VoronoidiagramsforVLSImanufacturing:
RobustnessandImplementation
EvanthiaPapadopoulou
IBMT.J.WatsonResearchCenter
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20021
TalkOverview
VLSILayout
VLSIData
:simplerVoronoidiagramforVLSIapplications(smalldegree)
CriticalAreaComputationProblem
ImportantprobleminVLSIYieldPrediction
Criticalarea
variationsof
VoDs
Planesweepconstructionof
VoD
Voronoi-basedCADtoolforCriticalAreaExtraction:tobeusedbyIBM
manufacturingin’03
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20022
VLSILayout
Layersofdifferentmaterials
Layer:shapesrealizingdevices(e.g.transistors,interconnect)
Gate:PC(polysilicon)
RX(diffusion)
Interconnect:M1,M2,M3.Contacts:CA,V1
VLSILayout:Compacthierarchicalformfollowinglogical(notphysical)hierarchy
Librarycell:groupofshapes(e.g.inverter).Mayappearinhundredsofplaces
ASICchip:thousandsofinterconnectedcells(Manyrepeatingcells)
HierarchicalLayout:compact,cellrepetitionusingtransistions.Logical,not
physicalhierarchy
FlatLayoutprintedinMask:millionsofshapes.Notavailable
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20023
VLSIData
Vastmajorityofsegmentsareaxisparallel
orientationsverycommon.Otherorientationspossible.Constantorientationscanbe
assumed.
Coordinatesonintegergrid
(verylargeintegers)
Degenerateconfigurationsaretherule
Noperturbationtechniquestoavoiddegeneracies:destroyaxisparallelproperty
Datavolumeofflatdata:orderofmillions(evenbillions)
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20024
VLSIProximityProblems
Width-spacinginteractionsessential.E.gnoise,yield,polishing
DesignRuleCheck:Checkwidth-dependentspacingrules
Lithography:Printingofshapesdependsonwidth/neighborinteractions
YieldPrediction:CriticalAreaestimation
Medialaxis/Voronoidiagram:
Addressrobustnessissue.AdapttoVLSIhierarchy
OurVoronoitoolforCriticalArea
–Robustnessissue:use
metric
–Constructionnothierarchical:Basedonplanesweep.
NeverkeepVoDinmemory:onlyactiveportionnearscanline
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20025
CriticalAreaProblem
VLSIYield:Percentageofmanufacturedchipsthatareworkingoverallchips
manufactured
Defects:Dustparticles,Contaminantsonmaterials
HighYield
Profit
LowYield
Loss
YieldPrediction:ControlCostofManufacturing
ave.#defectsperunitarea,
aclusteringparameter,
thecriticalareaoflayout
CriticalArea:measurereflectingsensitivityoflayouttodefects
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20026
DefectTypes
Twotypesofdefects:
Extramaterial
Shorts
Missingmaterial
Opencircuits
MissingMaterialdefects:Breaks,Via
Blocks
Defectofsize
=circle/squareofradius
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20027
CriticalArea
CriticalArea:
:
areaofcriticalregionfordefectsize
Criticalregion:
locusofpointswhereifadefectofradius
iscenteredcausesacircuitfailure
:
densityfunctionofthedefectsize
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20028
CriticalRegion(
CriticalRegion(
CriticalRegion(
CriticalAreaviaVoronoidiagrams
Shorts
2ndorder
Voronoidiagramofpolys
Papadopoulou&Lee99
Opens
(weightd)
VordgrmofMAsegmts
Papadopoulou01
ViaBlocks
min-Max(Hausdorff)Voronoidgrmofpolys
Combinatorialstructureofindependentinterest
Papadopoulou01
termsderivedfromVoronoiedges
Bound:
metric
squaredefectmodel
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200212
Why
?
Algorithmicdegree
Liotta,Preparata,Tamassia96
Testcomputationseval.multivariatepolynomialsofarithmeticdegree
.Testcomputationsbitprecision:
(input
-bitintegers)
in-circletest(segments):
degree
v
(Burnikel96)
in-circletest(segments):
degree
l
i
l
k
l
query
j
(Papadopoulou&Lee99,01)
VLSIshapes:slopesaresmallconstants
degree1
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200213
Quadrants
p
north,southquadrant:
dist=verticaldist
eastwestquadrant:
horizontaldistance
Non-orthogonalsegments:
Considersix
rays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200214
Bisector
q
p
q
p
q
p
Degeneracy:Collinearaxisparallelpoints
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200215
Bisectors:Straight-linesegments
L
p
L
p
Bisectorofapointandalinein
Consistsof
parts,oneforeachquadrantof
Theunboundedportionsarealways
rays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200216
VoronoiDiagram
Straight-lineskeleton
Ortho45polygons:8orientations:
-slope
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200217
Voronoidiagramofsegments
Straight-lineskeleton
Maintainsproximityinformationin
(goodenoughforpracticalapplications)
VoronoiverticesonRationalcoordinates
(assuminginputonrationalcoordinates)
Ortho-45shapes:integercoordinatesaftermultiplyingby12
In-circletest,degree5
(
:degree40)
VLSIshapes:slopessmallconstants
degree1
-timeplanesweepalgorithm
(basedonwavefrontparadigmofDehneandKlein97andoriginalplanesweepofFortune87
Algorithmicdegree:7
Papadopoulou&Lee99,01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200218
PlaneSweep
L
t
PlaneSweep:
Sweep-line
sweepingfromlefttoright
Papadopoulou&Lee99,01
basedonFortune87,Dehne&Klein97
:
polygons,portionsofpolygonstotheleftof
Maintain:Voronoi
Wavefront:
boundaryofVoronoicellof
SpikeBisectors:
bisectorsincidenttowavefront
Boundary:
treatedasspikebisector
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200219
Data-Structures
scanline face
PlanarSubdivisionDataStructure(PSD):Half-edgedatastructure
(const.degree)
Wavefront:Red-Blacktree
Parameterizespikeendpoints
ascenterofimpliedsquare.Key:
.TiesresolvedbyPSDorder
Eventlist:Priorityqueue
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200220
Events
SiteEvents:endpointsofsegments
Priority:
SpikeEvents:intersectionsofneighboringspikebisectors
Priority:rightsideofimpliedsquare
HandlingofEvents:
PSDOperations:CreateFace,SplitFace,SplitEdge,ContractEdge
AssignmentofVoronoi/GeometricData:slope,owner,spikeequation
constantorientations
lookuptables
Wavefrontsearchesby
-degrays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200221
ExampleEventHandling
b
a
Searchwavefrontfor
,
FixPSDverticesbetween
,
using
UpdatePSD:createfournewfaces
AssignVoronoidata:slopes,owners,newspikeequations
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200222
ExampleofSiteEvent
F1
F3
F2
F2
F3
F1
UpdatePSD
AssignVoronoidata:newslope,spikeequation
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200223
VoronoiCriticalAreaTool
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200224
SomeExperimentalResults
Machine:RS6000/PPCbi-processormachine,processorspeed200MHz,memory
8GBs(single-processorrun)
Window:
micronsabout
chip
ScanLineProcessing
Level
M1M2
Time(min:sec)
Peakmemory(MBs)
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200225
SomeExperimentalResults(continued)
FirstOrderVoronoi(nofacedeletions)
Level
M1M2
Time(min:sec)
Peakmemory(MBs)
N.VoronoiFaces
TotalN.Edges
ShortsVoronoi
Level
M1M2
Time(min:sec)
Peakmemory(MBs)
N.2ndOrderVoronoiFaces
N.Polygons
TotalN.Edges
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200226
Voronoidiagramforshorts
2ndorderVoronoidiagramofpolygons:everyregionhasauniqueownerwhichis
responsibleforshortswithintheregion
,
2ndorderVoronoicell
(Papadopoulou&Lee99,01)
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200227
CriticalAreaIntegral
k
r
k
r
j
r
e
j
r
e
lengthofverticalside,
maxcriticalradius,
mincriticalradius
Addupformulas
internalterms
,
cancelout
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200228
CriticalArea
SummationofVoronoiedges
+
r
i
r
j
k
r
e
-
Criticalareawithin
:
ExactCriticalAreaintegrationinoneLayoutpass
Papadopoulou&Lee99,01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200229
VoronoidiagramforOpens
Papadopoulou01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200230
VoronoidiagramforVia-Blocks
Min-MaxVDofrectangles
Papadopoulou01
Voronoidiagramunder
VoronoidiagramofpolygonsunderHausdorffdistance
Papadopoulou&Lee02
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200231
Conclusion
CriticalAreaExtractionTool
–Basedon
Voronoidiagramofsegments
–PlaneSweep:unfoldslayouthierarchyonfly
–DataVolumeofChip:breakintosmallwindows
Advantageof
metricforVLSIapplications
Current/FutureWork
–Usestatisticsonmanysmallwindows
–TakeadvantageofLayoutrepetition
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200232
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