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Domain

Node Element

TimeStep LinearSystem

Pidaperti and Hudl

Archer

Miller and Rucki

Analysis

Map

Domain

ConstraintHandler ReorderHandler

MatrixHandler

EigenSolution

Wilson NewmarkJacobi Subspace

DirectIntegration

DynamicAnalysis

AlgorithmicAgent

Algorithm

AlgorithmManager

DOF

Partition

PartitionManager

Element

GeneralElement

Zimmermann and co-workers

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GaussElimination

GaussSeidel

Container

Algorithm

Newton Newmark

CentralDifference

Algorithm

Jacobi ConjugateGradient

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EigenvalueAlgo EigenvalueIntegrator

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NewtonRaphsonModalLinearModalDOF_Group

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SupstructureType

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Domain

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Node

Domain

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SymmPosDef_SOE

Band_SPD_SOE

LinearSOESubdomain

FETIAnalysis

AnalysisModelConstraintHandler DOF_Numberer

Domain

PartitionedDomain DomainPartitioner GraphPartitioner

DomainDecompAnalysis

DomainDecompAlgo

DomainSolver

DOF_Group FE_Element

SolverBand_SPD_SOE

Band_SPD_SOESubstrSolver

SolverSubstructuring IterativeSubString

Solver

LinearSOESolver

Analysis

IncrementalIntegrator

Metis

Element

SubstructuringAnalysis

IterativeSubStringAnalysis

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ActorSubdomain ProcessActorSubdomain ProcessMain Actor Process

formTangent()

FE_Element ShadowSubdomainIncrementalIntegrator

formTang()

getTang()

formTang()

getTang()

getID()

getTang()

formTang()formTang()

getTang()

formTang()

getID()

formTang()

process busy in object

ActorSubdomain ActorSubdomain

getTang()

getTang()

process idle in object waiting for result to be processed remotely

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ActorSubdomain Process

formTangent()

Main Actor Process ActorSubdomain Process

FE_Element ShadowSubdomainIncrementalIntegrator

formTang() formTang()

getID()

formTang()getTang()

getTang()

ActorSubdomain ActorSubdomain

getTang()

getTang()

getID()

getTang()

getTang()

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'+_ $tz ¥ Y¨ `Cv Y ¨ ¨K D ' ` v ` >|Y`Cv v ` D

t ¥ `Iz uy Y ` Y a '8` m z`v R ¥ ` ` w 3 Y' z ¥ Y v |¥ Fv' zD`lv L ¥ IY ` z vTv` . Y a t> ¥ D Cv D `#|a 0 k|D C£)r D ` m uy m m !#"%$'&)(*+ , .-/0/1 32*4526 879"%$'&W :-7;926 8"<$ a & Y a = + >?' ;"<$ = A@CBMB@=GDE9MK:GFIH@JH ¥LK a `,£) ' D D ` 8M ` D SION QP Y` A ¥ R V S

5vT N ^ 5 E F OT < @DN OT# S9 j nVU S f S W < i t < Ig f S W < i t < Ig XZY\[Z]#]XZY\[J T x u uIX HG[ uIX Hmu H HDX|ZI HDX|Z M H uIX M [J T x J J Xou M J X HH H uIX uIX Du H uIX J J T x M M X|Z M M X M H M X J M XouO H uIXou JJ T x mXIH mX J H M Xb M M X[IH H uIXou JJ T x YX J mXb[ H mX[ mX H uIXou JJ T x YX HG YX|ZI H YX J YX HG[ H uIXou MM T x M bYX Hmu [YX|ZI H :2@ < @ g n 7 _^ OS @3U OTM T x uuIXou M uHDX|ZZ H :2@ < @ g n 7 _^ OS @3U OTM T x u J Xb[ u J XDu H :2@ < @ g n 7 _^ OS @3U OTM T x u mXbIH u mX M [ H :2@ < @ g n 7 _^ OS @3U OT

` ¥ ¨ a µ Y cb M ` NYª3SP ed

` Of c 9M ¥ ` e ¥LK ¤ D m ` ` `LM'` ¥ ` ` ZM ` ` ¢¥ ;M ` ¥LK D` m ` ` D`ef Ea ` M D ¤a K 9M) ',` ` ZM`Y ` & ,` £ M'` ¨ © ` V`w M §¥ , I D U0 `ef 9M` ¥ ¥LK Y` m µ

$ ` ¥LK D` m D Kq D` D ` K` ` ZM ` `

W U u[

5vT N ^ 5 E F OT < @DN OT# S9 j n U S f S W < i t < Ig f S W < i t < Ig XZY\[Z]#]XZY\[J T x u J Xou uIXb H uIX|Z uIX H uIX JMJ T x J mXou M X[ H M X|ZH M X H uIX HG[J T x M JM Xou YXb b[ YXb YX H uIX H ZJ T x GZX J [YX JJ H [YX|Z [YX H H uIX HGbJ T x [DuIX uHDXou M ZZ uHDX H bYX M H uIX HGbJ T x uHGYX H uuIX [ JJ H u M X H uHDX|Z IH uIX H ZM T x M [ M X J uZX H JM :2@ < @ g n 7 _^ OS @3U OTM T x u M HDX H J HDX|Z uHmu M :2@ < @ g n 7 _^ OS @3U OTM T x u ZX J uIX[ uu M :2@ < @ g n 7 _^ OS @3U OTM T x uZ M X J YX|Z u M ZI[ M :2@ < @ g n 7 _^ OS @3U OT

` ¥ ¨ S µ > D cbr M SIeN

5vT N ^ 5 E F OT < @YN OTa S9 j nVU S f S W < i t < Ig f S W < i t < Ig XZY\[Z]#]XZY\[J T x u J J H J H J H H uIX J HJ T x J [GZ M u GZ H uIXou JJ T x M M J u M u #u u JM uuO[ H uIXouuJ T x IH uIH [bIHG[ uOIH u J H uIXou JJ T x ZI[ uO[[ u M [[b uO[ uGZ H uIXou MJ T x [b J uO u[ M Z M bDu uOb M bb uIXouuM T x M [GZI M #u u M uO :W@ < @ g n 7 _^ OS @3U OTM T x uuuOb #uOb uZIb[ M :W@ < @ g n 7 _^ OS @3U OTM T x 5 @~ P n @ :W@ < @ g n 7 _^ OS @3U OTM T x 5 @~ P n @ :W@ < @ g n 7 _^ OS @3U OT

` ¥ ¨ ¨ µ > D cb M' P

W U ub

3dF63dF53dF3 3dF4

set up C++ C++C++form equations

form equations C solve equations C

% C

PU

2dF1 2dF2 2dF3 2dF4 2dF5 2dF6

Example

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solve equations

100

¡z M` ¨ © µ `,£) ef ` De, ` ` SeN

W U uZH ` M D ¤a 9M) ¡)DU0' 8& ¥LK D m ` ` M Y © > |ef Y ` #` ZM)Y D ' ` ` Y` ` D ` ` ZM`Y `

© ` ` Y` ¥ D` ` a Y , ¥LK ¤ D m ` ZM`Y Y ` ZM`Y ` K _¡)DrU0' 8& `I . ` ZM`Y ` D K ¨ C; ZM'` Y 2 Y V ' ©¡' ©¡ ¨ & ` ¥LK Y` m ` 9M = © ¨ ¨ Y V ` Z a

` Z Y% ¥LK ¤ D m ` w ZM ` ` ` Y D = ¥d & K ` &/ ¥LK D` m K ` ` MY ` w |¡Y.U0' 8& ` a = c . ¥LK D` m ` / 832 "%$2 = ?L .:B8;NO8LECB ¥LK ` . ` ;M /0\+ 8 832 9 '2 "<$ ¥ = ` E B8RCIMB4RC ¥9K & / M ;M c?' 8/1 9"<$ 26 9"<$ ¥ . = ` 3L 3: 8;NO8LECB. ¥9K eI26 Z"%$W = ; 2*452 ."%$ L 3: 8;NO8LECB ` .:B8;NO8LECB ¥LK ` M = 2 /1#"%$ )8;NO8LECB 47698A ¥LK 3 ` ZM ` & ` ` ZM Y Yr # ¡)D ¥ K 9= D = D m Y M & ` ` = Y' Y # Y M ¥LK D` m = mK ` ZM`Yr = ` # ¥ K

R 3 ` ¥LK D` m ` Y D ad a ¥ = & ` = ¥LK ` a a ¦¥ R ZM M ¥= ` ¡)DU0' 2*452 ."%$r = I .:B8;NO8LECB ¥LK Y D a ' 3: 8;NO8LECB M ¥8 '2*452 "<$ ` a , = C

` A ¥9K ¤ D # ` D D ` ' ZM`Y ` ;¡)DrU0' 8&= ` ZM'Y ` ` Y , Y M ++ 8"<$ D, ` U0'

W U uZYu3 V` ¥9K ¤ D # ` ` ©¡'`9M` 2 I = S5& ¥LK ` a6© ¡ ¨ = Y©$©8& ¥LK

` LZ Y ¥LK Y` m K w LM' . = r a Y A' M ¥LK M V ¥LK D` m ` ` K & M y D `. ¥' ` M K¡)D20' ;&a ZM'Y ` ` .Y D K ¥ ,Y D D' Y ` # ) R 3 ` ¥9K ¤ D # ` Y` D \+ 32 ¥LK 3 Y DI 3: 8;NO8LECB, ¥LK D Y D Y' Y/ ¥LK m Y 3: 8;NO8LECB& M

m MV` ' P2 M5BC@ ¥LK

` V M' D ` K D SIeN P ` 2 ¥LK D` m ` w `9M` z D` D & z ¥ ` R ;M ¥ ¨ S ¨ ¨ ` I D 9M m ` ¥LK D` m ` I ' . ` ZM ¥ ¥ Y ` ¥ K ¥LK Y` m ` D £0 M # q Y 3 & ;M M' &' Y D ` ` ` Y & ` . IY D ` Y Y' ¥ ` `Y` M §¥ ` ' D Y D `9M m , ¥LK ¤ D m ` µ

$ ` M A L 3: 8;NO8LECBA ¥9K ` `9M` $25MBC @ ¥LK

¥LK ¥ I DI L 3: 8;NO8LECB ¥LK 3 Y & ` I CIM ¥LK & = ` @JEA8M C@ ¥9K ` D ¥ M ` ' ' = Z &=`9M` 8C3B8 ¥LK Y¥ ` D ` ` 8C3B8 `LM'` ¥LK Y`A Y K 'E ' D ` Y D 9M A ` ZM`Y

© PM ` Y M §¥ , Y I ¥LK & ` K' % ¥ I D ¥LK D` m ` w2` A D ` MY K

R PM Y M §¥ ` Y ¥ . = `

W U uZ J = V ¥LK & ` `9M` #` ` ? Y` I Dq ¥LK D m ` A D ZM`Y

other

2dF1 2dF2 2dF4 2dF5 2dF6 3dF3 3dF4 3dF5 3dF62dF3

10

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100

% P

age

Faul

ts

form equations solve equations

Example

¡z M ¨ µ q C£) > ¡;M % De, SIeN

' M & M §¥ ;M ` 9M Y ¥ £ M'` ¨ a & 8M q ` ` ` Y = Y = ¤a 9M) ` D `%ZM' ` L` ` w ` m `D = = MYY'` ` rY I = p ¥9K ` & ` M D>) Y Z 6¡YzU0' ;& W L '` 'Of m /0 32 "<$y = ¨ '` ;M M `C ` ¥ ;M & = ;M M &Y = R M' a ¥LK ` Y` cY> = = MYY'` , K 2` ` ` `M

W U uZ M M §¥ L ;M' & Y M ¥ %` ` = ` z D # s M d M &"$R< \ ` = h M ¥ ` ¥ Ya L m MM D = D &mY Y D , ¥LK ` $ ¥LK M ¥pA` ¥LK ¡)D%0 ;& ` = M)YY 3:B8;NO8LECB% ¥LK Y ` &#Y Y ` *25MBC @ & 8G;B4 C% ¥LK & M' ¥ % m MM Y = M)Y Y L C

!#(- -% 9% &% % %3/ ##%3# + -%3/

!#"%$'&)(*,+.-/$)01(2"3 ` y ,` D L ` ¥LK D` m ` ` U0 M

Y`Y = Y M) ¥ ` R K ` ` qU0 M ` W ¥LK D` m ` U0 M ` M Y , ` Y ` Y M Y Z D SIeN P m ¥ ¨ R ` D ' W` ` = M) ¥ M K IY D Y a ` M §¥ ;M ` ME ` KY` ` ` IY r ef K D = ¥ R ` ` M ` , IY . M % ` D ,Y Y` ` r M ef D ` ZM µ

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!('*),+ %#

` ef 2 M D M ¥ ;M ;M > ;M `M `Y) D . ` m Y D = % M' ¥ M D = ` .9M) & M`ZM ` ¥ c M §¥ ;M' ` = = I M` ` m Y D = ;M ¥ w ef Y ` U0' Y V & ad w 9M) V = . = d m Y ` ¥ `,£) &m 2` D ef 2 Y a `z` ` % D `Y ¥ ;M >a = IK

W U uZ ` D d a Y`Y ` M ¥d M M` Y

a M K ` 2 Y Y M M = C ` Y¥ e aK ` M ¥ Y LM m VLM Y D`M C£) d D` ` C ` w m ` ¥ , D = ` ` ` A , ` ` ¥ µ!"!$# %'&)(*,+.-/!"!"0 V)W_&EX?>Y:EXd5BmM2<B345G*):)%Q<5B2mKX?:)Q<@,2$%'&)(*EX WV!"!"8 ( *^3'*):,%'z:,Q<@)2$%'&)( *^X?7!"!N =?2$:E3¤R*^3[5Bvqr:)%Q<5B2mz:,Q<@)2$%'&)( *^Xd7!"!"T =( * *):,%'z:,Q<@)2$%'&)( *^X?7!"!"U!"!"f V)WG=B5\&E3a5345Gt&B3( 3( 2$*5?@ @,2$%'&)(*G&,*E@ %'2,@,5HQ,:<( H @)5B6WV!"!"j FG5c3( X34E5svBD&,t)4 % &"B3¤( 3¤( 2$*E5BL7!$#\! 12$%'&)(* % &B3( 3( 2$*E5B345 % &B3( 3( 2$*5BL+M345sBD&,t)4 % &B3( 3( 2$*5B-7!$#"# % &"B3¤( 3¤( 2$*E5?@162$%'&)(*Y34E5"162$% &<( *)+M34E5 % &B3¤( 3¤( 2$*E5B-7!$#\0!$#\8 V)WG=B5\&E3a5345 X?:)Q<@)2$% &)(*^X&,*@ &@)@ 3a2G34E5@,2$%'&)(*!$#cN u;Qa5?=c3?IKB2$>¤5BK34E5^u Q a5?=c3cIKBD2$>¤5BL7!$#\T xhHSt)4&$F_&E=4)(*5<IKB2$>¤5B345<F_&E=4)(*5<IKB2$>¤5BL7!$#\U mM2<Br+[(*E3(G#7(Y*):,%'z:,Q<@)2$%'&)( *^X?7R(e-/!$#\f ghn % z2C=>¤5c334E5nK4&,*)*5HS7!$#\j z4&@)2z:)Q<@,2$%'&)(*e345\z:)Q<@,2$%'&)(*,+M345nl4&)*,*E5H345^u Qa5?=c3?IKB2$>¤5B-7!$#\k glBD&,*^X[mM2<BP%'&E3¤( 2$*34E5n2$*^XZ3BD&)(*E3o&,*E@wH 5BL7!"0"! n6F 34E5svBD&,t)4,qr:,%eQ<5B5BL7!"0$# 1vuA qv:,%eQ<5B5B345"1vu;Aqr:)%eQ<5BD5BL+M345sBD&,t)4Cqr:)%eQ<5BD5B-7!"0"0 x*&,H `EXc( X^FG2,@,5H"345<FG2,@,5HS7!"0"8 % B2yH 5\z % 1zwu z:,Q^XZ3B z2$H ~5B34E5\zp2$H ~5BP7!"0N % B2yH 5\z % 1zwu 34E5",( *zwu+M34E5\zp2$H ~5B-7!"0"T z$3Z&E3¤( ="J*^3[5c]BD&E3a2)B34E5"J*^3[5c]BD&E3[2<BL7!"0"U 162$% &)(*,15?=?2$%etExhH ]<2_34E5\z2$H*ExH ]<2$7!"0"f 162$% &)(*,15?=?2$%etExh*&,H `Xc( X 345?x*&,H `EXc( X?+M34E5\z:)Q<@)2$% &)(*)$34E5n2$*^XZ3BD&)(*E3o&,*E@wH 5BL!"0"j 345"1vu;Aqr:)%eQ<5BD5BLp34E5<FG2C@)5HS34E5\zp2$HS*ExhH ]<2$p34E5"J*E3a5c]BD&E3[2<BL345",(*zu;p-7!"0"k 34E5"162$% &)(*,O&@,@,z:,Q<@)2$%'&)( *)+M34E5\z:,Q<@)2$%'&)( *-7!"8"! !"8$#!"8"0 V)WG=B5\&E3a5& %'2,@,5HwQ,:<(H @)5B&,*@ Q):)(H @34E5_%'2,@,5HpWV!"8"8 9;:<( =>?0@ACBD&,%534E5<FG2C@)5HJIK:<( H @,5BL+M345"162$%'&)( *-7!"8N 34E5<FG2C@)5HJIK:<( H @,5BPOJQ):)(H @RFG2,@,5HS+.-7!"8"T!"8"U V)WG=B5\&E3a5&e3(%'5B&,*E@ XZ3[&B3( 3BL:,*)*)(*^] WV!"8"f gh(%'5Bi34E5?g( %5BL7!"8"j 34E5?gh(%'5BPO XZ3[&"B3+.-7!"8"k!N)! V)W|t&B3¤( 3¤( 2$*e34E5;@)2$% &)(* (*^3[2G34E5 X?:)Q<@,2$%'&)(*EX WV!Np# 34E5"12$%'&)(*COJt&B3( 3( 2$*,+Z*):,%'z:,Q<@)2$%'&)( *^X-7!N)0!N)8 V)WG=B5\&E3a5345&)*&,H `EXc( X'WV!N"N glBD&,*EX[m2)BL%'&E3( 2$*Y345n2$*EXZ3BD&)( *^3coK&,*@pH 5BP7

W U uZI!N)T n6F 345sBD&,t)4Cqr:)%eQ<5BD5BL7!N)U 1ru;A qr:)%Q<5B5Bi34E5"1vuAwqr:)%Q<5B5BL+M34E5sB&)t,4,qr:)%Q<5B5B-7!N)f xh*&,H `Xc( X^FG2C@)5H<34E5<FG2C@)5H7!N)j 1(B5?=c3 % BD2yH 5\z % 1zwu;|34E5\z2$H ~5BL7!N)k % B2yH 5\z % 1zu;|345",(*zu;R+M345\z2$H ~5B-7!"T"! z3[&E3( ="J*^3[5c]BD&E3[2<Bi34E5"J*E3a5c]B&^3[2<BP7!"T$# ,( *E5\&B345\z2$HS*xH ]<2$7!"T"0 z3[&E3( =?x*&)H `EXc( Xv34E5?x*&,H `EXc( X?+M345"162$%'&)( *) 34E5n2$*^XZ3BD&)(*^3co&,*E@pH 5BL!"T"8 34E5"1vuAwqr:)%Q<5B5BLp34E5<FG2C@)5H345\z2$HS*xH ]<2$w345"J*^3[5c]BD&E3a2)BLw34E5",(*zwu;p-7!"TN!"T"T V)W|t<5BmM2<BL%345&,*&,H `Xc( X'WV!"T"U 34E5"12$%'&)(*CO Xd5c32<&@n&EXd5+d#^-7!"T"f 34E5?xh*&,H `Xc( X\O&,*&,H `\57!"T"j!"T"k V)WYXZ3a2$t345;3(%5Bl&)*@ tBM( *^3 2$:^3 345GB5cX?:)H 3.X WV!"T"k 34E5?gh(%'5BPOJt&,:^Xd5+.-7!"U"! 34E5?gh(%'5BPOJtBM(*E3+L=?2$:E3Z-7!"U$#

Z '+_ $` ` M ` D ' ¥ Y ` .`

¥ ¨ ¨ R ` ZM Y Y` ` M Y w ` Y £ M ¢¨ a ¨ S ` £ M ` ` KmY §¥ Y ' w M ¥ Y ' & ¥ ` = 0 M ` w L ` M = ; 2 (*2 "%$r D G;B4RC@7>)6A4 NOM @0Ee ¥9K ` &/ ` = z` m L Y ¥ K ` ¥ M ;M R & K M M 3 ¥ ¨ `M Y J© ` D ` M = ' m w Y Y Y Y a 3 Y V ¥ Y ` = ' ' D ¥ M ` ef 9M Y = m ` ¥ 9M` ¥A ' = ZMY M ¥ Y ` ` ZM Y a

` M ` ' D µ$ Y`Y = ` ' # Y ¥ w ¥ V ¥

Y V R M ` ` D ;& ¥ 8M % D ¡)DU0' ;& ` ¥ w Y¡)S Y¡ ¨ & ¥ , Y V VP D & ¥ M 2 V ¥ . Y V M M §¥ 2 y` Y`Y.

© SY ` ¥ wA b¥ V d Y`Y. = ` ' # `

WU

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W U uZI[

2dF2 2dF3 2dF4 2dF5 2dF62dF1

1

2

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5

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7

3dF3 3dF4

10

9

8

Alg

orith

mic

Spe

edup

(A

S )

3dF63dF5

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

1

2

3

Spee

dup

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

4

5

6

7

8

1

2

3

Fact

or U

nbal

ance

in S

ubdo

mai

ns

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

20

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% A

naly

sis

Tim

e

Page Faults CPU Communication

NP=3 NP=4 NP=5 NP=7

Example

Example

Example

Example

¡> M ¨ µ zY`Y > D , SIeN

W U uZIb

2dF2 2dF3 2dF4 2dF5 2dF62dF1

1

2

3

4

5

6

7

3dF3 3dF4

10

9

8

Alg

orith

mic

Spe

edup

(A

S )

3dF63dF5

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

1

2

3

Spee

dup

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

4

5

6

7

8

1

2

3

Fact

or U

nbal

ance

in S

ubdo

mai

ns

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

20

40

50

60

80

90

70

30

10

% A

naly

sis

Tim

e

Page Faults CPU Communication

NP=7NP=5NP=4NP=3

Example

Example

Example

Example

¡z M` ¨ S µ >Y > D , P

W U uO[IH 9M m Y# = m ` ' W = IY R ¥ `9M e ` Yq Y a & M m & = ;& Y = S Y Y a h¡)D2U0' DY80 ¥ M ©¡' A SION M d !&Y D` ZM ¥' Y K

` ` Y `I D Y A I . D K` .`ZM M |¥ ;M ` M c Y`Yq = m ` ' ` ¥ ` w` Y D` w ZM w ¥ D` w` ZM ;M M &a ' ` K £ M ¨ a ¨ S ` £ M Y . ' ` ` L` 8 `ZM . M §¥ ;M Y `I D ` SIeN ` ` P , ` ZM' ` A` ` D ` § `I Y SIeN ` cP ,

R ` ` ` ` Y ` , w M D £ m M ' , D . YY C % ` ` ` Y K M `ZM & ` M ¥' Y Y `I C ` d ¥` %` ` ZM = YM 2 D P Y` I ` Y SIeN

` D K' Y = Y` ` ` Y Y Y'K` `Y`, M §¥ `I D ` µ r D ` ` Y Y` Y' & ` . 9M Y V ¥

I &q ;M m R D, = S Y `9M Y ` M D U0' ` Y Y ZM ¥ ` VU0 ` w`LM'` Y V m§ ` VU0 ` `9M` d M r , ` ` ` = A , I = Z ` = ` ¥ Y`Y Y ? ;M ¥ M &C zY1 q Y.&"$ &z D` # ` ` r M ¥ , = K` = IY ` K` Y ¥ ` = D M e C D£ ¥p Y ad & D0 YD m Y a ` ` D r Y

W U uO[Du K 8& ` R M D = % V9M) Y M `M ` £ M` ¨ ¨ & ` ` AY ' M M = Y M e ©S Y` Y ` SIeN ` m K ` .` M % 2 M = M ¥' Y Y`.Y M ¥ ` ¥' Y D V ;& m £ M'` ¨

¥ Y` ` ` Y Y & ` V .` m ` ¥ ` ` Y m ¥ `LM'` Y' Yef ^a . D = Y ` M ;M ` = ` ¥ ` V m ¥ M ¥ D = Y.

D` D Y cY' & ` ;M m K M `LM'` Y Y a `I & w` £ M'` ¨ e ¨ S ¡)D Y ` ¥ w w M Z Y # & M ¥ ¥V` w M & Y%U0' w M M #`K Dw = ¨ ` m Y V ©¡' ` 0 M ¥ ' SION ,

M D` Z ` D z ` z ` ¥ mY §¥ Y ` ` D 3 Y ` ` D Y £ = ` ` D D ` M ¥ Y Y M ' ` M ¥ R = Y & ` ¥ 2589B@0HW ¥LK &Y ` M |¥ ` R ` M Y 9M) M §¥ ` ` ¤ ' & ` ¥ 2 ¥9K ` ' W q # M m2` & M R ` M ¥d M M` ` & M m D w ¥ . M ¥ Y r M §¥ m` M ¥ Y & ¥ M Y ` w` ¥ Z m `Y. 0 D ' d ` ` wY D` ¥ 2 ¥LK ' m M m, & M R M ¥ M M & w ` ¤ Y ¥ M # Z `M &) # D U0 `Y!& M §¥ D`

W U uO[ J

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4 3dF5 3dF6

1

2

3

Spee

dup

Example

NP=3 NP=4 NP=5 NP=7

2

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8

10

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16

12

20

22

24

26

28

2dF2 2dF3 2dF4 2dF5 2dF62dF1 3dF3 3dF4

Alg

orith

mic

Spe

edup

(A

S )

3dF63dF5

Example

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* ~¦ 9:* $#¨V©È¨ *-.'A &3& m¨ v¨ 'hÑ34+¨ .'$C&39:,P4>*-.C23&J,E. Ó 2;4:0F9 = *-7?&3A ; 0F.'2 ¦ 9:9?&3.G4= &3254 9?,E&;.Y4:&3A *=. #¦ * &37 . LD*,,B$#:<"H76bnNp7pLrq c -¨ÑC<Ñ.Ñ@ : ,; *=( 9?,EA & .', § &39?7:,/4 ¡ 9?&37?7 ¨

*-DE& 1¨|¨ *=.'A 9?,E7?$'.N*-. A¨ 'hÑ"C4+¨ ;6ÎÓ ? Ò Ó 0F9^4>* DE& ; 0F.'2 ¦ 9 9:&;.G4 E &32;4 9:,/&3.G46&;A A¡ 7?46&;( *-7?&3A 0F. ;Q 0 3 %+YLox 0 j jz;:*,, 0FD ¨ 3 ' 0 ¨Ñ<4+ÒĨ ½Ñ.Ñ<3½¨

*-(+*-D ¨ *-.CA Ó A'&3D/, v¨ 'YÑ3<4+ ¨ HÓI¦ 460F(+*=4:,E2 *-9?4:,/46,/0F.',/. 0 ~ : 96*-(R& A 469 ¦ 2 4 ¦ 9:&37 ~ 0F9 ; 0F.'2 ¦ 9:9:&;.G4 9?0W2;&37?7:,E. Q 3 :*D*,+A :<" i:<uD:<5|n"H#:<"8,):<"H 0FD ¨ : ' 0 ¨ @,+ Ò <¨ >D. 3&C<¨

%:dR2G 7+, *-9 ¡ ,/7 <©È¨ *-.CA E¦ (*=9 ¨ 'hÑ : +¨N 9 r AÒ .'7?4:9 ¦ 254 ¦ 9?&3A © 96* $ *-9^46,P46,E0-.',E. *-.CA

A½ *=9:7:& *t469:, 9?A'&39?,E. BA¡ 7^46&3( &;9:7?,E0F. ¨A<< #"8"9CS'6) \Ay6) 8l(*,2lI2 #"$#:( lKHu:<"M<J9 o5?N-@ g 9A8@, J9"$#gH#:($2l92l ¨

*-9 ¡ ,/7 L©¨ *-.'A E¦ (*=9 ¨ 'hÑ : +¨ ¿¦ DP46,/DE& § &3D #« * ¡ *-9^46,/4:,E0F.C,E. A 2>$C&3(R& ~ 0F9 9:9?& #¦ D\*-9 © 96* $C7 ^ A9)jz#lyvWBfv .', § , § &;9:7:,P4 ¡ 0 ~/ ,/.'.'&;7:0-46* # & 46*-9?4:(+&;.Y40 ~ ; 0F( ¦ 46&;9 A 23,/&3.'2;& ,/.'.'&<* 0FD/,E7 ±¨

&39?.', $'*-. =I¨ *-.'A ,/. A¨ 'hÑ-Ð <4+¨HÓ .1& 8 &;2;4:, § &[$C& ¦ 9?,E7?4:,E2 9:0Z23&3A ¦ 9:& ~ 0F9 *-9?4:,/4:,E0F.',/. 9:* $'7 Q _J9 5<58%'CD ,*,J"9:( #5 k9)" #5 ¨ <Ñ). C3<3<¨

,E9 *=469?,E2 A¨(#© &;D\*=4?4 ;)¨ #¨ *-.'A &;232>$', ¨ ¨ 'hÑ73&C4+ ¨V = 46,/(+, *=4:,E0F. -¡ A ,E( ¦ D\*=4:&3AÓ .C.'&<*-D/,E. <Q %_*:"8*, " <<ÒĨ >-нÑ). >33<<¨

E¦ (+*-9 A¨ *=.'A Ó A'&3D/, ±¨'hÑ : +¨E # ,E7^469?, ¦ 4:&3A : ,/.',/4:& 9 D/&3(R&3.G4 Ó .'*-D ¡ 7:,/70F. &;4 « 0F9 0 ~ 0F9 7?46*=46,/0F.'7 ( D/&3(R&3.G4>*=4:,E0F.*-.'A Ó)< DE,/2<*=4:,E0F. k9" #5V6%'(9*(+O #58n"9WH#:<"8,):<"H + 0FD ¨Ñ Ñ4' 0 ¨Ñ<4+ Ò ¨Ñ@ : >D.Ñ@ > ¨

E¦ (+*-9 ¨© 96*-(+* Ó¨©L¦< 4>* ½Ó¨ *-.'A *-9 ¡ ,E7 ©¨( &;A',/4:0F9:7 'hÑ @,+¨ Y"9(2$#'*(:" ALb #O #5<5?5o+(:<"H1 = ,:IH#"h #"$ 0 " #5KCD:?Z6 0 5H):< JE = &;. *-(R,E. ;¦ (+(R,E. 7g¦ D/,E7:$C,E. 2; 0F( *-. ¡8 ? &;A « 0Z0ZA ;Ó¨

*-, C©È¨ *=.'A ; $'&3. ±¨'hÑ +¨M *=96*-D/DE&3D/, *=46,/0F.z0 ~6 ,E.C&<*-9 : ,E.',P46& 9 D/&3(R&3.G4 Ó .N*=D ¡ 7:,/7 . © 0Z0ZA'.'0 =)¨y<¨ *-.CA 9:, $G4 y<¨ ?±¨ &3AC,/460-9:7 +(:<"H|:<" i:<uD:<5 n"H#:<"8,):<"H`n:IHJ J "[6["8*,NJ95I$E:<"M*," )+"8*(:" P:< J 0 9ny %'CDAE|q #e#`V=! #5<5I ,` ,`k+"8&s[W <¨ > :3: . >3> ½Ó@A";69O¨

* «L¨v¨ *=.'A *-2 * ¡8 #¨?L¨*'hÑ7&C4+ ¨ HÓ *-9:*-DED/&3D ? 0 «¥ 9:,/&3.G46&;A A½ *-9:7?& A 0-D ¦ 4:,E0F. &546$'0ZA ~ 0F9 : ,/.',/4:& 9 DE&;(+&;.Y4 A 4:9 ¦ 254 ¦ 9:*-D Ó .N*-D ¡ 7:,E7 Q Y"9A)" #;:" #5 k')" #56[j+S):*B #5 3 J92$>:<" n"H#:<"8,):<"H + 0-D ¨ C3>"' 0 ¨Ñ?Ð+ Ò ¨ 3 : . <ѽ¨

* «L_¨ ±¨ 'YÑ33>4+ ¨ hÓ *=96*-D/DE&3D : ,/.',/4:& 9 D/&3(R&3.G4 A 0-D ¦ 4:,E0F. &;4:$'0ZA Q Z #"$%'(9*(+, - 0FD ¨ C ' 0 ¨ >+ÒĨ 3;@ : . 3 : 3<¨

&3& <¨'Ȩ *=.'A © *-.'.C0F. #¨'YÑ<Ñ + ¨z = &3254 9?,E&3.G4:&3A *-9:*-DED/&3D 9?0 96*=(+(R,E. 9 ½ &39 ,E(R&3.G4:7[*-.'A ? &37 ¦ D/4:7 . ; 0 &3D\*=.'A Óv¨( &;A',/4:0F9 %'/9*,;:<"HVq a`j!u adcdWAee#`0 5I@2 ' 9,`Pj! 3 d:*, ¨ Ð"C/. 3 999 ; 0F( ¦ 46&;9 A 0Z23,E&54 ¡ 9:&;7:7 ¨

, ¨ *-.CA I¦ A'* ¨ 'hÑ34+¨S &;(+0F9 ¡ ; 0-$'&39?&3.'2;&q,E. A $N*-9:&;A ,/9?4 ¦ *-D &3(R0F9 ¡ * 2>$',/.'&37 Q ]0 3 gB #"g2 *(:"gE"]+A7%'C/AE g 0FD ¨Ð"' 0 ¨ @,+ Ò <¨ C Ñ.C : <¨

, ¨ A¨ 'hÑ3>4+¨&,A *-9:7?& © * ¦ 7:7?,\*-. 9 DE,E(R,E.'*=46,/0F.10F. , $ &;9 ~ 0F9?(*-.C23& ; 0F( ¦ 4:&39?7 $ # 46$'&;7:,E7 .', § &39?7:,P4 ¡ 0 ~ ; *-DE, ~ 0F9:.',E**=4 = &;9 &3DE& ¡# A & 46&;( &;9 ¨

,E. :¨ ;)¨ v¨ *=.'A &;DEDE&;9 ?L¨=¨/'YÑ3-Ð+¨ r%$'& © 9:*-A',E&;.G4 0ZA'&3D 0G*-A = *=D *-.'2;,E. &;46$C0WA Y,nonon -O #"g2 *;:"g " %_6) # n"H#:<"8,):<"H M 0FD ¨>A 9ÑC ' 0 ¨Ñ +ÒĨ C . C3<¨

%:dR2G 7+R+C¦ <¨( $',P46& #¨*ƨ *-.'A ; $'&3. ƨ :¨'hÑ7&C4+ ¨ZHÓI D ¡ ,/. E &32;4 9:,/&3.G46&;A # &37?, .460 : ,/.',/4:& 9 D/&3(R&3.G4 9:0 9:*-(R(+,/. . LD*,,B$#:<"H 6r<J9Sa 0 3 '%+Y 0 LoL%'C9D:<+" 0 5K:B$M;:<"H`VY"$#:( #" 95K:?,` Y"$#:( #" <<¨ @ @. @4 ½Ó$;¨

C¦><¨ $',P46& #¨ ƨ( ; $C&3. ƨ :¨ *-.CA #=¦ .C7:(R0F9:& :v¨ 9O¨'hÑ : +¨ hÓ *=469?, ; D\*=7:7 , 96*-9 ¡ ,/. ; ~ 0-9 A 4:9 ¦ 254 ¦ 9:*-D 9 . ,/.'&3&;9:,/. ; 0F( ¦ 4:,E. > #"$%'(9*(+, - 0FD ¨ :3: ' 0 ¨Ñ+ÒĨ : .Ñ#Ñ#Ñ#¨

*-2 &39?DE& Z<¨ 'hÑ73>4+ ¨ ( DE&3(R&3.G4:,E. : ,E.',P46& 9 DE&3(R&3.G4 &;46$C0WA'7J0-. A½¦ &39:2;0F( ¦ 4 &39?7 0-9 7?46*=46,/0F.'7*=.'A *; 7 Ò·Ó = , DE,E0 96* $ ¡8Q n"H#:<"8,):<"H #(:"g 0FD ¨ÑC ' 0 ¨Ñ +ÒĨ C&CD. 3 : ¨

*-2 ,/& ?L¨&<¨ 'YÑ + ¨ = &3254 9?,E&;.Y4:&3A 9?0 96*=(+(R,E. 0 ~ 46$'& : ,E.',P46& 9 DE&;(+&;.Y4 &546$'0ZA Y)"9)" #(:" #5Mk9+)" #5Z6[4jmS):*B #5 3 <J92$ :<" n"H#:<"8,):<"H 0FD ¨*C : ' 0 ¨ +ÒĨ @ ;: . @C3>½¨

*-2 ,/& ?±¨ ¨ 'hÑ : +¨ E &32;4 9:,/&3.G46&;A &;4:$'0WAC7 .: ,E.',P46& 9 D/&3(R&3.G4 9:0 9:*-(R(+,/. *-.CA9 . ,/.'&3&;9:,E. A 0 ~ 4 « *-9:& # &37:, . . *-$'D ¨-<¨ *-.CA &;9:.'&;9 ±¨( &3A',P460F9?7 +(W:<"HM:<" i:<uD:<5 #"$ :<5I$#:<"H n"H#:<"8,):<"H1SLD*,,B$#:<"H 6& J9\%': d<JhY)"9)" #(:" #5"[6["8*,"h;:<"HE:<"Mi:<uD:<5 #"$ :<5I$#:<"H>n"H#:<"8,:<"H` 5K:<"g`)\ #"9CD`k+5KCSa#eW a a <¨CÑC&C/.ÑC3<½Ó¨Ó¨ = *-D &3(+* = 9?9:0 - &;DEA g r < : <&C3><¨

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&;.'&;4:9:& ¡8:¨ *-.'A ~ ,E(R(+&;9:(+*-.'. r ¨:'hÑ"C4+¨ = &3254 9?,E&;.Y4:&3A 0-. ,E.C&<*-9 : ,E.',P46&9 DE&;(+&;.Y4 Ó .N*-D ¡ 7?,E7 ÒNÓ) D/,E23*=46,/0F.460 D\*-7^46,E2;,/4 ¡8Q +A #"$h%'()'*(,, 0FD ¨ @4 ' 0 ¨ : +ÒĨÐ;>#ÐD.<Ð#Ð#Ш

,/DEDE&;9 ©È¨6?±¨ *=.'A ?Φ 2 , 4¨ #¨ 'YÑ&C+¨ HÓ 9:0 96*-( Ó 9?2>$',/4:&3254 ¦ 9?& ~ 0-9 .G46&39:*-2;4:, § & 0-.'DE,/.'&<*-9 #)¡ .N*-(R,E2 Ó .N*-D ¡ 7:,E7a0 ~ A 4:9 ¦ 254 ¦ 9?&37 . ; 0F$'. ¨$:¨ &3AC,/460-9 (:<"H:<"i:<uD:<5y #"$ :<5I$#:<"Hrn"H#:<"8,):<"H1 L,D*,,B$#:<"HE6> J97To: 6)<JSY"9A)" #;:" #5i"9W6["8*, YNN no` 0 " J9:< ` 0 `k+"8Ss[W ¨ :" /. : C;><Ó@A ;69V & « U0F9 Ñ<;<<Ñ?н¨

,/DEDE&;9 ©¨E?L¨ 'hÑ7<Ñ + ¨ HÓ . = &;2;4 9:,E&;.G46&3A ÓI 9:0F*-2>$H4:0 A 469 ¦ 2;4 ¦ 96*-D Ó .N*-D ¡ 7?,E7`*-.CA# &37?, . Q E #"$\%'()'*(+, 9 0FD ¨ @ < ' 0 ¨Ñ +ÒĨР: . 3 ¨

0ZAN* 8A¨(A 0-46&;DE,/.'0 9O¨#¨ *=.'A 7:,E&;$ 8A¨ ±¨"'YÑ-Ð+¨VHÓ *-9:*-DED/&3D *t469:, ; D\*=7:7 , 96*-9 ¡,E. ; ~ 0F9 ; 0F( <¦ 4>*=4:,E0F.'*-D &;2>$N*-.',/237 ÓI DE,/2<*=4:,E0F.C7 3 :*D*,!:<" i:<uD:<5n"H#:<"8,):<"H + 0FD ¨Ñ ' 0 ¨Ñ+ÒĨ 3&CD. <¨

¿¦ ¦ .CAN* i©È¨6?±¨ A 0-4:&3D/,E.'0 9O¨ #¨ *-.'A 7?,E&;$ A¨ ±¨ 'YÑ3>4+ ¨XHÓ . E &32;4 9:,/&3.G46&;A: ,/.',/4:& 9 DE&;(+&;.Y4 Ó .N*-D ¡ 7:,E7 : 9:*-(R& « 0F9 A9)jz#l!nW% _=>W vW ; , § ,ED 9 . , .'&;&39:,/. 2g¦ 9:A ¦ & .', § &39:7?,/4 ¡8* &;7?4 * ~ * ¡ &;4?46& ±¨

%:dR2G 7+ & § ,E. ±¨( &3A',P460F9 'hÑ3>4+¨ _J9rL <6[\ #"8*, 6Zx 0m3 vD`Afh #"$ 3 LPY F al(f#l<a#e]"G

D #(:"i5K A ?$gÒ $Y4?4 Ò «O«O«L¨ 0F7?2 ¨ &3A ¦ *=( =DE*-( =D\*-( >3< &;9 ~Y¨ $G46(RD ¨ , g¨ ¨( )¦;)¨/ƨ( *=.'A © &;.'A'9?&<* ¦ r ¨=)¨ 'hÑ3 : +¨HÓ # ,E7?4:9:, ¦ 46&;A # 9:* ~ 46,/. ¿Ó D

0F9?,/4:$'( ~ 0F9 0G*-A = *-D\*-.C23,E. Q Y,nonon -O #"g2 *(:"gh" %_6) #|n"H#:<"8,):<"H 0FD ¨ A 9]Ñ8Ñ,' 0 ¨Ñ<4+ Ò <¨CÑ#Ñ : CD.Ñ#Ñ><Ñ8¨

,/230FD #¨ ¨ *=.'A ? & ¡ .'0FDEAC7 9 ¨8¨ :¨ 'YÑ3<+¨ = 46,/(*-D #)¡ .N*=(+,/2 ? &3(+* ½ ,/. 0 ~(# *=46* *=96*-D/DE&3D ; 0F( ¦ 46*=46,/0F.'7 Y,nonon gO #"g2 *;:"g " 0FD ¨*C ' 0 ¨ +ÒĨ <;>/. ѽ¨

0 ¦ 9 (R,EA 6=)¨ *-.'A *-9 iȨ ;)¨ 'hÑ3#Ð+¨,A 0FD § ,E. A 469 ¦ 2;4 ¦ 96*=D &;2>$N*-.',/237 9:0 DE&;(+70F.4:$'& ; *=D/46&;2>$ Ρ &39:2 ¦ & *-2>$',/.'& 3 <J92$7:<" 0 5K:B$ 3 ,*,Jg #"9:*, #"$rn"H#:<"8,):<"H + 0FD ¨ ><Ñ,' 0 ¨ +ÒĨCÑ>½Ñ.Ñ?Ð;>½¨

9?4:, 8.¨ *-.'A 0 ¦ 9 (R,EA =I¨ 'YÑ33>+¨ .'2;0F.'A',P46,E0-.N*-DED ¡ A 4>* D/& ; 0-.'2 ¦ 9?9:&;.Y4 9:0Z23& A ¦ 9?&37 ~ 0F9vr9:*-.'7?,E&3.G4 : ,E.',P46& 9 D/&3(R&3.G4 Ó .N*-D ¡ 7?,E7 3 <J92$S:<" 0 5K:B$3 ,*,Jg #"9:*,! #"$rn"H#:<"8,):<"H + 0FD ¨ : 3 ' 0 ¨ +ÒĨCÑ : Ñ).Ñ?Ð @¨

7?4:&39:(+*-.'. 7ƨ(¦ .'A'&;9:DE,/2>$ 7ƨ *-.'A ; 96*-(R&39 v¨*'YÑ : + ¨M E &32;4 9:,/&3.G46&;Ar@0Z0FDE7~ 0F9s46$'& # & § &3DE0 (+&;.G4_0 ~ 7:&;9 .G46&;9 ~ *-2;&37 ~ 0F9 .G46&39:*-2;4:, § &¥r&3*-2>$ « *=9:& . *-$'D !¨d<¨*-.'A &;9:.'&;9 4±¨( &3A',P460F9?7 +(:<"H7:<"i:<uD:<5g #"$ :<5I$#:<"Hn"H#:<"8,):<"H1LD*,,B$#W:<"H 67<J9>%': d<J&Y"9A)" #;:" #5P"[6["8*,!" +(:<"H&:<" i:<uD:<5 #"$ :<5I$#:<"Hn"H#:<"8,):<"H` )5K:<"g`m)\ #"9C/`k+5KC|a#eW a|a ¨ÑC/.Ñ@ : Ó¨Ó¨ = *-D &3(+* = 9?9:0 - &;DEA - r < : <"C3><¨

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,EAN* *-9?4:, ?L¨ 7¨\¨ *=.'A I¦ A'D ¿Ó¨Z¨ 'hÑ7&C4+ ¨ #)¡ .N*-(R,E2 Ó .'*-D ¡ 7:,/7 0 ~ A 469 ¦ 2 4 ¦ 9:&37 ¦ 7?,E. = &;2;4 9:,E&;.G46&3A r@&32>$'.', Ϧ &37 Q A, #"$ %'()'*(,, 0FD ¨ @4 ' 0 ¨Ñ +ÒĨCÑ@/.Ñ : >½¨

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¦ ,E.C. &7¨<¨( &3A',P460F9 'YÑ;@,+ ¨ Lb #B #5<5?5;:<"H1 _J9,Cm #"$LO *(:*, & 2 © 96* «¥ ,EDED .'2 ¨

%:dR2G 7+ ? * $N*-&;D =)¨ *-.'A 9?,E7?$'.N*-(R0W0-9?46$ ¡86;)¨ A¨.'hÑ7&C4+ ¨ HÓI¦ 460-(*=4:,E2 : ,/.',/4:& 9 DE&3(R&3.G4 # &

§ &;DE0 (+&;.Y4 ¦ 7:,E. = &;2;4 9:,/&3.G46&;Adr@&32>$C.', Ϧ &;7 n"H#:<"8,):<"H #(:"g 0FD ¨Ñ< ' 0 ¨*C4+ÒĨ >#ÐD. Ð3½¨

? ,E$N*-2 & ;)¨ *-.CA 9?0 D/,E. %=)¨4'hÑ"C4+¨ = &;2;4 9:,E&;.G46&3A : ,/.',/4:& 9 DE&;(+&;.Y4 0WAC&3DE,/. .; 0F$'. g¨ :¨ &3A',P460F9 ;:<"HS:<"i:<uD:<5V #"$ :<5I$#:<"H\n"H#:<"8,):<"H1,D*,,B$#:<"H6 J996) JrY"9A" #(:" #5V"[6["8*, ^YNN no` 0 " J9:< ` 0 `k+"8rs[W ½¨: @ : . :3:& ½Ó@A";69O & « U0F9 Ñ<3<<Ñн¨

? 0G* ¨ 0 *-9:*=46$'*-. 7¨ *-.'A ? *-(+*-. !v¨N¨'hÑ7;@,+ ¨ ¦ D/4:, ~ 9:0-.Y46*-D = *=7:&3A Ó)½ 9?0G*-2>$ ~ 0F9 ; 0F.C2 ¦ 9?9:&3.G4 : ,/.',/4:& 9 DE&;(+&;.G4 Ó .N*-D ¡ 7:,E7 Q A, #"$^%'()'*(,, 0FD ¨ :& ' 0 ¨ @,+ÒĨ 3;@<Ñ. 3;@ >½¨

? 0F7:7 r ¨d<¨" * .'&;9 4g¨2?±¨ *=.'A C¦ &;9 ©¨":i¨ 'YÑ +¨ = &3254 9?,E&3.G4:&3A 9:0 96*-(R(+,/. ~ 0F9 A 23,/&3.G46, 2 ; 0WA'&;7 ¨0 Ò*9 *=( D/&37+,E. ; k')" #5m6 +(:<"HU:<"i:<uD:<5n"H#:<"8,):<"H + 0FD ¨ > ' 0 ¨ @,+ Ò <¨ @4-ÐD. : Ñ@¨

? 0-46$ &39 < 9O¨ *-.'A ©L¦ 4>* Ó¨'YÑ&C4+ ¨ HÓ . 9 XR23,E&;.G4 = D/0W2 # 9?,E&;.Y4:&3A Ó) 9?0G*-2>$4:0 *-9 *-D/DE&3D A *-9:7?& ; $'0-DE&37 -¡ : *-25460F9?, *=46,/0F. . &;9:.'&;9 ¨ &;A',/4:0F9 %'D9,*,(:<"HVq w`jzu\a#W a`iL (5I #"$` pbH" ¨ : <&C/. : Ñ 999 ; 0F( ¦ 4:&39 A 0Z23,/&;4 ¡ 9:&;7:7 ¨

? 0-46$ &39 <69O¨ *-.'A A 2>$'9:&;, &39 "?±¨$'YÑ;@4+¨ ( 9:0 § &;A 0G*-A # ,E7^469?, ¦ 4:,E0F.,/. *-9:*-DED/&3DA½ *=9:7:& ; $'0FD/&37 ¡ : *-25460F9?, *t46,E0-. Q . &39?.'&39 =)¨ &3A',P460F9 %'/9*,;:<"HVq g` jzua9W adcw`ag`| ,J:<"H#A"g`i=!lol ¨Ð3&C/.<Ð 999 ; 0F( ¦ 4:&39 A 0Z23,/&;4 ¡ 9:&;7:7 ¨

?V¦ 2 , ¨ #¨ *=.'A ,ED/DE&;9 ]©È¨ ?±¨ 'hÑ73>4+ ¨ HÓ . Ó D 0F9:,P46$'(R,E2 : 96*=(+& « 0F9 ¤~ 0F9 : D/& , DE&: ,/.',/4:& 9 D/&3(R&3.G4 9 DE&;(+&;.G4 = *-7:&;A A 469 ¦ 2;4 ¦ 96*=D 0ZA'&3D/,E. <Q 3 <J92$:<"0 5K:B$ 3 ,*,Jg #"9:*,z #"$rn"H#:<"8,:<"H + 0FD ¨ÑC3> ' 0 ¨ C? @,+ Ò ¨ C3>&C/.C3;@¨

?V¦ 2 , ¨#¨ 'hÑ3>4+¨bHÓ . Ó D 0F9?,/46$C(+,/2 : 96*-(R& « 0F9 ~ 0F9 : DE& , DE& : ,E.',P46& 9 DE&;(+&;.G4 0WA &3D/DE,/. / $ # 4:$'&37?,E7 .', § &39?7:,/4 ¡ 0 ~. *-7:$C,E. 4:0F. * ¡8¨

?V¦ ( * ¦< $ N<¨"= DE*-$N* ¨ 9:&3(R&39?$N*-.', 4ƨ 9 A'A ¡8 :¨( *-.'A 0-9:&3.C7:&3. ƨ 'YÑ<Ñ+¨p @[,*(W)pb:"9AB$ 3 2$5K:<"H #"$ = ,:IH#" 9:&;.G46,E2;& *-D/D .9 . DE& « 0Z0WA ; DE, 8 7 ! & « &39?7:& ¡&<-Ð;>&C ¨

A *F* ¨-©¨ *-.CA ? *=0 w¨0=)¨"'hÑ<Ñ +¨ ; 0F( ,E.N*t460F9?,\*-D = 4:,E(R, *=46,/0F. ¡ A 4:0W2>$N*=7?46,/2 9§ 0 D ¦ 46,E0-. Q Y,nonon -O #"g2 *;:"g\"A 0 :($B$R= ,:IH#" 6zY"9AH#O #B$ i:<*+:<< #"$\%'CDE - 0-D ¨Ñ< ' 0 ¨ @,+ Ò ¨ :&3: . : C : ¨

A *-.G4:,\* 0 69O¨*#¨ *=.'A * «L ¨0±¨ 'YÑ3>+¨ HÓ . ( DE&;(+&;.G4>*=4:,E0F. 0 ~ : ,E.',P46& 9 DE&;(+&;.Y4 &546$'0ZA0-. # ,/7?469?, ¦ 46&3A 0F9 7^4>*=4:,E0F.C7 . ; $C&3. .:¨ ¨( &3AC,/460-9 0 " #5KC/:?\ #"$ #(:"1 ,D*,,B$#:<"Hh6 J9M 5 6)<J4*,"[6["8*, J95I$ :<"*," )"8*(:" P:<<J%'(9*(+,S"H#,,Y 8`J:*B DHD`Y5<5K:<"8:?,` 0 ):<5Va#W adc ¨CÑ33/.Ñ7<<Ó A";69O¨

%:dR2G 7+ A * ¦ 7:& ?L¨ *-.'A A 0F. +<¨ 'YÑ;@,+ ¨b E &32;4 9:,/&3.G46&;A A 469 ¦ 2;4 ¦ 96*-D Ó .'*-D ¡ 7:,/7 « ,/4:$ A½¦< 7?4:9 ¦ 2 4 ¦ 9:&37 Q . $'0 &3,/(+&;$ +Ȩ &;A',/4:0F9 ;:<"HZ:<"]i:<uD:<59n"H#:<"8,:<"H1LD*,,B$#:<"H6r<J9ZTo:<!"[6["8*, J95I$|:<"X" )+"8*(:" P:<<J 0 9ny %'CDE q g` | ,JW:<"H#A"g`i=!lolI`Pk"8zegfWAee#`a ¨CÑ : C/.Ñ>;<<<Ó A";69O¨

A 2>$'D/0W& &3D =Ó¨> *-9 ¡ ,/7 L©¨ *-.'A E¦ (+*-9 ¨ 'YÑ-Ð+ ¨ Lb # 3 (:?21 Lb #O #5<5?5x:(@2B #)C6[ V"g()'*(,B$ 3 ,,J9, DLb #)(:<;:"9:<"HM #"$%dg #D 3 #;): pbO$):<"H ?.gÒ$G4:4 Ò «O«O«¥¦ 7?&39?7 ¨ 237 ¨Ä¦ (R. ¨ &3A ¦ *-9 ¡ ,E7 t4>*-D 7 *=9:(R&;46,/7 =,/.'A'& ¨ $G46( ¨

A 2>$'0-D 8 A¨ ¨ 'hÑ +¨ 9 DE&;(+&;.Y4:7%0 ~ *-. = &3254 9?,E&3.G4:&3A : 9 9:0 96*-( ,E. ; ! #"$&%'()'*(+, - 0FD ¨ @C ' 0 ¨*C4+ÒĨ : Ñ?ÐD. :& <¨

A $N*=9:(+* A¨VȨ *=.'A = * ¦ $ 9 ¨(<¨Æ¨0'hÑ +¨U CÓ! 0 A 469 ¦ 2;4 ¦ 96*=D Ó .N*-D ¡ 7?,E7O0F.* &54 « 0-9 Q . © 0W0ZA'.'0 =I¨o<¨ *-.'A 9:, $Y4 <¨E?±¨ &3A',P460F9?7 +(:<"H :<"4i:<uD:<5n"H#:<"8,):<"H`n:IHJ J "[6["8*,!J95I$ :<"G*," )+"8*(:" P:< J 0 9ny %'CDEUq #e#`=! #5<5I ,` ,`Pk"8&s[W ¨ >&C/. >;@ >½<Ó@A ;69V¨

A ,/(+0F. ±¨ #¨'hÑ<Ñ +¨ *-9^46,P46,E0-.',E. 0 ~ .C7?469 ¦ 2;4 ¦ 9:&;A 9:0 DE&3(R7 ~ 0F9 *-9:*-DED/&3D 9?0 DE&;(+7 Q ;:<"H\%'CDAEz:<"n"H#:<"8,:<"H 0FD ¨ ' 0 ¨ C+ÒĨCÑC : .Ñ@43<¨

A 4:& § &;.'7 ƨ2?±¨ &;A',/4:0F9 'YÑ3<+¨ y"9: >jzL,BH#O #ZZ:<"H 9?&3.G46,/23& *-DED . DE& « 0Z0ZA; DE, 8 7 & « &;9:7:& ¡#¨

A 4:9:0 ¦ 7^469 ¦ =)¨'YÑ<Ñ + ¨ _J9\ LBH#O #ZZ:<"Hrx_ #"H#- DH <Ó A'A',/7:0F. &37:D/& ¡8¨A½¦ .CA'&39:*-( -¨ A¨(© &3,/7?4 ©È¨Óv¨( *-.'A # 0F. *-9:9:* '<¨ <¨'hÑ;@,+¨! r%$C& i ; 0F.'2 ¦ 9:9?&3.G4

; 0F( ¦ 4:,E. A¡ 7?4:&3( Ò69§ 0-D ¦ 4:,E0F. "9 ½ &39?,E&3.C23&37S*-.'A¿r9?&3.'A'7 Q Lb #O #5<5?5o+(:<"H 0FD ¨ < ' 0 ¨ @,+ÒĨ : C½Ñ. : @ >½¨

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%:dR2G 7+

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