Spectral Bloom Filters - Stanford CS Theory

19 downloads 197 Views 275KB Size Report
School of Computer Science. Tel Aviv University [email protected]. Yossi Matias. *. School of Computer Science. Tel Av
Spectral Bloom Filters Saar Cohen

Yossi Matias

School of Computer Science Tel Aviv University

School of Computer Science Tel Aviv University

[email protected]

[email protected]

-012B7;+ C 324  %*+ 56 #DFEG H   H7824  $)*9* $AI# !:0* 0;"#-)J $  %&NEG]*'\0* ' Š9d- 9*%* $%?/ $)*78+ 5+X(( 7; #%1AI 

AJ %;5+0*$@;8 A+ • J:&0*=> ˆ RZJ+%;3)‹J9* A $5+ 9Š65+0;(+7*%&/0*(+o3% A $O+IRAIg+% K)k+ %Š Mˆ6 K9%‹N9;9; + 5+ $%;+W)*(-D  A#0*Z78%;T -+39; #< $+%Y ,R )*%;)*(*+q?7; K+ %”R15 => %K324)*

%* J(5*  R  =>CZ0; %;5K(9+ # %q+ j)*%\(IJ-3/B0; %Q0*)*(D $#-)2If%&%;  -%6)‹+I 780;K #0; K-3).I'24+ AM89* $9*%Š4?);9*( $Z(/+(9&-)JIP +%MMV -'+(9Jd* K-+a& %I(DW+[\ %i9*   +[\9;-C24]1H0*Mp #+H KC-Io™\7;+(49;C-++(9;5

%*5j A9; 9*1 # 801+ $- )#, %* $%#5X9* ( #*Z ;= + 8  D 2Bu+›3vw4R $24 # A*9kq #=+%- $3)J $ +%q%k9*+'M]*9;: 0;' $)J+5 2\ 7 A9;* † -a&-IK$--)k9*™\j+(+9;MoO#2Bœj3 K5  t /š& R

 #EO;%@ #%!-)KG %Q '$/3I1/ C #=  RY + %@% "#MH]* :0;% $')VD 5 + A9* ž X%;-)1-) S# R\/+(%;7;)Š 9d9*69*+'0*%;-q3@/C-%*+&()*9‡' -9;D‡+$)*EG%Ÿ %;5Kt u+u3#vwR4+P&Iq/+0*MH 0*-q:0*-+/--)) 7%;  \%& '-+S;%&A0;I?-%?)k3H6($$Iq+78 ¡+MT- ).   #A24 ; A¢?'9k N9;)*  A Ž   =>78 A-H)V  $Dd%P (+8 žY  # C1 Aq -=

ADjX #EG%i+% #t u\ƒ*vW9-9*=>X-)?)*78 ( ,#%Q /Rc*(+0;8 9*  01j-)QX9;2B3+=>+%dC< A j*9*%;X  #+  K, 5 -+ A)   78 ,O+2\Rl Q%Q$-2 =>MnR  Q'9*9* )*)*(((R+ %;).9*5j %.+);o)1(9;&/#RF+$);+Mo3/(N%/ #)c-)Dc;m  %DK9;9*EO%%   AG9*I6 X+4+N87*0;X 2\A!# $%q9;+-7*2^ $ AGI9;R*0*X%)19; J]*\9*j g+10*3=1 # $)1*j  %K9; j93Mn0*\%;9;j +);%+(A 

 --=& $%*5X9;4 o)1 %;5X24 %)*-2‹ $H-=  o7;ZMp+\)*   %cR>249; Z+A Mn -%k24\t $u+ %*%˜35i)1v!D =1_w+ $)1*0*+;-a1  $ K #3DT3`ZZ #%#- N378 $ \9;=>  @+ +%;%6)@ NMY0;9*A F ;2 + AOL#I@:;0;8-3 $ / -I1+%;[\+A9;Iq;  $% YN2489* + $( #9j11  RB%; H%g]1\ # r%+MTMp') (  % ,J+_ …'?"`N  #Rc 9;RT24]*9;   * +_ '?&` (#,J-_ Y`N7* AJ-+%‡7867;0; A# . '0/ x 1 24365  75 89 DN ;+'%Id5

=>%@:  *3=& o)1B :% ,J_/ƒ-` $ #C+%q/  K? = R 7S;+% A)  GI.+ %;%N_ˆ5Pa*/+8Mj + K%; $ % #\ $+%AIQ+243  $= 9@ D 78[\ $6,9;-6)QUV]* $ # j=>Mp+A+Ik%kX7* 78%*0;J A K %* RV%& 0;(V # )*%;-%)@(Z+ $A% +AI@%;,J)?Mn_/+ƒ3)*`j/  a1 8- +%* # Rc -%*)P+5d%;+)Q); #-(+&%  D mk]*\j i1/0;%40* a1 8D    #%\/ %;5J=+ +0; F0*+**  +(9;+%;)k+5++ A9; ##0*-)  mk.9*-2†O2\‹+5  A9; #?MnP]*\ b K $%%;+%*k%) $++L&010;(+Y R-I24Df9* [\9j9*qS;0*/Al-7R 24B A9X %;  K0;7*+/4(+EG%%; $+-AIX

R\ # *J3 => #-)9QMp+0;$%;#† $+%;Rc $y %*z A{( oy |{-I‡}}}  {y8~J. K˜&+Dg*;[\ %;9*5 L H $I1 %  7889*#+ Mn0* A+5

$+9?q%;…8DF"@+mk AjR8 $=  A6MT++Y9*9*+ $ C( L '* ),+-9;I1+K4./Mo7; A   e [9*'\9*'* 

 7;%@+7*+ $MF AOIq9;'MnX3(+Mˆ # N8Z+ 0 €@+  B{ =>xV+DZ%)q4Mp)* C%*Z+ Z9;X)*' A4 8%;7%;2I )*14 % 3Z7;I*+7;J;  A#OI6( Mn+0;%;C$)*3N+C ¡7;7* 4 P  ‘Z˜9* Q 4 Za*( +%I‡ A+9*IPMj _ J ! `R  % K9*X249*  9Q %* ‘/+( +X% AC 9;O4jDj 4 *+9;C+ 5 4O9*(A/$IK' 1 UVR8 'Mp+ _/X"ƒ =>7 ja;u+` #=>/^;%K _wMn˜&R+} 419* ƒ€ X  Mˆ/P+` O Nd8DH0; [\  A%;9&5 0;=> R 9*%- -R;++%dMp0;]1%;\ i $+%%$7; AGI#5N 7&I  $%*%*ƒC55J9;%q9;Z' A  0*%  !D %;" 4 ( +Z L Mo^ %_* ’` Mnb{0C9*Z X%*^-c-]1` \b  . }%;} 56B}{ +!lZ (R*^9;%d ` = #b -%)6DTAIEO %6 #)1* $IJ)*- %;7&/I5  24 j %-=+)1+3 24$+B7;q _w%+3C $ + M8X(=>>  R]*D\ DNW%*[\-9;F %Bj78:*0;\+ 0*8%;IZ A YO-Id)'-: +0; #%@ A;78-A)I  0*; Mn0*4+MnMWB9*j3 $ + %;

%c%;D 5N'9*  oB3Mn  %60* $%ZMW=> 9;(']1(\+ k7*$3= =>%< %*5 )1 / $7*0*-)6 a101 $+%Q+M /

%*C3X*%-)6 %d]1 $+% &D ›&D

Distributed processing

 N]&+0;%”=>A (78 ;i -  #;*; +$ %d- #3 %& :&%;0* A) RFZ0*"#9*9. +]1%N\A ‡I7 KIŸ 78/ + 0* $%;)*A $%*_ˆA I* -6 +%*l]15 $%? ]* 9*+ %.Dj AC &[\D;›>9*`  R ( #AIK   0*+ %;*).$ -9)9.5 Mn0*%;%;  (3 +C%;Dq%Q[\]19;\K @+0;1%'%=> %;5X+9; /+K %q %*M  %;NAI? $ ##1 K3=>-$)@cMp]*(\ g  ;%@(+MT #9;' +# _ˆ5+ $+%; #+W%&0; N)d78   -9;%Q $9;+%'% =>%Š %&+24BO2\+< #+j  0;% (`D

[l+7;Šƒ@9;-24Q a18   #%(+40*q249;%0* $%*5‡iS;A  24=+ A 9f+0;x  / T &= { #$0*/6+ƒ-%;˜+˜ )˜&R\V 1S8++%%g)*);I‡(.]124\ A r9g ,L 6 2 +MZ˜1€ D &a D /ŽH€\+0*7+u& R 9*9*- 0*24'%k -%;5X] #T _%*RlJ $0* N_ 1 0;0 H@ _H>J$`_`@ +R; 9;``£TRTdMp(;( (14 + 3 #%#RcMp(MVZ /Mp +  KV9* sZ   0 5  ++$+V AN9*  ’%¢?)QH2\$I1+L& $ †%*H5J7£WINK o)1+%;#%01_nMn A+Mp(I1+ %; ,!5X8N8 +9* A  Q=>%*  D )*(+ƒWl(`Z AC %;)* N0; AjC+9;%d %* ,IQ   )*M)1+Zj Mp+ ' # AZ24 #9;Z (249d9*+ *9@8-+*X8 -%Qj9* $%  9* A9* ); $( ¢#7I+9;C A #D A9; e 24+ %)   9    %

AB+ \L10*9;q+%;Mp+)* -!l)6R + \ %# $B9;(9;Z +L> 5 )# %c+)* 5   #9* $ Q%*  %* 5d%*5E9*!#q24 9*%* %K-0;+6 %;5 8Œ (0*  % %*5E B A $/%*K  )10; - q  AJ% o 0;%I#

 A  (24+†R TŠB= EH80; Mp+D6 # ;4 78a* +0* #; X Rc  #24 A\9k78 1 A

5 % A 9; # žB jMT%* +'+(/+ ll8 #  Mp0*+%& #+-MW)6 #78  # +BIJ9Mp+4%-9; 9qB? #] +9;&5 )V R 0;9; A 9;%* †Œ5NB†_ˆ9;7*C+0*5+++B # AE\C9;+3 ‘ 0*8 Mp+ #4 A`D . + 0/  21 3!4%," [\9* Œ B 5 + A9*  Y9*T # /c #* aj # 9*&)cR Mn789;6 -\ 9;0; + #B*8M24%;9;F A\Mn9g+9;B%;+++9*78%* $ w%*015C4 01u%˜HCQ’ $%*O5P2MYX #]19* \%*Z $ - N0;R> Q7*0*R&D‡W24 ((I?D 5 [\d 9;%Q  B?9* # ]q*Œ  # Ba1 G9;I‡&5 +)q+Mj  AB9*9*d Q9;C+R5

$9; #* 9* / J Q-D D 8*56+/-&N)1+ 0;(9;!TMˆ*q7(8+ !%; $2+5 !%;3.DŠ  =>#[\" 9* +B96E AM*+9*0;5++-B?)V A]6D9;£W ¡a1%8) )* Œ  Be  #J%* #%+-YRH9;+9*&%;3)*)i2.H K0;9-1I 

+%?7* 9;DB ;Nu&uƒ ˜&D ˜+˜ š

J_ @  ` J_ 1   @  ` T a

˜&˜&DD  š  ˜&˜&DD 1+š ƒ3šu ˜&D š  š

˜1˜1DD ˜+˜+˜+˜>–4u ˜1˜1DD ˜+˜+˜>˜1uƒ3u

˜1˜1DD  +˜ u ˜1˜1DD uu + ›ƒ ˜1D ˜ ›&ƒ

1 3a

˜1˜1DD ˜ ˜ ›˜+š ˜1˜1DD ˜ ˜ ˜+˜+˜˜  – u&}  CM ƒ˜ G

1

˜&˜&DD ˜1˜ ˜+ƒ-–› u ˜&˜&DD ˜ ˜ ˜1˜1 ƒƒ  &} u&:ƒ M ƒ-˜ G z 

1:3071 D š

+ƒ ƒ D  uƒ ˜1 1DD –› ˜ ƒ+ƒ– + ˜ › u      : (!  ! )  :      + 3! )  !  !#" 1 3   ! )    3 .  :  J_ @`   ! )6 ! 3   :       J_ 1   @  `   ! ) 2!    : . '.  :      T a { 1 3a  :! )    . +  *%  / 8. 2!: 3! )     7!"# 1    13ƒhfi    $        +! )    ! #.%    ! )  8 " # #+IJ $%* - ƒ ˜1D ˜1D › › ˜1D u ˜1D u ˜1Dƒ £T Œ 3 $ ˜1D +–1ƒ ›1D ›+–1ƒ –*D – ›1D >u8 u&D –>š  ˜1D >u B&)* AS)Qx ƒ-˜     x /< "    !& (' #!   +  $8*%) /:  *%+*,-8 + # + 8 %.*,-/0  ! ):   6 String-Array Index 9;M%:&0*%; MN+ # F” A #R4+(+ %;5 Hs)t  +VU4q"‡AH  U @H:Hu0„ ,J_/ƒ3`= ‡ us   .+ 0 >=" L p.  ‚  #CŠŸ7*   ƒ D4 $™%;>)* aVR  +F%;)  AFW8+$9;4  WW j9; UV\7;+( $= L&78AIN %;H-)1+0;M9*4Z/5  => %;%J5G1( 71I  )(13 %;[\ 5? 9*%YqR9;7;#+ %*+ q0;%01* $=  Z AIŠ%& ( $ 3%f%k $+9;%f'0 A *MZ\5 9* q 05 Z ]*i\& 9 Ÿ!+7*MC 

$%*,7* $%; $/'— #IP7* MC*DŸ 78_w ;-%*)& $- 9;+ (C9Y5+RZ  %=>+d%;%)gQ*:& 0*3N=&+%; $)*M4 j6 K %&+UV$E   FfN7GF 0;%*C 71Mn!0** K7; 7*-: 0; #%g %;DQMp+qBE!N;HM9;dF C 78 A A 5  # %*%1  ( #*+$- D `i AGIdEO%?+M\9;Na*8/(+ A J $+%Y-Rc2\J9;24N F  0* #%& AX78X-2^J*$9;+N-)d 24 $%* A59; +01* %;+UV %;5X MlD‹-[\9;9#Q0* >7;-:0*=>%; +R#0; -%;:50; A%KQI_(FfI#7GF +C MV`S* a*S5 Š‘)1!7; , A-) R )&%I1Iq%+5 #+q #78( 9O2%;5  %qN2\9*q Q R>)*)(BD+Cp€• %;F5 $'e 7* 7; AR&24789*  O2-+B%iN+0;0;%;8  R +  (  k L 24)10;9*  CE #B C (‰g9+˜N $Z%*5 J $ K%d+ a1c  0*%*/ (%;+5J%-9*D[\9; + B +#%6Mn78+ Q+Š  CD60*mk9‡#L 0;71  * 5H1+7;$+75 #'P%D< D\%; *Z5‡+%Zi-%&+0;(7;9?I60;:&J(0*9Q9;%;X % HI+R1L1C C'0*9;+'%() +7* L&%0; Q $ %;%;)*( 7;a /XZRN24+MFƒ $lJ / 0* %; %5 +9;MW9*+CUV j9QMp A+  9;%  78D 5+ $%*%; %;56+7M HC CF %d9*J]1\ ‹7;+N(IR [\9;$+L&0;f(7;QC C+%; /K+M'P $ #*Q("I !NŠR249; 

%%)1 $ $J# *C C { &%R;%K%)#+Mn=" +‹] +. nu +qƒV  ]Z  us s  € [\p„ 4p„V  + )Vqƒ.'  $+‹n pVs

 +V pn4 .n+  B C € € ‹*+ /B C / B07 u F ‚  „V  € p„VVqƒ.'  n+‹ˆ   .ˆˆ'0  …U0 %Z  [%p„VS 0 Œ ),H'*)t 0),Hu @ € H [+   € s     _   F` / ,J_/ƒ-75B(`}  %*5 13$+Z ,8(50L#RV56  A4.4MF9 -9*= ]1N0* $;$%&80*7; ZB7824 S* A/9;& %K9M39;6 9;+9*ˆ% %DFMp+]&4 %;%&\I A- #:0;T3%*B49+MB 9;C €-)  %;%Z $9*+\%;7R &

Zq78%;   0;R %;)*- _ )g & 7` Ig k7B7;C € %;+71 #_ B o€?CF` 24/ A9 7B C 75B(Dg #[\9;Q 1@ _ 7BDCp€P7; {  A71OIi_ B€?9`3`  D

A #T24 A 9; %nl(9&0;%*L124 %*-)NX #3=>B0*8 %#%# $%*   %# &   2478+ A0;9g%;)*-9*)?@7$I / & %;/ :& 0;+* _ A OI‹& > )*0*GQ` / ‡ ™*9; _  & %*&> +U

78  _0; %;& )*`` D$C_ Fz  %* & `  RR  _   `  1&4 3 z  75B*M  &  1& 3 z _  75B( ` M$_ F ` /r-ƒ 75B 1& 3 z (_ | `  u75E!BBD  K+ %*4J++0;%BMnZ 8--)q a1+%*  %;4MW3 $ 0* $!@3' a1 0*%;%&  %*DJDj]& *0;+* )K  %;( %q+5  *#_ˆ€?$ $+`%Q+L&7*–*0; AD  u  -%q78)*

%; $)*5N)c9;DFj ;%&0;4 N a*78+ #4;u+` 2\R&jJ%* %;-)q&–)1C7; A9; D EO+%?MF˜&D &R;R89* #X -/=>Z ( 0*5 %X  j AI0; $%*A /%; ';4 9;-%;)NC+= 91%;) $0*@‹B9;+  FT$)o-Š'8+%*D %;Œ --)V)  R # 8+(\+9;-+R%#+9;%; 6B  % #@N %;DAmgI‡d9;%J K0*+ B%;Mp5 (   ‰4 ¢# 2% _ˆ Hk  +>4M  ++ 249; A9'%P0;- %;$561+Ml9;Vƒ 5(  1/C3A¢Qhj iH8R> (%3)'+8-R G2\ 7;GAI $)1 -)*).)* A  %  Mˆ[\ 9;G+J$]*\ #U  D X7;  AOI?K K+ %(+ %d 0*% j-%.$6780*-)

%?:0;  \249* 9Q8Mn+ ‘%*NSA  %*5*R0*9+B9;jMp -24 $%*5    !4,J.0

01( !-.A "# !4,2%37,689:!; `M RR 24019*-) Oi2e X B(3+(L>+ #%q 0;%* *Mp+W  #9;\AIqS;/4H( +F%;)*9*+Be  #Mp-++% 5 :&t ˜&0;{e ƒ vw-DF)mk );j)1 # A- $= 3  +-R;249* 9Q 4-$ 0;$-)Q7I D |

1 & 2 "; &j5 & K 6.

1 /   9;X[\Mo9&(+0;  R [\ +9*_ %”1)N+ MN & `%;9;)di/ :--)c)dRl785 9%*- =&( +j %*2\5@+' K %*$q /_w%%X).247*  5+9  9*XEG%# a19;8B S;  #/B%\+Mc8/-)R&9;9;4 L>D 2+Mc9;Z);3(X2\+B(9%;5 )cR Mo   0;AW/H˜_ˆ9;0;-%*24 AMn%'+ %N )* 5+(0* `W\›+ 1R76/Ÿ_ˆ u $_ˆ)=   %;I(L>`DW2\Zl));+3%'(78>`HDF[\9*%Y R +9*);)1 B =>EH4 5 + A\9* ’%)9;+H+9*4(378 */R&8%)JMn+ H K=  %;I/4(787;B9#24 $%J A9J #5>H3() M B?]@(9++5 %*5  A9; $%# 9* %.Z78L>+2X9@DF[\3(9;+ #Œ  BŽ+R5 7;01 X9*  b  01+/8X -+Mn # H'%;9* . % # K3EG%969;B9;  %;B)JEB +5 MV A9;/ QR>D 9;B/(+5+Z ,- #)1 %&x.Z+24MB% # $F%* %&  K9;H-+2g9V3EO B%*$ 9;- A+IJŒ X9; #401 %;9; $%*&65)c-)*R1B 7;% %;Ik)1   %k*9++R Qž+MB9*# A #X2\ #(+%;)*+ #I@(9; 9;%‹-24+%‡%)Š %f2\   5+?0*Q%&›& ARA If #)*$36-)‡9*Moq  ‰+J9;?(3]*\ $P +DH%;[\)k9;Q9;q  0;);A)1 A R 8  =>QMp+  K+6%*+MX249; A?9;]*+0*\6 #)124 9;   %‹%;D‹0;7 EG/K K#i=& $)*)*% J +9;%;#J.9* 9;B E MnMˆ+5+ #+ !- A%;)c9;DN5 ‘[\)19* =>N6 9;  A+(+()d35+D (3j8 $= J\_ (B?9; ]? +4%;24) 9;Œ  %qB)19- =>  %;%*   %*%1-#-Rc)cKRW(+24(( ALf+9iT+MX)*M8(19;Q  -1?- =& #7 $ .%*C5Q #2\ 9;/ 24 d%)1 N320;(9‹a*78* $  6AI@)193q -9*)c DXB[\9;EK6+B?5 ]d A+%;9;)Q 9;Mn+ qŒ 9;B 65 +% A39*  *R 24 A9? )1= %(+5 jN9; Œ BPD Encoding £T%*$ $&B)* # %; 59; #&methods.  )VR&9;+&%;)1)K')* => mk (+ '788- )@ /%* S;-%Š)?5 01]1(9;N  $ +/%*%.+\(–1++D 5+M›&JRW/%;+ #-*)*¢-)dMp+ %*74IQ5)*9;9;( 24N A 9J =+5 01I1G %; &5XR2-=>  (#5 B+ #Mp:&0*)@%;INMV9* A  #  RY+DT5d[\™9; \  0*% 0;A¢*RRW249;9*-24 $(%9

B  K #;kAI ƒd1 _  J& 3z #$+ 65 /NZ &/DF ;¢`j4);9;3#(N/*N #24 A9q9;&-)*=>' (3+5+KC #Mp+:&#0*%;I  %;%*+ #& )*- +%;ˆ5R)* $ #0*@13‡=>X9;j Ad9;-#2-=>3 =>( 5 9;J Mo)c-D :0;C%*- 2Id=> %; -Rj-9*kRT£T o%)  78-349*j8 Mp+ K+%*X+MW9*j/*< #9;&)*D / [\0; 9; ž J0* ; $+A80*  ;C8*7&IX/ 0;%&+V );%;)Š356]18X+0; %%()@4)*C %* /

 - •%* )d Rl+ 24AMH9* 7; $ AJ RH1 +a&5+ DX%)1=&[\  %;%;9;55 /9; +0*( 5 /j 6 ,CMp++qM '$%;( *  _+'?%*` ( %,J)‡_ˆ€?$+`HL&7*0; ;6DW m.%Ci1]1\ %D -B)# %* $ N=> 0;(  ]1B $%;   K  +%kEG%;0*X+9;\# WA  -I D Œ -2 N(3RT7;01%;Q0*;8+'Mp+N0;V);3+%;)k)*  +%;D +);M 3 0*1# 24 7;%* A5 9g KB%;3 i%;   ¢.0*-01 ž(+0; IfR4 %)ŠDr Jm. %;0;?);;8+IPi/#1+)*( $5 #% $K+%;K9;(QL>%/#)Š- 0;%;*5  =+(78 $+I.Z7*0;$ $%;G-)*)K %*aV 5+%6RF?9Š9*)*'%*]1(Q8&)*/ -)Š(0;+)*V0*(?#+2424 ‘9;9;   ( $9iA78*- DT%;3=&5dm. o)1C#+ #9;Mˆ+-/2 J J+ AB %*/ N0*0;5 9  0*q+%;)k K+ %%%;#Mp+J/(3 Q);3(d+%)i)10* %;5@)&I*%;+ # 

7. CONCLUSIONS

0

50

45

120

10

Minimum Selection Recurring Minimum Minimal Increase

Minimum Selection Recurring Minimum Minimal Increase

Minimum Selection Recurring Minimum Minimal Increase

100

Additive Error

35

30

25

20

15

−1

10

80

Additive Error

Error Ratio (logarithmic scale)

40

−2

10

60

40 −3

10

10 20 5

0 0.2

−4

0.4

0.6

0.8

1

1.2

1.4

1.6

#%   ! 3 ! ) T ˜&}

 + 3  '  x ' !A*"

1.8

10

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1

2

3

4

5

6

k

Gamma

   N!      * 38   "# 8     2   "#  8:    $0  /   A" 

0

35

0.95

10

Minimum Selection Recurring Minimum Minimal Increase MS with deletions RM with deletions MI with deletions

0.9

False Negatives/Total Errors

30

log(Error Ratio)

25

Additive Error

0

2

+  8 ,: ! )   '8:   ' 8%   T (  ! ) x / (  ! ),+ + !  '.A  + , ! ! +   A2/ ! ,/ !   T "  + +  !  '.  :+Z % a1(+ %;0* %* 5'+%6)* #Ml+a*5+ (/+* %;9;5 T;+;Mc j-3 N $0;+A%; D F]*+F \ --R&+%K%78)J+91E! (]19\ k%;5+BT++%K %9*)1\ );a#3()*` %  u m – m)1u S  &+RT.4ƒ-‡š+qƒš  & D

tAƒ ƒv ŒBPD DB +%;G2B51VR Y%; ˆD8RV]&+9;%; ) = L&D*0;œ KD*hC-R  KD +„ %cD™+ $ # B;01+$ $ %*%;51R 785 :&S 0*+ -‹-   4 ["#u0ˆ +-%AV, I D1 0EO/% f. 6  +-u  ‚ +‹   -+Ap>4„ ;0sV   0,0[. +>4[R*) +5+  €  u+š š&›&ƒ-˜1RWƒš š 1D tAƒ3u-vN]VD D;„ ]1 +%;785+ 0;A(I9R*D,+ZD8D A„ Mn 7;+78+%; #R1N;D %;BQ5Mp+ $ %2B) w /(+% /r ;# "%Qm ,% '/;0V  K  3 0[ $+ %Y +D*[EO% ) 6S ++--u ‹-‚  + ‹C [p„Vu0ˆ2+ '335mX4