HAL Id: inria-00071988
https://hal.inria.fr/inria-00071988
Submitted on 23 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Computing connected dominated sets with multipoint relays
Cédric Adjih, Philippe Jacquet, Laurent Viennot
To cite this version:
Cédric Adjih, Philippe Jacquet, Laurent Viennot. Computing connected dominated sets with multi- point relays. [Research Report] RR-4597, INRIA. 2002. �inria-00071988�
ISRN INRIA/RR--4597--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 1
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Computing connected dominated sets with multipoint relays
Cedric Adjih — Philippe Jacquet — Laurent Viennot
N° 4597
Octobre 2002
+
',.-/
02143$57698;:/3=<>6@?BADCFEG?H69I@69JHJ1;KMLN8PORQ$1PSTAUCBV.LNQH5>14WXSY691PWHWHZNS[A
\^]=_,`babcedgfh>ija>kmlonapqijroispt_,`ba>i
uFvtwyxsa,pqz.{}|~a>vt,wm`
f.ky|=|~wPvp.oaGvta>"]a>vt"]=aN $4m4;depjwTvtayPPedc[|=kyPa>i
$>,TmmG lpj{}|Nwm{}Pp^vta,k[riFwyMa,v^kywP|opj{}`b{ >a>/¡^k[r*wm¢X£=wwoo{=|=kP"¤map"iF{}kGv"km{w=aps¡wmvt¤N¥
zqwT¡¦a>§ma>v>¨mpj]{i¦pja>"]={}©4laªvja7©4l{vta>iFpj]aekmip¦]wP|¤wT¡}a>oPam«¬ptw2a>{oaª¡¦a,pj]a>v¦wPv¦wmpk£=ww{
|~km"¤ma,p{}ivja,pjv"ky=ij`b{pjpja7X¨$k;wooaa>a>ipjw;¤wT¡®¢¯vjwP`°¡]{"]±wooapj]a|~km"¤ma,p¡kPiGvta>a>{§Pa>X¥
²³]=a,DwP=i{oa>vj{}2vjw4km>kmipj{}*kyp´µu¶}a,§Pa,9¨pj]{i{}o¢¯wmvt`/kTpt{wP`/k[rNao{·,lpptwbwmop"ky{}¬¥B²¸a
pt]l=i¬|vtwm|NwPijaFkit"]a,`baB¢¯wPvRwP`2|=lopj{}qkmewP|opj{}`b{ >a>,wm=a>(pta>owP`2{}=kypj{}qijap¬¢¯vtwm`¹`lpj{}|~wP{4p
vta,k[roi,¥\^]{}iia,pky}wT¡.i$ptwGa·{}a,4pj}r=vjw4km>kmip$|=kP"¤ma,ptiH¡{pj]wPlopBpj]aq}kPispB]wP|2{}o¢¯wPvj`/kTpt{wP*¡{pj]
|Na,vj¢¯wmvt`/ky=,a>i^}wPijaepjwb`l=pt{|Nwm{}4p.vja>}k[rb£=wwoo{}=¥
ºD»4¼M½t¾¿ TÀ$m `*lpj{}|~wP{4p.vta,k[ri>¨owPa>pja7wm`b{~kTpj{}/ijap>¨kP]woª=aps¡wmvt¤
ÁÃÂÅÄmÆyÂÅÇÉÈRÊ,ËsÌ>Í[ÎsÏ[ËsÊ(Í[ЯÑÒÔÓÊ(Õ.Ö"רÏ[ÎÚÙ[ÎÛ~Ê(ÜØÍ[ËsÎjÖ"ÍPÒÞÝ"ß7àá(â ãÎäMå[Îsæ9ÏTÖsç,ÒéèTÐ9Ö"Ï[ËsÎ(Ò¹ê7ë[ì>í[îmï[ðNñóò[íyôTï(õ¬ö
÷ õ4ï7øTï"ù[ùmìXñµú,ûyð,ü>ýmì>þXöÿTû,ýyîTìTþ¬ñPï>ì>þ yïyîmï,ûXñ[î
+ +
&'( (¸*
aG|=ky|{}a,v|=vjh7ia>PptalDky}mwmvt{pj]`ba|~wPlv.,km},l}a,vlDa>=ia>`}aGowm`b{}=ky4pq,wma,noa
lUmv"ky|]a|=kmvpt{vea>=ija,`=a7iqoa*vja>}km{}iq`lpj{}|~wP{4p7¥ a,pªa,~ia>`}a|Na,lopptvja*lopj{}{ijh|Nwmlv
{Ml=ija,va·>kma>`ba,4plaG{}o¢¯wmvt`/kTpj{}wm*pjwPl=i^a7iGl==il=vth>ija>kmlv"kmo{}w=¥
¿ 7 ½ vta,ky{i¦`*lpj{}|~wP{4p>¨~o{Ml=ij{wP¬¨vth>ija>kmlkm9]wom¨oa,=ija,`*aG,wma,noawm`b{~ky4p
. /
*
+
/
0 é]=w;aps¡wmvt¤oi2143ewmNa>va,¡ |~kyv"kmo{}m`/i2¢¯wmvvjwPlopj{}~¥65q{,kmipvtwmlopt{= ]=kmi~a>a, ¡{oa,}r
ipjl~o{a7X¥ªª pj]=awypt]a,vª]=ky=¬¨=vtwPkP,kPispt{b{}Uky;kP]woa,ps¡¦wPvj¤{iqipj{}}$kmDwm|Na,D{ijijlam¥\^]a
`/km{¢¯wol=iq]a,vtaG¡{}~a*wm wm|opt{`b{} ,{}/pt]a=km=o¡{pj]Dlopt{}{} >kTpt{wP ¢¯wmv.=vjw4km>kmipj{}/kb|=km"¤Pap>¥
²³{}vta,}a>iti{}4pja,vj¢ókm,a>i]=k[§maky {}]a>vja>Pp/,km|=km,{psr¸¢¯wmv2vtwPkP,kPispt{~¨ "787 ¡{pt] wPaa>`2{iti{}wm¬¨Fk
=wa,ky vta>kP"])kmBpt]aa7kyvtr;wooa>i>¥zqwT¡¦a>§ma,v7¨¬{p{iG]=kyv"¸{±|v"kmpj{abptwD~a>a:9p¢¯vjwP` pj]={}i
>ky|=kP{psr/{ wPvtoa>vpjwk[§mwP{}vja7ol==ky4pvja,pjv"ky=ij`b{}iti{}wm~i,¥
0 wypt]a,vª|vjwP}a,`É{}iqpjwk[§Pwm{wwm|~i.¡]a>Dvjw4km>kmipj{}=¥²)a¡{}Hkmitil=`2a{ pt]a¢¯wP}wT¡{=
pt]=kTpijwm`ba.`ba>"]=km{i` ky}}wT¡.ik=waqptw¤wT¡³{¢R{p^]=kPi¦kmvta>kPor*vta>,a,{}§ma7/k|=kP"¤map¦wmvwyp7¥H\^]={}i
>kyl=ijl=ky}r~akP"]{a>§ma7rl=ij{}/ia7©Pl=a,=,ael`*~a>vti^wPvwm`b|lpj{}i{}m~kTpjl=vja7i¦wm¢|=kP"¤ma,pti>¥
\^]ai{}`b|}a>ip*¡^k[rUwy¢.=vjw4km>kmipj{}Dk;|=kP"¤mapGptw¸km¦wooa>i{ km km ]=waps¡wmvt¤U{i|=lvja
£~w4woo{}~¥
;
R
»=<H¿¬¿
À?>"@?ACB
¿ ¾
oTÀ?>"@?AÔ
»
kDwooavtaptvtkm=ij`2{ptiGpj]a|=km"¤PapwPr)wm~a
kT¢pta,v.]=k[§{}2vja7a,{}§ma7/pt]aG|=km"¤Pap7¥
}a>kmvj}r2kmNwooa7ivta>kP"]=ky}a^¢¯vtwm` pj]aijwmlv"a.¡{}}~vta>,a,{}§mapt]aq|=kP"¤ma,p>¥Hz.wT¡a,§ma>v>¨Pa,§Pa,vtr*wooa
¡{}}Rvtapjv"ky~i`b{ppj]a|~km"¤ma,p.¡]a,D{pª{i|NwPiti{}}am¨Nkmi.¡aG¡{}$ia>am¨ptwmvta>kypj}rvta>ol~apj]=al`Na,v
wm¢vtaptvtkm=i`b{ijij{}wm=i>¥
D
a>§ma>vtkm~pta>"]={}©4la7i¦]~k[§mae~a>a,|vjwP|~w4ia7/ptw2wP|opj{}`b{ >ae£=ww{2¡{pj]ps¡¦wb]wP|a>{P]4Nwmvt]wwo
¤wT¡}a>maFE
"787
k wooa¸¤wT¡.i{ptia>{P]4Nwmv"iky=épt]a)a,{}m]Nwmv"iwy¢*{p"ia>{P]~wPvtiG(¥IH¬{}¤
iptkypjavtwmlopt{2|vtwyptw,wmil=il~ky}r|=vjwT§{oaqpt]{i^{o¢¯wPvj`/kypj{}wm¬¥H´ p.{i^l=il~ky}rwPoptkm{=a>4rvja>mlkyvtr
a>`b{pjpj{}iw >ky}a7zqa,}wb|=kP"¤map"iwPPp"ky{}{}*}{isp"iwy¢$a,{}m]Nwmv"i,¥
J=vjwP` Pvtkm|]bpj]=a,wmvtr2|~wP{4p¦wm¢¬§{a>¡G¨mpt]aeijapwy¢¬wooa7iBpt]=kTp^¡{}Nvja,pjv"ky=ij`b{p¦km{}§ma>/vjw4km>kmip
|~km"¤ma,pEó{=,l~o{=pt]a/iwPlvt,aKG.`l=ipª¢¯wPvj` kwPa7(pja7UowP`b{=kypj{}ijap7¥ 0 owm`b{}=kTpt{ ijap{}i
kDijapwm¢=wa>iijl="])pj]=kyp*kyr¸wooa/{)pt]aa,ps¡¦wPvj¤;{iGa>{P]~wPvGwy¢.iwP`ba2a>a>`ba,4pwy¢pj]=aia,p>¥
´p{ie,wm=a>(pta>U{¢¦pj]a/ijlmv"ky|=]D¢¯wmvt`ba>;rpj]={}iGijap{}i,wma7(pta>X¥\^]=a2,wma7(pta>oa7ijiªwy¢¦pj]a
wm`b{~kTpj{}GijapF{}=il=vja7i$pj]=kyp¦ky}owooa>iFwy¢Npj]a.,wma7(pta>bowm`b{}=kTpt{=ia,pF¡{}vta>a>{§Papj]a|=kP"¤ma,p
km= ¡{}pj]l=iª~abky=aGptwvtaptvtkm=i`b{pq{p>¥ª\^]=a*owP`2{}=kypj{}ijapª|=vjwP|~a>vpsr{=ijlvta>iqpj]=kypekmwooa>i
¡{}}¬vja7a>{§Paqpt]aG|=km"¤Pap$EókPijijl`b{=2=w*ptvtkm=ij`2{iti{}wma,vtvjwPvG¥
wP`2|=lopj{}qk^`b{}{`*l`Úi{} ,a,wm=a>(pta>owP`2{}=kypj{}.ia,pR{iML.u;]~kyv"X¥ wPvja>wT§ma,v7¨"pj]=aia>a7(pt{wP
wm¢pj]a)wm=a>pja>Þowm`b{}=kTpt{ ia,p`l=ipNa¸{}ipjvt{=lopja7X¥ON^kmija>éwPéps¡¦w ]=wm|³a>{P]4Nwmvt]wwo
¤wT¡}a>mam¨kwooaG]=kmi.pjwoa7{oa¡apt]a,v.wPv.wmpq{pª{}i{} pj]=aowm`b{}=kTpt{=ia,p>¥
D
wm`baG]a>lvt{}ipj{,i
~kmija>³wP a,{}m]Nwmvt]wwoé{=,l~i{}wm=i]=k[§Pa;Na,a,¹|vtwm|NwPija>61}cmcy¨^=¨^¨.¨c7439¥Ã´ ¢*k wPa>pja7
wm`b{~kTpj{}/ijap]=kmi^Na,a> a,}a>(pta>X¨pj]a>pt]a¢¯wmvt¡^kyv"o{=*vtl}aeNa>,wm`ba>i>«
P ¿ @?@ » m » ÀUÀ ¿ F>"@$o8>@QA/ » RB ¿ ¾ oTÀ?>"@?A » k.wooavja,pjv"ky=ij`b{p"i¬pt]a|=km"¤Pap
wm}r±wm~a kT¢pja>vb]=k[§{¸vja7a>{§Pa> pt]a |=km"¤Pap2km= {¢ª{pb~a>wPPi*pjw¸pj]=a ,wm=a>(pta>
owm`b{}=kTpt{=bijap7¥
ªGpt]awmpj]a>vR]=km=X¨7pj]=a¦`*lpt{|Nwm{}4pvta,k[r.pta>"]{©4la]~kmi¬Na,a>|vtwm|NwPija>eptwqwm|pj{}`2{} ,aF£=ww{
¡]=a,pt]a^}kPispH]wP|{}o¢¯wmvt`/kTpj{}wm{}iHP{§Pa,1S¨cm¨m439¥$\^]a^{}oa7kqNa,]={=pj]={}ipja7"]{©4la^{}ipjwwP`b|lopja
ijwm`ba/¤{}= wy¢}w>kywm`b{~kTpj{};ijap"i,¥km"] wooawm`b|lpja>i*k `lpj{}|Nwm{}Pp*vja>}k[r)ia,p¡{pt]±pj]a
¢¯wP}wT¡{}*|=vjwP|~a>vpt{a7i,«
pj]a/`*lpj{}|~wP{4pGvja>}k[rDijapG{i{=,l~oa>){}Upt]aba,{}m]Nwmvt]wwUwy¢Fpt]a/wooaP¨Xpj]aba>a>`2a>4pwm¢
pj]aG`*lpj{}|~wP{4p.vta,k[ria,pqkyvta>ky}a7
V 4
E¯wmv
uf¹¢¯wmv.ij]wmvjpG¦wy¢$pj]=aGwooam¨
a>kP"]ps¡w/]wm|a>{P]4Nwmvwm¢pt]aGwooaG]=kPi.k2a,{}m]Nwmv{}pj]aG`*lpt{|Nwm{}4pqvta,k[ria,p E¯¡aijk[r
pj]=kypqiwP`bae`*lpj{}|~wP{4p.vta,k[r 4 pj]aps¡wb]wm| a>{P]~wPvG¥
\^]=aq`l=pt{|Nwm{}4p¦vta,k[r2ijap|l~iFpt]aqwooaq¢¯wmvt`/i¦kwm`b{~kTpj{}2ia,p¦wy¢Xpt]a.ps¡w]wP|a>{P]4Nwmvt]wwo
wm¢Rpt]aGwooam¥
wPvja¢¯wPvj`/km}rm¨}ap .oa,wmpjaGpt]aijapqwm¢Fky}¬wooa>i.pj]=kypªkyvtaG{};kbm{}§ma,Dijap ÃwPv.]=k[§Pak
=a,{}m]~wPvª{}!/¥²)abitk[rpj]=kyp" kia,p"# ¡]a,$#&%'((¥ H¬a,p *)( +,-/.0
a,wmpja*pj]a2wooa>ikypo{iptky~ac¢¯vtwm`1Éky=22345+6789.:oa>wyptapt]a2wooa>i
kyp^{}iptkm=aGG¢¯vtwm`;/¥B²³]=a,<+=?>A@e,wm4ptkm{=iwPrbwPaª=wam¨*)B=C>D@{i¦pj]aea>{P]4Nwmvt]wwo
wm¢E>Dky~-3F=?>D@¦{}ipt]aps¡¦w2]wP|a>{P]~wPvj]wwo/wm¢E>R¥
0
uf¹ijapHGwy¢$k*wooa">D{ipj]l=ikyr
ijl=ijap¦wy¢<)FB=C>D@IFil="]2pt]=kTpE3F=?>A@I %$(G(¥H²)aªitk[rGpj]~kTp9Gg{iFkijl=ijapwm¢Na>{P]~wPvtiHpt]=kTp
,wT§ma>vti.pj]a2ps¡w]wP|Ua>{P]4Nwmvt]wwoDwy¢9>¸wPva>©4l{}§[kma>4pj}rG ¡]{"]U{iekowP`2{}=kypj{}ijapwy¢¦pj]a
ijlPvtkm|]{}=ol~a>r7B=?>A@I8(¥
²)a¸>ky}q`l=pt{|Nwm{}4pvta,k[r ija,}a>pjwmvwy¢k±wooaKJ°k =wa;¡]{"] ]=kmiia>a7(pta>éwooa$J°kmi
`*lpj{}|~wP{4pGvja>}k[rP¥G´ p>kyUpj]a>¸Nab|vjwT§Pa,Dpt]=kTppt]a¢¯wP}wT¡{}¢¯wPvj¡^kyv"o{}vtl}a2km}wT¡.i.ptwvta>kP"]
kmMpt]aGwooa7i^{pj]=aGaps¡wmvt¤ EókPijijl`b{=2=w*ptvtkm=ij`2{iti{}wma,vtvjwPvG«
¯8>L ¿ >"@MG » ó ¼ <$¿¬¿ ÀQ>@QA2B ¿ ¾ oTÀQ>@QA¸ » k=waqvtapjv"ky~i`b{ptipj]aª|=km"¤Pap
wm}r wP=a {¢{p/]=kmibvta>,a,{}§ma7 pt]aD|=km"¤Pap2pj]a 9~vtipbpj{}`2a ¢¯vtwm`k)`lpj{}|~wP{4pvta,k[r
ia>a7(ptwmv7¥
\^]ai`/km}a>ippt]a`*lpt{|Nwm{}4p2vta,k[r¸ijap"i*kmvjaP¨$pj]a¢¯a,¡a,vvtaptvtkm=ij`2{iti{}wm=i¡{}¦wo>lv7¥U´p2{}i
Lqu]=kmvt±pjw¸,wm`b|loptakU`lpj{}|~wP{4pbvja>}k[r ijapb¡{pt] `b{{}`l=` i{} ,a 1Ø43^=lop*pt]a,vtaa,no{}ipti*mwwo
]=a,lvt{}ipj{,i^~kmija>wm|vja,¢¯a,vtvj{}2a,{}m]Nwmv"i¡{pt]kyvtmaGoa>mvta,akmi^`*lpt{|Nwm{}4p.vja>}k[roi>¥
´µ±pt]{i*|~ky|Na,v7¨$¡a|vtwm|NwPijakmPwmvt{pt]`/i¢¯wmvbwP`2|=lopj{} u¦f ijapti2ky~±kUwPa7(pja7±owm`
{}=kypj{};ia,p=kPia7¸wP±`lpj{}|Nwm{}Pp*vja>}k[roi>¥\^]awP}rU¤wT¡}a>oPakmitil`ba7U¢¯wPvk m{}§ma> wooa/pjw
apja>vj`b{}a{pti uf®ia,pª{iq{ptiqps¡w]wm|Ua,{}m]Nwmvt]ww¬¨~ky~ pjwoapta,vt`2{}apj]a2owm`b{}=kTpt{ia,p>¨
pt]a{ispbwy¢ªa,{}m]Nwmv"iGpj]~kTpb]=k[§maia>a7(pja7 pj]=awooa kmi*`l=pt{|Nwm{}4pbvta,k[r Eóijl="] a>{P]~wPvti*kyvta
>ky}a7`*lpt{|Nwm{}4p.vja>}k[r/ija,}a>pjwmv"iG(¥H\^]a>ija{o¢¯wPvj`/kypj{}wm>ky~a,wm4ptkm{a7{}]=a,}w2|=km"¤Pap"ipt]=kTp
=wa>i2|~a>vj{}woo{,ky}}r)=vjw4km>kmippjw¸pj]a>{v/=a,{}m]~wPv*{} wPvta,vjpjwU`bwP{ptwmv2{}¤oi2§Tky}{}o{psrm¥ \^]=a km
Pwmvt{pt]`/iowa>iGwmpa,a7¸kmrUo{isptvj{}lopta>)¤wT¡a7omabwm¢pj]=a/m}wm=kmBaps¡wmvt¤ pjwP|~wPwPmrm¥/\^]a7ia
kPijijl`b|opt{wP=i.`/ky¤Papt]a2kmPwmvt{pt]`ɧma,vtrDkTpjpjv"km(pt{§PaG¢¯wmve`bwm{}}a*kmU]woa,ps¡¦wPvj¤oiªi{}=a*{pa>a>i
xsl~isp.}wo,kyXl=|Nkypja7i.kTpa>kP"]apja7(pta>pjwP|~wPwPmr"]=kmmaP¥
M
,ONQPR
´µ |v"kmpj{aªpj]a¢¯wP}wT¡{}bmvta,a>rky}mwmvt{pj]` ¡wmvt¤oi¦§Pa,vtr¡a,}@¥
²)aisp"kyvjp¡{pt] km a,`b|opsr u¦fia,p>¥
D
pja>|UcP«?J${}=/pt]a.ps¡w ]wm|/=a,{}m]~wPvtiFpj]=kypkmvja.=a,{}m]~wPvFptw*kij{}m}aqa>{P]~wPvwy¢AJ i>¥HuFlop
{pt]a
ufÚijappj]=a>ijaG}kyppja>va,{}m]Nwmv"i Eó,km}a>ij{PaG,wma7(ptwmva>{P]4Nwmv"iG(¥
D pja>| o« 0 = {} pt]a uf ijap2pj]a a>{P]4Nwmvbwooapt]=kTpwT§Pa,v"iGpt]a }kmvjPa>ip2l`*~a>v2wm¢
ps¡¦w9]=wm|a>{P]~wPvtiwy¢AJÞpj]=kyp^kmvjaªwyprmapwT§Pa,vta>2rbpj]al=vjvta,4p u¦f¹ijap7¥Hf.a,|Na>kTppj]={}i
ispta,| l4pj{}RkmMps¡w ]wm|a,{}m]Nwmvt]wwo{iwT§Pa,vta>X¥
´ pG]=kPiªNa,a>)ij]wT¡ 1ÅK3Hpt]=kTppj]=a2Pvja>a>orDkmPwmvt{pt]` |vjwT§{oa>ieky ufÃijap¡]wPija2ij{} ,a2{}ikTp
`bw4isp }kmvjPa,vFpt]=kypj]aewm|pj{}`bkm ufÞijap7¨o¡]a,vta {iFpt]ai{} ,aewy¢¬pt]aea,{}m]Nwmvt]ww¬¥$\^]a
l`*~a>vwy¢Bispta,|vja7©4l{vta>r/pj]=aGmvta,a>rky}mwmvt{pj]`!{i
3
(¥
\^]a>vja{ikbilNwm|pj{}`bkmRky}mwPvj{pj]=`!,ky}}a>pj]a`b{9{ ky}mwmvt{pj]`Ãpt]=kTpq,wm=ij{}ipti{}4pjw/ia>a7(pt{
pt]aGwooa7i^{pj]=aG{=,vja7kmij{=*wPvtoa>v^wy¢$pj]a>{v´i E¯wmv.kmrkyvt{pjv"kyvtr2{}=,vja7kmij{2wmv"oa>vG¥
D p"kyvjp.¡{pt]Dkya,`b|opsr ufÚia,p>¥
]a>"¤¸pj]=aa,{}m]Nwmv=wa>iG{} pj]a{~vta>kmij{} wmv"oa,vGwm¢pt]a,{}v{oa>Ppt{9~a,v"i,¥´ ¢pj]alvtvja>4p
wooabwT§ma>vtiªkps¡w ]wP|;a,{}m]Nwmve¡]{"]U¡^kmiªwyprma,p,wT§ma,vta> rpj]=ab,lvtvja>Pp uf ia,p>¨
pj]a>Dkmpj]=alvtvja>4pwooaptw2pj]a u¦fia,p>¥
0
pt]wml=m]*pj]a^`b{9{*kmPwmvt{pt]`{i$¢ókyv$¢¯vjwP` wm|pj{}`bkmMEpt]a,vta^{}iHweNwml=2¡{pj]*pj]a^wP|opj{}`/ky
uf
ijapij{} ,aKG¨m{pF]=kmiBpj]aqkPo§Tky4ptkmma¦pjwG¡wmvt¤{} <Fispta,|¬¥H´ pkyijwe]~kmiHpt]aqkm§[km4ptkyPapt]=kTpBpt]a.wooa
>kyDoa,pja>pqr{ptija,¢B¡]apt]a,vªwmvwmpq{pª~a>wPPi^ptwbpj]a ufijapqwm¢Bkb=a,{}m]~wPv$139¨~lopq{} pj]={}i
>kmijaGpj]=a*ky}mwPvj{pj]=` vta>©4l{}vja7i (¥ª²¸a2,ky}pj]a2ia,pªwy¢F=wa>iqpj]=kypeija,}a>(pk/m{}§ma>Dwooa-J kmi
uf ijap>¨Npt]a u¦f ia>a7(ptwmvªijapªwy¢Fwooa J¥q\^]a*vja>§ma>vtija uf ija,}a>pj{}wm;kmPwmvt{pt]` ¢¯wPvqwooa
JÚwP|~a>vtkypja7ikPi¢¯wm}wT¡G«
iptkmvp¡{pj]Dky a,`b|opsr ufija,}a>(ptwmvijap>¥
]a>"¤¢¯wmv.a7km"]|=ky{}vwy¢Ba>{P]~wPvwooa>i km=2¥´¢Bwooa JÚ]=kPi^pj]=ai`/ky}}a>ip.{}oa>4pj{9=a,v
wy¢kmpj]a*wooa>iqpj]~kTpekmvjaa,{}m]Nwmv.ptwNwypt] km=2¨Npj]a> ky=Ãkmvja*kmoa7 pjwpj]a
ufija,}a>pjwmvijap7¥
, '
+
" #
\^]=a 9=vtipowP`2{}=kypj{}ia,pFwma.>kyb{}`bkmm{}a{iHpj]aªia,pFwy¢Xky} uf³=wa>i>¥
0
wooaq~a>wPPi$ptwGpj]a
wm`b{~kTpj{}ijapª{¢F{pª{i ufwy¢kypq}a>kmipqwPawooa*{Dpt]aa,ps¡¦wPvj¤M¥5qo¢¯wmvjpjl~kTpja>rP¨NkPi.¡a¡{}
ija,a{Dpt]a|Na,vj¢¯wmvt`/ky=,aa,§Tkml=kypj{}wm;ija>pj{}wm pj]={}ieowm`b{}=kTpt{=ia,pª{i.wmpªvta>km}rwP|opj{}`/ky$i{}=a*{p
,wm4ptkm{~iFptw4w2`/kyrbwooa>i>¥H´µwPvta,vFptw*Papk`*l="]ij`/ky}a>vwm`b{~kTpj{}bia,p¡ae`l=ipa>{}`b{~kTpja
ijwm`bawy¢pj]a ufÚwooa>i>¥H²¸aG{}4pjvtwool=aGkm kmPwmvt{pt]`â¯wmv^pt]{i|lvt|~w4iaP¥
\w;ky}wT¡ wwmv"o{}=kTpt{wP¸Naps¡a,a> wooa>i>¨wPlvkmPwmvt{pt]` vta>©4l{}vja7iqpt]a¤wT¡}a>oPabwy¢kpjwyp"ky
wPvta,vqwm¢Fpj]=a*wooa7i,¥*ª=a*,kmU|NwPiti{}}r l=ia*pj]=a2ij`/ky}a7ispe´µu¹km=ovja7ijiªwy¢kwooabkmieky¸´¨Xky~
¡a¡{}}$itk[rpt]=kTpkwooa*{}ieij`bkm}a,vqpj]=km¸km;wypt]a,vª{¢{pª]~kmiekij`bkm}a,vª´¥ 0 4rptwyptkmRwPvtoa>vqwP
¡]={}"] ky}¬wooa7ikymvta,a,kmNaGl=ija>X¥
0 =waªpt]a,Doa7{oa>ipj]=kyp.{p.{i^{pt]awPa>pja7owP`b{=kypj{}/ia,p.{¢s«
fl}a/cm«Hpt]aGwooa]=kmikbij`/ky}a>v´Úpj]=kmDky}X{p"ia,{}m]Nwmv"i,¨
fl}ao«BwPv{p.{i`lpj{}|Nwm{}Pp.vta,k[r/wy¢${p"ia>{P]4Nwmv¡{pj]pt]ai`/km}a>ip´¥
P +
/ é
+$+
R
HRap/l=i/>ky} pt]a;ia,p/wy¢ekm=wa>i2pj]~kTp/]=k[§ma oa7{oa> pjw NaD{ pj]a;,wm=a>(pta> owm`b{}=kTpt{
ijap7¥\^]a/ij`bkm}a>ipwooa/wm¢¦pt]a/aps¡wmvt¤D{i}a>kmvj}r;{} r;vtl}aDcm¥H¬a,p ~abpt]awPa>pja7
,wm`b|Nwma>Ppwm¢pt]aij`bkm}a>ipwooaG{} pj]aijlPvtkm|]{}=ol~a>r;¥F²¸a*kyvtaePwm{}2pjwij]wT¡ pt]=kTp
{iqkowm`b{}=kTpt{ia,pq¢¯wmvqpj]aaps¡wmvt¤ E@kmitil`b{}/pt]=kTpªpj]a*aps¡wmvt¤{iq,wma7(pta> wy¢,wmlv"ijaKG(¥
\^]={}ib¡{}|=vjwT§Pa{}¶|~kyvjpj{lkyv2pj]~kTpkyr±wooa {} ]=kPi2k)a>{P]~wPv2{} 2¨¦{}`b|r{})pj]=kyp
{}=oa>a>a>©4l=kyi D¥B\^]{i^¡{}Npj]l=i|vtwT§ma.pt]=kTp {}i^wPa7(pja7wPpj]=aewPae]=ky=ky=pj]=kyp!{ik
wm`b{~kTpj{}/ijapwmpj]=aGwypj]=a,v]=km=X¥
0 ijijl`bar,wm4pjv"km{}pj{}wmpj]=kyp ®{i.wmpªkowP`2{}=kypj{}ijap>¥.\^]=a,vtaG`l=ipªa,n{ipªiwP`baGwooa>i
pt]=kTpbkyvtawmp{$ ¥ wm~i{oa,vpj]aia,p wy¢.wooa7iwPa7(pj{})iwP`2awooa{} pjw)iwP`ba
=wa{} >¨¦pj]a;,wm`b|}a,`ba,4ptkmvjr±wm¢ (¥°{ibpt]a;ia,pwy¢ewooa>i/¡]={}"]¶]=k[§maDkypb}a>kPisp
wPa/a>{P]4Nwmv{ Éky=±kypa7kmipwm=a/a,{}m]NwmvG{} ,¥
0
iGpj]=aaps¡wmvt¤;{i,wma7(pta>X¨$wmlv
kPijijl`b|opt{wPD{`b|}{a7i.pt]=kTpÃ{iqwmpªa>`2|psrm¥RLqwypj{apj]=kyp +br wP=isptvjl~(pj{}wmR¥.²¸a*wT¡
,wm=ij{oa,vpj]=ai`/ky}}a>ip^=wa { (¥
F{pj]a>v°{i{} ,¥ 0 i (¨pt]a,vtaGano{isp"i.kba>{P]4Nwmvwy¢ °{}b¥ H¬a,pe~ak
a,{}m]Nwmvwy¢b{} 2¥ wm=ij{oa,vpj]ae`lpj{}|Nwm{}Pp^vta,k[r2ijapwy¢ )«HkPi.{}ikGps¡w]wP|a>{P]4Nwmv
wy¢ )¨=pt]a,vta`l~ispqa,no{}ipªijwm`ba`l=pt{|Nwm{}4pqvja>}k[rbwy¢ °¡]{"]D{iqk/a>{P]~wPv.wy¢T¥Lqwypj{a
pj]=kyp/`*l=ispe~a*{b¥ 0 ipj]=a2ij`/ky}a7ispq=a,{}m]~wPv.wy¢b{i)¨ i]wPl ]=k[§maa>a7(pta> {ptija,¢
{!Érvtla¨=wPPptvtkPo{(pj{}"# +$o¥
F{pj]a>v {i^{ ¡]{}"]D>ky~aG|=kmvpt{pt{wPa>{}! km= .%/«
& ´ ¢ {i{¨ ij]wPl}]=k[§ma.a>a7(pta>{ptija,¢Rkmi¦Na,{}*{}' i{}=aeky}~{ptia>{P]~wPvti¦kyvta
Pvja7kTpta,v7¥$\^]{i{iky4ky{}kbwPPptvtkPo{(pj{}wm¡{pj](# +$o¥
&
ª;pj]a2wypt]a,ve]=ky~D{¢ {iªwyp{b¨X{p>kywmpª]=k[§Pakyr a>{P]4Nwmvª{ ,¥$H¬ap
Nak/a>{P]4Nwmvªwy¢ °{})¨Nkm= a,p >¸~aka,{}m]Nwmvqwm¢*{} ,¥
0
i°]=kPi.w
=a,{}m]~wPvG{ ,¨E> {iGwypk =a,{}m]~wPvGwy¢ ky~)ijwm`ba/`*lpt{|Nwm{}4pvja>}k[rwm¢
`*l=ip.~a,wma7(pta>pjw >R¥ 0 i °>ky=wyp]=k[§maGkm4ra,{}m]Nwmv{} ¨+*`*l=ispNa{}
(¥ 0 i]=kPiGka,{}m]Nwmv{ ¨¬{p{i{,/¥/\^]aij`/ky}a7ispa>{P]4NwmvGwm¢{}i
pt]l=iÉ{}`b|}r4{}*pj]=kyp-*i]wPlNaª{} 4r/vtl}ae¨,wm4pjv"km{}pj{}*kmPky{}. +/o¥
´µUkm$,kPia7i.¡ama,pªkwP4pjv"kmo{(pt{wP¬¥®pj]l=iª]=k[§maptwNa*kowP`b{=kypj{}ijap7¥ 0 iª`2a>4pj{}wma7
Na¢¯wPvjaP¨pj]{i{}`b|{}a>ipj]=kyp;+ {i.kbwPa7(pja7wm`b{~kTpj{}/ijap>¥
0
+
,R !( # '" $(¸,
+
'q-21//
²³]=a,k`lpj{}|Nwm{}Ppvta,k[r£~w4woo{}*{}i^,kmvjvt{}a>bwmlp>¨4pj]aijap^wy¢Rwooa7iFpt]=kTp^vtapjv"ky~i`b{ppj]=aª|=kP"¤ma,p
Eó{~}l=o{}pj]a/ijwmlv"a4G¢¯wPvj`/ik,wm=a>(pta>¸owm`b{}=kTpt{ ijap>¥2²)a2vta>>ky}Rpt]ab|vtw4wm¢wy¢¦pj]=kyp¢ókm(p
¢¯wPv^pj]aG|~ky|Na,v^pjwbNaia>¢BwP4ptky{}a7X¥
\^]a2wooa7i.pt]=kTpvja,pjv"ky=ij`b{pkyvta*,a7kyvtr wPa>pja7X¥²¸abkyvtaPwm{}/pjwi]=wT¡Úpj]=kypkm4r wooa
kyp.o{}iptkm=a3/¢¯vjwP`!pj]=aiwPlvt,aª{ia>{P]~wPvwy¢BiwP`2avta 9pjv"ky=ij`b{pjpja>v^kypqo{isp"ky=,a3 .4ª¢¯vtwm` pj]a
ijwml=vt,a2r¸vja7lvtvja>=a2wm%3~¥\^]=a/wooa>ikypo{isp"ky=,aDcb¢¯vjwP` pj]=aiwPlv"a/kyvtaba,{}m]Nwmv"iwy¢^pj]a
ijwml=vt,am¥
0
itil`ba3 4ekm=pj]a|vtwm|Na,vjpsrb§ma,vt{9=a>¢¯wmv 3 3~¥
wm~i{oa,v^k2wooa5>;kTpo{}iptkm=a3
¢¯vtwm` pt]aeijwml=vt,am¥ H¬apNaªpj]aea,{}m]Nwmv"iwm¢D>kyp^{}iptkm=a 3H.4.¢¯vtwm` pt]aeijwml=vt,a E{i¦,a7kyvtr
=wypBa,`b|opsrVG(¥FªlvHvta>l=vjvta,=,a¦]r|Nwypj]=a>ij{}i${}`2|={}a>i$pt]=kTpFkyr=wa'!¹]=kPiHiwP`2avta 9pjv"ky=ij`b{pjpja,v
=a,{}m]~wPvkTpo{iptky~a"3 . ¢¯vtwm` pt]aijwmlv"aP¥ 0 `bwm;kmFpj]w4iabvta 9pjv"ky=ij`b{pjpja,v"i}ap Na/pj]a
wPapt]=kTpa,`b{ppta> 9=vtip>¥ D wP`ba`l=pt{|Nwm{}4pvta,k[r wm¢`l=ipeNabka>{P]4Nwmvewy¢>)ij{~a> {iªk
ps¡w]wP|Da,{}m]Nwmvªwy¢=¥ {ipt]4l~iq{})/¥ 0 ia,`b{ppta>29~vtipªky`bwP/pj]=aa>{P]4Nwmv"i )¨Nkm=
{i`*lpt{|Nwm{}4p.vja>}k[rwy¢=¨ `*l=ip.]=k[§maa,`b{ppta>Dkm>wPvto{}*pjwbpt]aG`lpj{}|Nwm{}Ppªvja>}k[rb¢¯wPvj¡^kyv"o{}
vtl}am¥ {ipj]l=i.k2vta 9pjv"ky=ij`b{pjpja>v¡]={}"]Dk2a,{}m]Nwmv^wy¢ >R¥F\^]={}i.kP"]{}a,§ma7iwmlv^vta>l=vjvta,=,aª|=vjwwy¢s¥
²)aGkyvtaªPwm{}ptw2ij]wT¡³pt]=kTp^pj]=a,wm=a>(pta>owm`b{}=kTpt{bijapwmop"ky{}a>rk`*lpt{|Nwm{}4pvja>}k[r
£~w4woo{}q>kyGNawPoptkm{a7rewPlvRkmPwmvt{pt]`¹{¢opj]=avt{P]Pp¬wPvta,vRwPpj]a¦wooa>iR{}iRptkm¤ma> 4 X X"X¥
wP=i{oa>vFwma`*lpj{}|~wP{4pvta,k[rG¢¯vtwm`®kGijwml=vt,aqky=b}apÞoa>wypta^pj]aqijapFwm¢Xvja @ptvtkm=i`b{ppta,v"i
{}=,l={P¥B²)aoa:9==akptwyp"kyXwmv"oa>vwmwooa>ikPi¢¯wm}wT¡.i>«
G{ii`/km}a,v^pt]=kyDkyr/wmpj]a>vwooam¨
kyrwooaG{}{ii`/ky}}a,v^pj]~kyDkyr/wooaGwmp.{2¨
¢¯wmv >'2¨>{¢Hkm=wP}r{¢ >;]=kPi^a,`b{ppja7~a,¢¯wmvtaM¨
pj]aG=wa>iwmp{ kmvjawmv"oa>vja7kyvt{pjv"kyvt{}rm¥
HRap^l=i^i]wT¡épt]=kTpwPlvkmPwmvt{pt]` wP`2|=lopja7ikPi¦,wma7(pta>wm`b{~kTpj{}bia,p>¥%H¬a,p oa,=wypja
pt]abia,pewP`b|lopja7;r wmlvky}mwPvj{pj]=`¥e²)a*]=k[§PaGpjw ij]wT¡+2¥$J{}v"ispwypt{},apt]=kTp"% r
vtl}acm¥
wm=ij{}a,vªkmrwooa->,. =F@P¥
D
{}=,a>)]=kmiqa,`b{pjpja7X¨Npj]a 9=vtipqpt{`ba*{pevja7a,{}§ma7pj]a
|~km"¤ma,p^¢¯vjwP` ijwm`baGa,{}m]NwmvM¨]aG¡^kmi.`lpj{}|~wP{4pqvta,k[rb¢¯vtwm`ky= oa7{oa>ptw/vja,pjv"ky=ij`b{p7¥
{i}a>kmvj}r/pt]ai`/km}a>ipa,{}m]Nwmv^wy¢ >R¨{}`b|}r4{} >( É4rvjl}ao¥F\^]{i^|vtwT§ma7i %.D¥
LqwT¡Ú,wm=ij{oa,vqijwm`ba=wa >D¨Mky= }ap °Napj]=a*ij`bkm}a>ipqwooa{ =?>A@I(¥ 0 iq|=vjwT§Pa,
Na¢¯wPvjaP¨ {ikDowP`2{}=kypj{}Uijap>¥ {}ipj]l=i{}!2¥<F{pt]a,v + > |vtwT§{=>/"!wmv $#+ >R¨
{}`b|}r4{}Dpt]=kTp > ¡^kmia,}a>(pta> r vjl=a«-> {}i2kD`*lpt{|Nwm{}4p*vta,k[rUwm¢¸¥ 0 i ¡kPipj]=a9=v"isp
=a,{}m]~wPvbwy¢">³vja @ptvtkm=i`b{ppt{¸pj]=aD|=km"¤Pap>¨>³`l=ip/]=k[§maDoa7{oa> pjw±vtaptvtkm=ij`2{p>¨ky=¶ky4ky{}
>!2¥B\^]{i|vtwT§ma>i- %2¨=kP"]{}a,§{=pt]aG|vtw4wm¢wm¢ +"2¥
\^]a`l=pt{|Nwm{}4p2vta,k[rU£~w4woo{}U¢¯wmvt¡kmvto{}Dvtl}a,km Naija,a> kmi2k;|~kyvjpj{lkyv2,kmijawm¢qwmlv
,wm=a>(pta> wm`b{~kTpj{}Uijapbky}mwPvj{pj]` ¡]a>vja/pt]awm}r wooaa>a7(pja7±r¸vtl}aUc{}ipj]aiwPlv"am¥
\^]={}iG{ipt]4l~iGwyp2ilvt|vt{}ij{}pt]=kTp*`lpj{}|~wP{4pvta,k[rD£=wwoo{} m{}§ma7i~a,ppja>vvta>ijlptiG¡]a> wooa>i
kmvjawPvta,vta>bkyvt{pjv"kyvt{}rij{=,aqkGwooaª{}ia,}a>pja>br2vjl=acq¡{pt]|vtwm=km{}{psr'4% '& Eó¡]a,vta(&{iF{pti
a,mvta,a4G(¥\^]ak[§ma>vtkmmael`Na,vªwy¢Hwooa>iqa,}a>(pta> rvtl}acm¨~{}ipj]a>;ky||vtw[no{`/kypja,}r*)%,+ ¡]a>
kmX=wa>i^]=k[§Pakboa>mvta,a}wPijaepjwbpj]=ak[§ma,v"kyPaqoa>mvta,a+¥
N^kmija>wPpt]akm~wT§PaGvja>`bkmvj¤M¨o¡a,kmD|vjwP|~w4iaGkbNapjpja,vªwmv"oa,v.pj]=kmv"ky~owm` ¢¯wmvq,wm`b|lopt{
pt]awPa7(pja7wm`b{~kTpj{}/ijap.¡]a>km¬=wa>i^¤wT¡ pt]a,{}v.o{isp"ky=,a¢¯vjwP` pj]=aiwPlvt,am«
¢¯wmvGps¡wDwooa7i >8>-kyp*o{isp"ky=,a>i3- 3.B¢¯vtwm`°pj]aijwml=vt,am¨R¡a/]=k[§Pa >/>-{¢.ky~ wm}r;{¢
3 3XwPv3-+/3¬km=pj]aG´ wy¢E>D{i.ij`/ky}a>v¦pt]=kypt]aG´wy¢ >-9¥
D
l="]ky/wPvta,vt{G¡wml=}/vta>ijlp|NwPiti{}}r2{}k*i`/ky}}a,v,wma7(pta>owP`b{=kypj{}*ijapi{}=aqpj]aªwm}r
=waGa,}a>(pta> rvtl}acG¡wml~aGpt]aijwmlv"aP¥¦zqa,vtapj]aG¢¯wPvj¡^kyv"o{}2vjl}a¡wmll=ijapj]aijwml=vt,a
wm¢~pt]aq=vjw4km>kmipH|~km"¤ma,pHpjwoa7{oa.¡apt]a,vBpt]a.wooaq{}iBwPvFwmp{bpt]aq,wma7(pta>/owP`2{}=kypj{}ia,p>¥
P + T
+
R,&' $
²)aGp"ky¤makPi`bwooa,aps¡wmvt¤wy¢¨Pm2wooa>i.vtkm=owm`b}rkm=l{¢¯wmvt`2}r{;k/it©4l=kyvta 9=a,.¥
0
|=km{vwm¢wooa{i^{ v"kyPaªwm¢a7km"]wmpj]a>v¡]a,pj]=ao{}iptkm=a{ii`/km}a,vpj]~kyDk 9noa>vtkPo{l~i2¥
²)a,ky}Xoa,~i{psr pt]a©Pl~ky4pj{psr*
¨pj]=kyp{i¦pj]ak[§ma>vtkmmaa>{P]4Nwmvt]wwoij{ >aqwy¢$kwooae{}
pt]aa,4pta,v^wy¢¬pt]aij©4l=kmvjaR9=a,X¥H²¸avtlo{Ma,vta,4p^ij{}`lkTpt{wP¡{pj]{Ma,vta,4p^§[kml=aªwm¢Ú{wmv"oa,v
ptwDa,p §Tkyvtrm¥\^]ak[§Pa,v"kyPaa>{P]4Nwmvt]wwoUij{ >ab¢¯wmvkm4rUwooa/{}ii`/ky}}a,vpj]~ky ij{=,aiwP`ba
=wa>i]~k[§mabpj]=a,{}va>{P]4Nwmvt]wwo lopr¸pj]a~wPvta,vwm¢pt]a9~a,X¥ J${PlvjaDci]wT¡.iGpj]akm(ptl=ky
k[§Pa,v"kyPa.a,{}m]Nwmvij{ >ae¡{pj]vta>ij|~a7(p^pjw/oa>=i{psr )¥B\^]aG~wPvta,v^aMa>p^Na>,wm`ba>ikm||=kmvja>Pp^¡]a>
/4¥
´µ=9=PlvjaD¸¡¦aDo{ij|}k[r pj]a;k[§Pa,v"kyma4l=`Na,v/wy¢ ufÉ|Na,vwooa l=ij{} mvta,a>r ky}mwmvt{pj]`
km=¸`2{} {} kmPwmvt{pt]` ¥2´pGptlvt=iwmlopGpt]=kTpGpt]a/l`Na,vGwm¢ uf wooa>i{iG`l~"])ij`/ky}a>vpj]=km
pt]ak[§ma>vtkmma.=a,{}m]~wPvj]=w4woi{} ,aP¨o{}l~isptvtkypj{}2pj]aG|Na,vj¢¯wmvt`/ky=,aªwm¢Rpj]=a
ufÚia>a7(pt{wP¬¥
0
r]wT¡
pt]aGmvta,a7orkmPwmvt{pt]`!{i`bwmvtae|Na,vj¢¯wmvt`/ky4ppj]~kypj]aG`b{}{}DkmPwmvt{pt]` ij{=,aG{p.a>{}`b{~kTpja7iwP
wP|opj{}`/kyXwooa7ipj]=kyp.¡¦wPlNaG{~ia>vpta>{pj]=aG`2{} {} kmPwmvt{pt]` ¥
J${}mlvta - ij]wT¡.i¦pj]aGk[§Pa,v"kyma.¢¯vtkP(pj{}wmwy¢$wooa>i^{}pt]aaps¡wmvt¤2pj]=kypvja,pjv"ky=ij`b{p^{Dkm uf
£~w4woo{}~¥?Lqwypj{a.pj]=kypFpj]aq|~a>v¢¯wPvj`/ky~a^vtkypj{}wG~a,ps¡¦a>a,/Pvja>a>orkmPwmvt{pt]`®km=2`b{ {/ky}mwmvt{pj]`
{iij{`b{}}kmv^pj]=kmpt]aGv"kTpj{}w2~a,ps¡¦a>a,Dk[§ma>vtkmma u¦fia,pij{} ,a>i>¥
J${}mlvta; ij]wT¡.ibpj]=a;pjwmptkm.l`Na,vwy¢Gwooa>i{épt]aU=aps¡wmvt¤±pj]=kypkmvja u¦f wy¢kTp}a>kPisp
wPa/wooa{} pj]=aa,ps¡¦wPvj¤M¥Lqwypt{},a/pj]ailvt|vt{}ij{} vja7ilptipj]=kypmvta,a7orUkmPwmvt{pt]` km= `b{{
kmPwmvt{pt]` |vtwT§{}a>i2§ma>vjr ij{}`2{}kyv/vta>ijlp"ikypj]=wmlP] pt]aUk[§ma>vtkmma4l=`Na,v"ibwy¢ uf |Na,vwooa
Pvja7kTptr/o{Na>v>¥F\^]=aªvta>kPiwP{}i¦pt]=kTp^pj]=ae`b{}{}ky}mwPvj{pj]` ,wml4pja>vj~kyky=,a>i¦{p"i¦}wT¡ |Na,vj¢¯wmvt`/ky=,a
r2¢¯wmv"{}*wooa>ipjw2~a ufqi¦wm¢Ria>§ma>vtkm=wooa7i,¥Bz.wT¡a,§Pa,vpj]aowm`b{}=kTpt{=*ijap^`bkPoaqr/ky}Npj]a
uf.i{i^wyp.|Na,vj¢¯wmvt`/ky4p$Epjww/`/kyr=wa>iG(¥
J${}mlvtaªi]=wT¡.iXpj]=a¦¢¯v"kmpj{}wmwm¢wooa>i${}pt]a¦=aps¡wmvt¤ªNa,}wmP{=.pjwªpj]=a u¦f owP`b{=kypj{}ªijap
¡{pj]*vjl=a7i¦c^ky~*o¥?L.wmpj{apj]=kyppt]a uf¶owm`b{}=kTpt{=ªijapBmvta>kypj}rewPlopj|Na,vj¢¯wmvt`/i¬pt]a^owm`b{}=kTpt{
ijapª`bkPoawm¢Hkm
uf.i>¥.\^]aPvja>a>orky}mwmvt{pj]`!wPlopj|Na,vj¢¯wmvt`/ipt]a`b{9{Dky}mwPvj{pj]` ¡{pt]{}Dpj]a
itky`bav"kTpj{}wbkmipj]=ak[§ma,v"kyPa
u¦fÔijapqij{} ,avtkypj{}w=¥
J${}mlvta$S2,wm`b|=kmvja7iFpt]a¢¯vtkP(pt{wPwm¢wooa7i^~a>wPm{}*pjw2pj]a ufÚowm`b{}=kTpt{/ijap¡{pj]pj]a
¢¯v"kmpj{}wmwy¢Hwooa>i^{}`b|}{a7 {Dky ufÔ£~w4woo{}2E¯Pvja>a>orky}mwmvt{pj]`!{} ~wmpj]D,kPia4G(¥Lqwypj{apt]=kTp
pt]a uf¹£=wwoo{}bwmlopt|~a>v¢¯wPvj`/ipt]a ufÚowP`2{}=kypj{}/ia,p.lop.wmpqi{}m{9~>ky4pj}rm¥
J${}mlvta.i]=wT¡.i¬pt]a¢¯v"kmpj{}wm*wy¢wooa7i{}pt]a^aps¡wmvt¤e¡]{}"]*~a>wPªpjw.pt]a uf owm`b{}=kTpt{
ijap¡{pt]vta>ij|~a7(pptwvtlao¥ L.wmpj{a.pt]=kTp¦pj]{i¦¢¯vtkP(pt{wP/{}i¦i{}`b{kyv¦pjwGpt]aª¢¯vtkP(pj{}wm/wm¢¬wooa.{}`b|{}a>
{}Dky uf¹£=wwoo{}b¡]a,pj]=aoa,=ij{psr{}i.{`b|Nwmvjptky4p7¥
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvtaDcm«*k[§ma,v"kyPa*=a,{}m]~wPvi{} ,a Eó{}v"}a>iGky= k[§ma>vtkmma2a,{}m]Nwmvt]wwo¸ij{ >awy¢kDa>4pjv"kyFwooa
E@ky]=a> G§Pa,v"il=i^oa>=ij{psrP¥
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvtao«Rk[§Pa,v"kyma uf¶i{} ,a^¡{pj]2mvta,a>rkmPwmvt{pt]` E¯|=}km{G$km=¡{pj]b{}9{kmPwmvt{pt]` Eó=kmij]a> G
§Pa,v"il~ia,=ij{psrP¥
0 20 40 60 80 100
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvta - «$k[§Pa,v"kyma¢¯v"kmpj{}wm/wy¢Xwooa7i{}/pt]aqa,ps¡¦wPvj¤*{`b|}{}a>/{}ky ufé£=wwoo{}=¨4¡{pj]Pvja>a>or
kmPwmvt{pt]` E¯|=}km{G^km=¡{pt] { {kmPwmvt{pt]` Eó=kmij]a> G¨o§ma>vtijl=ioa>=i{psrm¥
0 20 40 60 80 100
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvta^=«Hk[§ma>vtkmmaF¢¯v"kmpj{}wmbwy¢Mwooa7iB¡]{"]bkmvja uféwy¢XkTpF}a>kPispFwPawooa{}bpj]a.=aps¡wmvt¤N¨T¡{pj]
Pvja>a>orkmPwmvt{pt]` E¯|=}km{G.ky=¡{pt] { {kmPwmvt{pt]` Eó=kmij]a> G¨§ma>vtijl=i^oa>=i{psrm¥
0 20 40 60 80 100
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvta;o«;k[§Pa,v"kyma¢¯vtkP(pt{wP wm¢ewooa7ib¡]{"]ékmvja {}¶pj]a u¦f owm`b{}=kTpt{±ijap7¨¡{pj]éPvja>a>or
kmPwmvt{pt]` E¯|=}km{G^km=¡{pt] { {kmPwmvt{pt]` Eó=kmij]a> G¨o§ma>vtijl=ioa>=i{psrm¥
0 10 20 30 40 50 60
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvtaS«Hk[§ma>vtkmma¦¢¯vtkP(pt{wP/wy¢Xwooa>i{`b|}{}a>/{}ky u¦fé£=wwoo{= E¯|ky{}GBkm=/k[§ma>vtkmma¢¯v"km(pt{wP
wm¢wooa7iNa,}wmP{*pjwbpt]a ufÚowP`b{=kypj{}/ia,p EókPi]=a> G¨§ma>vtijl=i^oa,~i{psrm¥
0 10 20 30 40 50 60
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
J${}mlvta2o«k[§Pa,v"kymae¢¯v"kmpj{}wm wy¢Fwooa7i.Na,}wmP{=/pjw/pt]a ufowP`b{=kypj{}ijap Eó|ky{}G(¨N¢¯v"km(pt{wP
wm¢wooa7iNa,}wmP{*pjwbpt]aowP`2{}=kypj{}/ia,p^¡{pj]Dvtl}aE@kmij]a7 G(¨§Pa,v"il~ia,=ij{psrP¥
¦'R,
²)ae]=k[§maª|vja7ia>4pja>k§Pa,vtr2|Na,vj¢¯wmvt`/ky4pkmPwmvt{pt]` ¢¯wmvwP`b|lopj{}bk,wma7(pta>wm`b{~kTpj{}bia,p>¥
wP4pjv"kyvtr/pjw|=vja>§4{}wml~i¡¦wPvj¤oiwP pj]a7ia*ilxsa>(pqpj]a2ky}mwmvt{pj]` wm}rvta>©4l{}vta>ips¡w ]wm|pjwP|~wPwPmr
{}o¢¯wPvj`/kTpt{wP=iFkm=bwijr4~"]vjwP{} >kTpt{wP2km=b{pja,v"kTpt{wP=i>¥$\^]a7ia.kPijijl`b|opt{wP=iF`/ky¤Pa^pj]aªky}mwmvt{pj]`
§Pa,vtr*kyppjv"kmpj{}§ma¢¯wPv`bwm={}aqkm]woaps¡wmvt¤oiFij{=,aq{p¦a>a>i¬xsl=ip¦}w>ky~l|N=kTpja7i¦kyp¦a7km"]oa,pja>pja7
ptwm|Nwm}wmPr "]=kyPam¥³ªlvb¢¯lopjl=vja ¡wmvt¤±¡{}~a ptw `bkm¤ma ©4l=ky4pj{ptkypj{}§maDwP`b|=kyvt{}ijwm~i¡{pj]éwypt]a,v
,wm=a>(pta>owm`b{}=kTpt{=bijap.]=a,lvt{}ipj{,i 1}cmcm¨ - 3
RÉ T
+
1c3"¦\
D ´ D \
µf D c7
wP`2`b{ppta,am¥f.kmo{}wa7©Pl={|`ba>Ppky~iroipja>`bi>«Hz.{}m]|Na,vj¢¯wmvt`/ky=,a.vtkPo{}w
}wo,kmkyvta>kªaps¡wmvt¤E¯]{}|Na,vt}kmGpsr|~acm¨y¢¯l=(pt{wP=ky~i|Na>,{9~>kTpt{wP=i,¨PlaGc7mS¥ \ D-
P SPm¥
13 ¥ D ¥ wPvtijwmDky~=¥ kP"¤ma,v7¥ wP{}akPD]woa,ps¡¦wPvj¤{} E 0 L ¦\G(«¬fwPlopj{}|vtwyptwowm
|Na,vj¢¯wmvt`/ky=,a{}itil=a>iky~a>§[kml~kTpj{}wm wP=ij{}oa>vtkypj{}wm=i>¥HfJ my=cm¨4kyl=kyvtrDc>Pm=¥
1- 3 D ¥ ªkypp"k¨R´(¥ D pjwyxs`ba>wT§{}m¨Rky=~¥X²¸lR¥/´µPpta,vt=kyHwooakm=)i]wPvp"lopG=kPia7Uvtwmlpj{}¡{pj]
Pl=kyv"ky4pta,a> oa,}{§ a,vtr±{é¡{}vja>a7iji*aps¡wmvt¤oi,¥ ´µ *9 8K7 : 7 :7$ X (X
) 4 X ¨ 0 |vt{FcKS "c>2ym=cm¥$uF]wa,={nX¨ 0 fG¥
1 3 D ¥ ªkypp"k¨R´(¥ D pjwyxs`ba>wT§{}m¨Rky=~¥X²¸lR¥/´µPpta,vt=kyHwooakm=)i]wPvp"lopG=kPia7Uvtwmlpj{}¡{pj]
Pl=kyv"ky4pta,a>³a,}{§Pa,vtr {}Þ¡{}vja>a7ijiaps¡wmvt¤oi,¥´µ * 8K7X 7:7 X (X
) 4 X
¨ 0
|vt{FcKS "c>2ym=cm¥$uF]wa,={nX¨
0
fG¥
13 D l=o{}|opjwel=]=kGky= D ky`b{v]l}}a,v7¥ 0 |=|vjw[no{}`/kTpj{}wm/kmPwmvt{pt]`/iR¢¯wPvwPa7(pja7bowm`b{}=kTpt{
ijap"i,¥B´µ 4 X X '& ¨=|=kmma7iªcTTNc> - ¨~c>mS¥
1S3u]{}{}||~a4km>©4lap7¨TuHkPij>ky}a {a,p>¨yuBkml l]}apt]=ky}a,v7¨mky~ L.{wP}kPiHf{§{}a,vtvjaP¥´µ=vta>kPi{}qvta
}{ky{}{psr2{>ky}a 9¢¯vta,a^vtkPo{}w "!$#¬i,« H¬wT¡¶a>§ma>o¢¯wmvt¡kmvt{{}&%('*),+.-/"!$#B¥10 W: *
" " ¨=EcKG«S4 P¨2Pkml=kyvtr c>m¥
1Ø43
0
¥Hkywml={pt{@¨
0
¥23ek[rrl` ¨mkm= HF¥"4ª{a>wmp>¥ lpj{}|~wP{4p¦vta,k[r{==«
0
/a·{}a,4pBpja7"]{©4la^¢¯wmv
£~w4woo{}/{} `2wP{}a¡{}vja>a7iji^a,ps¡¦wPvj¤oi>¥H´µ576 '& .8 # 9X X ":
4 :"X ;<89.>=?2@"@BADC ¥=´B / wm`b|lopta,v D wo,{a,psrm¨=yPcP¥
13Gf¥XªP{a>v>¨ J¥N\Ra>`2|={}¬¨ Ne¥Na,}l=v>¨Nky= ¥H¬a>¡{}i>¥\RwP|~wPwPmrvtwPkP,kPisp.~kmija>wmDvta,§Pa,v"ia
|~kTpj] ¢¯wPvj¡^kyv"o{}=¥¦´µ4pja,vta,pqv"kT¢p7¨~ov"kT¢p 9{}apj¢9`/ky=ap 9pj=vj|o¢µP¥pjnp>¨
kmvt"]UbmmP¨²)wmvt¤{}
|=vjwPmvta>iti,¥
13´(¥ D ptwyxs`ba,wT§{y¨ ¥ D a7o{}m]¬¨ky~=¥7EMl{y¥X´µ4pja>vj=kmwooa.~kmija>2vtwPkm=,kmipj{}Gky}mwPvj{pj]=`bi
{}±¡{}vja>a7ijiG=aps¡wmvt¤i>¥/´µ *9 KK7 FG8 # H: 7,:7 4 X I": : ¨94kyl=kmvjr
mm=cm¥
1}c>3euH¥J~¥m² kyR¨Bb¥ ¥ 0 } ,wPl{9¨Pky~/*¥Jvt{a7oa,v7¥ wPa7(pja7bowm`b{}=kTpt{ijapF{}/¡{}vja>a7ijiBkP*]=w
=aps¡wmvt¤i>¥$´µ KMLONQP >P ( ¨,PlayPP¥?Lqa,¡SRFwPvj¤M¥
1}cmc3 ~¥X²)l¸ky~¸z¥HR{@¥bª¸,km},lkTpj{} wPa>pja7UowP`2{}=kypj{}ijape¢¯wmva,·/,{a>4pevtwmlopt{={}¸kP
]=w¡{}vta,}a>iti¦=aps¡wmvt¤i>¥$´µ
* 8K7(1
(
¨|=kyPa>i.Mc,~¨
0
lml~ispc7mm=¥
D
a>kTptaP¨V5 D 0 ¥
Unité de recherche INRIA Rocquencourt
Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
Éditeur
INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399