
Low-VTH High-VTH Low-VTH High-VTH Low-VTH High-VTH
INVX10 1.00 0.90 0.57 0.49 0.36 0.27
NAND2X2 1.00 0.85 0.54 0.45 0.34 0.23
NAND3X6 1.00 0.88 0.55 0.47 0.35 0.24
NOR2X1 1.00 0.86 0.52 0.39 0.30 0.19
NOR3X4 1.00 0.85 0.50 0.37 0.29 0.18
AVERAGE 1.00 0.88 0.54 0.44 0.33 0.23
Single-VDD Dual-VDD: VDDL=0.8V Dual-VDD: VDDL=0.6V
VDD
ECVS
Package level resultsPackage level results Two VRMs on board to supply VDDL and VDDH
Ground path can be shared by VDDL and VDDH
Decoupling capacitance divided in the ratio of current loads
Similar power supply noise with same resources as single-VDD case (decoupling capacitance, C4s)
VDDH
Lmb1 Rmb1 Lmb2 Rmb2 Lskt Rskt LpkgHRpkgH
RblkH
LblkH
CblkH
RhfH
LhfH
ChfH
Rpkg_capH
Lpkg_capH
Cpkg_capH
RdieH
CdieH
VDDHLoad
1
2
RblkL
LblkL
CblkL
RhfL
LhfL
ChfL
Rpkg_capL
Lpkg_capL
Cpkg_capL
RdieL
CdieL
VDDLLoad
3
VDDL
+
+
-
-
Lmb1 Rmb1 Lmb2Rmb2Lskt Rskt LpkgLRpkgL
I(VDDH)
I(VDDL)
Intel, “Intel Pentium 4 processor in the 432 pin/Intel 850 Chipset Platform,” 2002.
Dual-VDD physical design alternativesDual-VDD physical design alternatives
Segregated placement constrains placer leading to higher core area and wirelength
Single-VDD Dual-VDD
Dual-VDD segregated Dual-VDD segregated
VDDH + VDDL row
VDDH + VDDL row
VDDH + VDDL row
VDDH + VDDL row
VDDH + VDDL row
VDDH + VDDL row
VDDH + VDDL row
VDDH VDDL GND
Dual-VDD fine-grained
C. Yeh, et al., “Layout techniques supporting the use of dual supply voltages for cell-based designs,” Proc. DAC, 1999.
M. Igarashi, et al., “A low-power design method using multiple supply voltages,” Proc. ISLPED, 1997.
Dual-VDD power grid alternativesDual-VDD power grid alternatives Routing the power supply rails
Dual-VDD Dual-GND requires two separate grounds off-chip and complicates timing analysis and design of the board itself
Multi-rail standard cells can be used to realize the Dual-VDD grids allows placer to operate with no constraints
VDDGND
VDDHGNDH
VDDLGNDL
VDDHVDDL
GND(shared)
Single-VDD Dual-VDD Shared-GND Dual-VDD Dual-GND
VDDHGNDH
VDDLGNDL
4-rail cell3-rail cell
VDDHVDDL
GND(shared)
Dual-VDD standard cells topologies
Dual-VDD on-chip power grid designDual-VDD on-chip power grid design
Guidelines while designing the dual-VDD grid: Scale wires with respect to the single-VDD considering how the
current demand has scaled VDDL gates more sensitive to grid noise important since
ground is shared 120mV noise is 10% for a 1.2V gate, but 20% for a 0.6V gate
Placement of VDDL and VDDH gates assign more wiring resources to VDDL grid in areas where there is more demand for VDDL current
Consider effects that arise from the board and package level such as shared C4s
Fewer C4s leads to higher effective package R, L
Proposed technique D-PlaceProposed technique D-Place Let = I(VDDH)/I(VDD) and = I(VDDL)/I(VDD)
Scale wires as follows
WVDDL
VDDHW
WVDDL
VDDHW
WW
GND
VDDL
VDDH
VDDH VDDL GND
GlobalRegional
Local
global
local
regional
local
global
localglobal
regional
localregionallocal
effective
Area
Area
Area
AreaArea
Area
Area
Area
1
Partition the chip floorplan
Obtain eff. and as follows
Obtain currentconsumption ofSingle/Dual VDD designs (SPICE)
Calculate local,regional, global& effective & for each wiresegment
Size each wiresegment in eachlocal area usingeffective , β &simulate grid
Measure voltagedroop/bounce
Measure wirecongestion
Break down dieinto “local” &“regional” areas
Placementdatabase(Cadence)
Original Single VDD design
SingleVDD Lib file
Obtain DualVDD design
DualVDD Lib file
Peak voltage drop comparisonsPeak voltage drop comparisons
D-Place grids better than single-VDD grids in AVG cases
Inferior by < 2.6% (≈15mV) in some MAX cases
0.6V VDDL as robust as 0.8V
0.6V also provides higher power savings
Proposed approach better by 2-7% (AVG) and 7-12% (MAX) compared to prior approaches
Single VDD DVDG D-Vanilla D-PlaceMAX 16.9% 30.9% 16.4% 18.6%AVG 9.5% 14.7% 9.6% 9.5%MAX 25.6% 35.5% 32.2% 25.5%AVG 15.9% 19.8% 15.2% 14.5%MAX 29.6% 38.2% 37.4% 32.0%AVG 21.6% 23.4% 20.2% 19.8%MAX 26.8% 34.2% 34.5% 29.4%AVG 22.2% 21.0% 21.1% 18.7%
c880
c2670
c5315
c7552
Single VDD DVDG D-Vanilla D-PlaceMAX 16.9% 30.3% 16.3% 19.5%AVG 9.5% 15.9% 9.7% 9.8%MAX 25.6% 36.1% 27.6% 27.0%AVG 15.9% 22.1% 15.8% 15.3%MAX 29.6% 38.1% 33.0% 31.8%AVG 21.6% 25.4% 20.1% 20.3%MAX 26.8% 31.4% 31.6% 28.7%AVG 22.2% 24.9% 22.3% 20.1%
c880
c2670
c5315
c7552
VDDL = 0.6V VDDL = 0.8V
Voltage variation across dieVoltage variation across die
Voltage drop contours
Wiring congestion similar for dual-Vdd vs. single Vdd grids
Lower current demands can lead to smaller amounts of decoupling cap; lower leakage (or use same decap for better performance)
Dual-VDD grid no less robust than single-VDD grid
TopicsTopics
A new dual-Vth assignment formulation
Dual-Vdd power distribution
Approaches to parametric yield optimization: statistical leakage + delay
IntroductionIntroductionOptical Proximity Effects Variation Chemical Mechanical Polishing Variations
Leff
V th
P
Process Parameter-space
Good TimingHigh Leakage
Power Yield Loss
Low Leakage PoorTiming
Timing Yield Loss
Power
Delay
P
Chip Performance-spaceThis Work: Optimize the timing and power yield using gate sizing
PconstTconst
Problem DescriptionProblem Description
Nonlinear Continuous Optimization Objective: Maximize Timing and Power Yield
Yield: A utility function defined w.r.t the JPDF of leakage and timing Decision Variables: Gate Size
Efficient implementation requires Computing yield as function of decision variables - gate size Fast and Accurate Gradient computation
Power and Timing Yield Analysis (see DAC05 for more detail)Power and Timing Yield Analysis (see DAC05 for more detail)
Timing Analysis [Sapatnekar03, Chandu05]
(d, d)
Delay
Log(Leakage)
Power Analysis (l, l)
Correlation (1 parameter)
Delay and Power Bivariate JPDF (d, d, l, l, )
RdXddDelay n
n
iii 1
10
n
iiild
1
RlXllLeakage n
n
iii 1
10log
d
l
Traditional Incremental TimingSize Up 7
Cut Set SSTA: IntuitionCut Set SSTA: Intuition
1
3
4
5
6
7
9
8
10
2
Consider Timing Graph
Unperturbed Sub Graph
Unperturbed
Left Sub Graph
Unperturbed
Right Sub Graph
Arrival Time (AT) Required Arrival Time (RT)
Max Cut Edge Time (CT)
Cut Edge Time(CT)
If Forward SSTA Reverse SSTA then Cut Set SSTA will give exact same sensitivities as naïve approach that recomputes yield relating to all nodes, most being unchanged
Statistical Yield Optimization ResultsStatistical Yield Optimization Results
D < Dμ,initial , P < Pμ,initial
Circuit Yield without L (%) Yield with L (%)
c432 45.4 80.2
c499 39.2 59.0
c880 49.3 83.2
c1908 47.9 82.8
c2670 51.1 85.3
c3540 51.2 87.1
c5315 50.0 87.3
c6288 50.3 86.5
c7552 51.2 80.8
Initial yield ~0-2% due to inverse correlation
Gate sizing alone provides good improvements
Combined with Lgate biasing, provides outstanding results
Chopra, et al., ICCAD05
Another approach to statistical optimizationAnother approach to statistical optimization
General statistical optimization
Method relies on efficient deterministic formulations and variation space sampling to drive statistical optimization
Applicable to many mainstream VLSI design problems: gate sizing, Vth assignment, Leff biasing as well as potential new levers
Statistically Optimized Body Bias Clusteringfor Post-Silicon TuningStatistically Optimized Body Bias Clusteringfor Post-Silicon Tuning
Concept:Speed up critical gates using FBBand slow down non-critical gatesusing RBB to meet timing andpower constraints
Traditional view:Centralized body bias generatorcontrolling different die regions Ineffective for compensating
intra-die variations Highly suboptimal power
Critical Non-critical
BBcontroller
FsbFthth VVV 220
Coarse Body Bias AssignmentCoarse Body Bias Assignment
Simplified assignment minimizing routing overheads
Biasing dictated by placement instead of gate criticality
Disregards complex dependence of gate criticality on: Circuit topology Correlations in process variations
Effective in tightening delay but leads to high power
ONE BIAS FOR ALL GATES
DELAY POWER
Critical
Correlated
Important to cluster gates to leverage ABB effectively
Proposed New Optimization FrameworkProposed New Optimization Framework
4
5
1
2
37
6
Leff_4.1 Scenario ‘1’
Leff_5.1
Leff_1.1
Leff_3.1Leff_2.1
Leff_6.1
Leff_7.1
Generate sample scenarios
4
5
1
2
37
6
Leff_4.2 Scenario ‘2’
Leff_5.2
Leff_1.2
Leff_3.2Leff_2.2
Leff_6.2
Leff_7.24
5
1
2
37
6
Leff_4.x Scenario ‘x’
Leff_5.x
Leff_1.x
Leff_3.xLeff_2.x
Leff_6.x
Leff_7.x
Generate PDFs of optimal actions
Clustering
BB-PDF ρi,jGate
DETERMINISTICALLY optimize each scenario (i.e., tune each gate for each die scenario)
Solve BB assignmentfor each scenario
Post-silicon tuning
Results vs. Traditional Dual-VthResults vs. Traditional Dual-Vth Leakage power
Dual-Vth vs. 2-4 ABB clusters Avg. 28-38% (51-59%) lower μ (95th)
Area Capo generates contiguous regions of similarly clustered cells while
minimally displacing cells 5-8% increase in wirelength and area
Delay 3-9X tighter σ
A few conclusionsA few conclusions
Parametric yield is a critical design objective going forward Requires accurate estimation and fast optimization
approaches to this key metric Envision all tools in 4-6 years being yield-driven, rather
than timing or power alone
Lots of room for improvement in many ‘well-studied’ CAD problems today Recent examples; dual-Vth+sizing, placement (Cong, et
al)