• Aucun résultat trouvé

Complexity Results on Election of Multipoint Relays in Wireless Networks

N/A
N/A
Protected

Academic year: 2021

Partager "Complexity Results on Election of Multipoint Relays in Wireless Networks"

Copied!
16
0
0

Texte intégral

(1)

HAL Id: inria-00073097

https://hal.inria.fr/inria-00073097

Submitted on 24 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.

Complexity Results on Election of Multipoint Relays in Wireless Networks

Laurent Viennot

To cite this version:

Laurent Viennot. Complexity Results on Election of Multipoint Relays in Wireless Networks. [Re- search Report] RR-3584, INRIA. 1998. �inria-00073097�

(2)

ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Complexity Results on Election of Multipoint Relays in Wireless Networks

Laurent Viennot

No 3584

Décembre 1998

TH `EME 1

(3)
(4)

(!)*+,! - ./( 1023546#.879

:3;5<8=?>A@CB9DFEG>A@/@8HBJI

K(LNM?OQP'RTS UV?WXPZY\[A]FP!^*WX_AWX^`M?OQP?W

acb`dJeXP ^gf*hiakjlU3m(n#o

U*Y\pNp5dqbX^3rNPbXP?stLNP!b`stLNPvuwyxyz\{}|~S gV!s?P?OQ€NbXP9R‚\q{#S R„ƒ…p†Y\‡qP!W

ˆ‰/Š!‹ZŒJNŽ\‹y

K(LNP9P?‘P!s!^`’‘dqu“d\”3O[N•^`’‘p5dq’‘uy^TbXP?‘Y‚_AWY\‘•d‚–W—^`d˜r™P?s?bXPZY\WXP9^`LNP9s!dqWX^d\”

€Nb`dyY}rNsZY\Wš^›’•uN‡…’‘u9–’‘b`P!‘P!W`W3uNP ^œ–d\b`žW?Ÿ  †dqbP?Y\stLW`dq[NbXs?P}¡™^›LNPT”GP –P!b(P?•P?OQP?uy^›W^`LNPTWXP!^

L†Y\W?¡5^›LNP‡qb`P?Y}^›P!b¢^›L™P…‡yY\’•u£’‘W!ŸThœu£^`LN’‘W¢p†Y\p5P?b?¡¤–P…p™b`d‚¥qP…^`L†Y}^g^`LNP…s!dqO9p™[™^tY}^`’‘dquFd\”

Y˜O…[N¦^›’•p¤dq’•uy^§b`P!¨YZ_“W`P!^–’•^`L©OQ’‘uN’•O+Y\8W`’•ª?PQ’‘W«ga/¬œs!dqO9p™‘P!^`P\Ÿ­®P+Y\•W`d˜O+Y\}PQ^`LNP

Y\u†Y\¦_¯WX’‘W(d\”/Y°LNP?[™b`’‘Wš^›’•s§pNb`dqp5dqWXP?rF€ž_±Ÿ5²§YZ_A_A[NOFŸ

³˜´yµC¶X·…¸ ŒJ¹8Š\ O…[N•’•^›p5dq’•už^ b`P!¨YZ_q¡N–’•b`P!‘P?WXW(uNP!^œ– dqbX5¡¤«ga/¬ºs?dqOQpN‘P ^›P}¡™LNP![Nb`’•WX^›’•s\Ÿ

»½¼8¾œ¿œÀ‚Á(¾kÂ8濜ÄÆÅ!Ç

È/ÉÊ(ËÌÎ͕ÏÐ‚Ñ Ò‚Ó„ÔՂÖC×ÙØqÚtÔ ÕZÕJÛ!ւÜJÚ`Õ?ÓqÚÑ×ÙÝ!Ó

Unit´e de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : 01 39 63 55 11 - International : +33 1 39 63 55 11

T´el´ecopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30

(5)

!&#?°.8(?)—F)#, ,.)#+ )F

Š V!‘P?s ^›’•dqu9r™PgO…[N¦^›’•p¤dq’•uy^›W bXP?¨Y}’5p5P?bXO9P ^lrNPgrN’•O9’•už[NP?bk [™^`’‘‘’•W›Y}^`’‘dqurNP

¨Y+€†Y}uNrNP°p†Y\WXW›Y\uy^`PQrNY\uNW§‘P!WTb`V!W`P?Y\[¯]~W`Y\uNW† Ÿac•[NW§s?P ^—P?uNWXP?O…€N•P…P?Wš^#p¤P ^›’•^gp¤d\[Nb

stL†Yž[NPW`dq[™b`s?P}¡pN•[NW…•P‡yY\’‘u©P?WX^Q’•už^`V?bXP?W`W`Y\uy^ZŸ «*dq[NW°pNbXdq[™¥qd\uNW'’‘s?’A[™PF•PsZY}‘s?[™

r [™u®P!uNW`P!O…€N‘PQrNP'O…[N¦^›’•p¤dq’•uy^›WgbXP?‘Y\’ rNP'^tY\’•‘•POQ’‘uN’•O+Y\•PP!WX^«ga/¬ºs?dqOQpN‘P ^ZŸ'«*dq[NW

”ÆY\’‘WXdquNW(rNP—pN•[NW( Y\u†Y\¦_AW`P§r [NuNP#LNP?[™b`’‘WX’•^`’ž[NPTpNbXdqp5dqW`V!Pp†Y}b(±Ÿ5²§YZ_A_A[NOFŸ

¸ ‹„Š Ž  O…[N¦^›’‘p5dq’•už^b`P!¨Y\’¡™b`V!W`P?Y\[˜W`Y\uNW † ¡†«ga/¬œs!dqOQpN‘P ^Z¡NLNP![Nb`’•W`’¦^›’ž[NP}Ÿ

(6)

; <

' .8 Fl!

K8d'W`[™pNp¤d\bX^(^›L™P§O9P!stL†Y\uN’•W`O d\”8”Gd\bX–Y\b`rN’•uN‡…’•u€Nb`dyY}rNsZY\Wš^dqb(O…[N¦^›’•sZY\Wš^3’‘u–’•b`P!‘P?WXW

uNP!^œ– dqb`žW!¡k^`LNP?bXPO…[NWš^°P ]¯’•WX^…WXdqOQP+”GdqbX–Y\bXrNP?bXW–LNd –’‘•c”GdqbX–Y\b`r®^›LNPpNY\st\P ^›W°d\”

dquNPuNdArNP^›d“Y\uNd\^`LNP?b'uNdArNP}Ÿ hœu W`[NstL uNP!^œ– dqbXAW!¡(Y~uNd¯r™PFLNY\W'O+Y\uy_ u™P?’‘‡\LA€5dq[NbXW

=

!>! ?>

uNd¯r™P?W3–’•^`L–LN’‘stL’¦^(sZY\us!dqO9O[NuN’‘s?Y}^›P¢rN’‘bXP?s ^›•_A@¡™W`d°^`L†Y}^Y…WX[N€NW`P ^*d\”^›LNP!O

L†YZ¥qPg^`d€5PgP?•P?s ^›P?rY\W ”Gdqbš–3Y}b`rNP!b`W ^`d…YZ¥qd\’‘rbXP?rN[Nu™r†Y\uy^3bXP!^›b`Y\uNWXO9’¦^›’‘d\u9d\”C^›LNP¢W›Y\OQP

O9P!W`W›Y}‡qP\ŸCB¯[NstLQP?‘P!s!^`P?r°uNdArNP?WkY}b`P3s?Y\‘•P?r ()**D45 -,./ ’‘u^`LNPf¢hiakjkU ±g«FE΃HG

WX^tY}uNr†Y\bXr£Y\u™rF’•u^›LNP—rNb`Y}”Ù^›Wd\”^›LNPo˜±g«gjcK ‡qbXdq[Npd\”^›LNP—hijcK IE|.GŸ

KdFP?uNWX[Nb`P'€NbXdyY\rNs?Y\WX^`’‘uN‡N¡ ^`LNP°O…[N¦^›’‘p5dq’•už^gb`P!¨YZ_AW#O[NWX^—€5P'W`P?•P?s ^›P?r“W`[™stL®^›LNY}^

^›LNP*O9P!W`W`Y\‡qPg’•Wl^`b›Y\u™W`OQ’•^`^`P?r'^›dY\‘N^`LNPgu™d¯rNP!W –’¦^›LN’•uQ^œ– d…LNdqp™Wl”GbXdqO ^›LNP¢W`dq[™b`s?PKJqŸ

hœu“d}^›LNP!bT–dqbXrNW?¡Y\uy_&u™P?’‘‡\LA€5dq[Nbgd\”3Yu™P?’‘‡\LA€5dq[Nb§d}”ML®O…[NWš^—€¤P°^`LNP°uNP?’•‡qLž€¤d\[NbTd\”

W`dqOQP°O…[N¦^›’‘p5dq’•už^¢b`P!¨YZ_ d}”LŸ°hœrNP?Y\‘¦_q¡ YOQ’‘uN’•O…[NO už[NO…€5P?b—d\”lO…[N¦^›’•p¤dq’•uy^Tb`P!¨YZ_AW

–dq[™‘r€5PgWXP?‘P!s!^`P?rW`d…^`L†Y}^ ^›LNP¢s?dqOQO…[NuN’•sZY}^`’‘dquQ€†Y\u™r™–’‘r™^`L’‘W d¯s!s?[Np™’‘P?rY\W•P?W`W3Y\W

p¤dqWXW`’•€N‘P}Ÿ

hœu®^›LN’•Wp†Y\p5P?b!¡/– PW`LNd‚– ^›LNY}^ †u™rN’‘uN‡£Y\u dqp¯^›’‘O9Y\cO…[™•^›’•p¤d\’‘uy^—b`P!¨YZ_“W`P!^

=

!>! ?>

Y W`P!^°–’¦^›L Y OQ’‘uN’•O+Y\cuž[NO€¤P!b°d\”*O…[N•^`’‘p5dq’‘uy^b`P!¨YZ_AW°P?uNWX[Nb`’•uN‡&€Nb`dqY\rNsZY}WX^›’•uN‡N@’•W

Y\u «¢a ¬ºs?dqOQpN•P!^›PpNbXdq€N•P?OFŸ%o˜dqbXP?d‚¥qP!b?¡3– P£rN’•W`s![NW`WQ^›LNP A[NY\‘’¦^œ_ d\”§Y}u LNP![Nb`’•WX^›’•s

pNb`dqp5dqWXP?rF€ž_²—YZ_ž_A[NOOE|.G Ÿ P ^¢[™W Nb`WX^*‡q’¦¥qP#Y…”Gdqb`O9Y\¤rNP †u™’•^›’•dqud\”^›LNP—pNbXdq€N‘P!O˜Ÿ

P Q ˜!

hº”RL ’•W…Y£u™d¯rNP+d\”(^›LNP+uNP!^œ– dqb`¡c– PrNP?u™d\^›P€y_TSTUVLXW—^›LNPWXP!^°d}”’•^›WuNP?’•‡qLž€¤d\[Nb`W!¡

!>! ?> ^`LNP&uNdArNP?W^`L†Y}^s?Y\us?dqOQO…[NuN’•sZY}^`P rN’•b`P?s ^›¦_ –’¦^›L%’•^?ŸYS#UVLXW+’‘Ws?Y\‘•P?r ^`LNP

uNP?’•‡qLž€¤dq[™b`LNdAd¯rFd\”CLŸ

=

f*P?bXP– Ps?d\uNW`’•rNP?b*^›L†Y}^ZL2[

\

S#UVLXW Ÿ)@FK(L™PuNP!’‘‡qLž€5dq[Nb`W*d\”CL

Y\b`P—Y\•W`d°s?Y\‘•P?rdqu™P§LNdqpuNdArNP?W*Y\uNr^`LNP§uNP?’•‡qLž€¤d\[Nb`W(d\”8uNP!’‘‡qLž€5dq[Nb`W3–L™’‘stLFY\b`P§u™d\^

uNP?’•‡qLž€¤dq[™b`W*d\”]L“Y\b`P#sZY}‘‘P!r˜^œ– dLNdqp£uNd¯r™P?Wg€5P?s?Y\[NWXP^L~L†Y\W*^›dWXP?uNr&LN’•W¢OQP?WXW›Y\‡\P?W

^›d#dquNP(”GdqbX–Y\bXrNP?bk^›d#b`P?Y\stL9^`LNP?OFŸ P!^9S*STULXWkr™P?uNd\^`P*^`LNP^œ–dLNdqpNWk”Gb`dqO_LuNdArNP?W!Ÿ

hº”a`’•WY'u™P?’‘‡\LA€5dq[Nb(d}”aL¡N– P#Y\‘WXd°W›YZ_^›L†YJ^RL s?d‚¥\P?b`WZ`5Ÿko˜dqbXP?d‚¥qP!b?¡’•”Cb Y\u™r3c

Y\b`P(WXP!^›Wkd\”¤uNdArNP?W!¡y–PW›YZ_…^`L†Y}^b~s?d‚¥qP!b`W+c ’ed+P!¥\P?bX_'u™d¯rNP’•uc ’‘Wcs!d‚¥qP?bXP?r9€ž_…W`dqOQP

uNd¯r™P’‘u'b3Ÿ*± W`P ^fbhgFSTULXW¢’‘W¢Y9O…[™•^›’•p¤d\’‘uy^b`P!¨YZ_W`P ^T”GdqbL~’•”Mb s!d„¥\P?bXWfS*STUVLXW¡

dqbP A[™’•¥}Y\‘P!uy^›•_jilkmon+prqs!STU!`AWutvS#UVLXWRgIilkmowS#U!`AW Ÿ ± O…[N•^`’‘p5dq’‘uy^lb`P!¨YZ_9W`P!^(”Gd\b3Y

uNd¯r™PL'’‘W8dqp™^`’‘O9Y\y’•”†’¦^›W8uA[™O…€¤P!b/d\”P!‘P!O9P!už^`W/’•W/OQ’‘uN’•O+Y\qY\O9d\uN‡gY}‘y^›LNPO…[N•^`’‘p5dq’‘uy^

b`P?‘YZ_W`P ^”GdqbxLŸc­©P§–’‘•Cs?Y\‘C^`LN’‘W(už[NO€¤P!b^`LNP—dqp™^`’‘O9Y\CO…[™•^›’•p¤d\’‘uy^bXP?‘Y‚_+už[NO…€5P?b

yzy|{M}~?€‚

(7)

|

”Gdqb B¯P?P“ /’•‡q[Nb`P®R?”GdqbY}u%P ]™Y\OQpN‘P}Ÿ­®P&sZY\uuNd‚– WX^`[Nr™_ ^`LNP&s?dqOQpN•P ]¯’•^œ_ d\”

s?dqOQpN[™^`’‘uN‡'Y\udqp¯^›’‘O9Y\¤O…[N•^`’‘p5dq’‘uy^b`P?‘YZ_W`P ^ZŸ

s N(s)

two hops from s nodes a

b

c

d e

f g

 /’‘‡q[™b`P°Rc± uNP ^œ–dqbX–LNP?bXPY°‘’•uNP§€5P!^œ– P?P?u˜^œ–dQuNdArNP?W*WX_AO…€5dq‘’•ª?P!W3^›L†Y}^^`LNP§^œ–d

uNd¯r™P?Wgs?Y\u s!dqO9O[NuN’‘s?Y}^›P—rN’•b`P!s!^›¦_qŸ(hœO9Y\‡q’•uNP§”Gdqb¢P]™Y\O9p™‘PY9b›Y}rN’‘d9uNP!^œ– dqb`–LNP!b`P

sZY\ubXPZY\stL

Ÿ Y\uNr

Y\b`P*O…[N¦^›’•p¤dq’•uy^cb`P!¨YZ_'W`P ^”Gdqb9JqŸ

’‘WuNd\^¢€¤P!sZY\[™W`P’¦^rNdAP?W*uNd\^¢s?d‚¥qP!b~Y\uNr5Ÿ K(LNPdqp¯^›’‘O9Y\¤O…[N•^`’‘p5dq’‘uy^(bXP?‘Y‚_

uA[™O…€¤P!b(”GdqbZJ#’•W™Ÿ ’‘WY}uFd\p™^›’•O+Y\¤O…[N¦^›’‘p5dq’•už^ b`P!¨YZ_WXP!^”GdqbZJqŸ

0! † 3

­©P—pNbXd„¥\P§^›L†Y}^^`LNP§”Gdq‘•d„–’•uN‡…pNbXdq€N•P?O’•W«ga/¬œs!dqOQpN‘P ^›P

G‹#"%$ ¸ "%&¤‹ ´ ƍ µ  ' )(. T,j !8945:+* !>! ?>0-,* f/ %$0 /.,10(4%$ ,( ,

32( 54#6l, 32(

L

%$),* !8945:^,.72 ,. *% 8.( 4:96a -,* 45 K,^!)**l45 -,.f/

$o4 L %$ /; f 0-,A,.<9>=

 8’‘b`Wš^gd}”lY\‘ ¡5u™d\^›’•s?P—^›L†Y}^¢^›LN’•WgpNbXdq€N•P?O ’‘W*PZY\WX’‘P!b¢^›LNY\u£^›L™P…pNb`d\€N‘P!O d}” †uNrN’•uN‡

Y\u+dqp¯^›’‘O9Y\†O…[™•^›’•p¤d\’‘uy^kb`P!¨YZ_'W`P ^ZŸchº”Y\u9dqp™^›’•O+Y}†W`P!^’‘WlžuNd‚–u ¡NW`’•O9p™•_°s!dqO9p™[™^›’•uN‡

?-@zyA?-B

(8)

’•^›W W`’•ª?P§Y\u™rFs!dqOQp†Y\b`’•uN‡’¦^^`d 9 Y\‘•d„–Wk^›d'Y}uNWX– P?b3^`LNP ž[NP?Wš^›’‘d\u Ÿ P!^([NW(u™d„– W`L™d„–

^›L†Y}^(^`LNPo˜[N¦^›’‘p5dq’•už^3U*P!¨YZ_acb`d\€N‘P!O’‘W«¢a/¬œs?d\O9pN•P!^`P\Ÿ

hº^’•Wd\€ž¥A’•dq[NW`¦_§’‘u#«¢a£W`’•uNs?Pk^tY\ž’•uN‡*Y*b`Y\uNrNdqO W`P!^8’‘u0S#UVLXW ¡}dquNPksZY}uP?Y\W`’••_§stLNP!st

’‘u“p¤dq¦_AuNdqOQ’¨Y\/^`’‘OQP'’•”3’¦^’•WYFO…[N¦^›’‘p5dq’•už^Tb`PZY}¨YZ_~W`P ^°Y\uNr©’•”3’¦^›W#W`’•ª?PQ’‘W#‘P?WXW^`L†Y\u

9 ŸKdpNb`d‚¥\P'^`L†Y}^—’¦^T’‘WT«ga/¬œs!dqO9p™‘P!^`P\¡C– P'pNb`d‚¥\P'^`L†Y}^T^`LNP…”Gdq•‘d‚–’‘uN‡+gdqOQ’‘u†YJ^›’‘u™‡

B¯P!^akbXdq€N•P?O –LN’‘stL ’‘W+žuNd‚–u ^`d €¤P£«ga/¬œs!dqOQpN‘P ^›PvEÎzG¢sZY}u €¤P˜b`P!rN[Ns?P!r%^›d®^`LNP

o˜[N•^`’‘p5dq’‘uy^3U*P!¨YZ_acb`d\€N‘P!O’‘up5dq•_AuNdqOQ’¨Y}^`’‘OQP

¸ "%&¯‹" & ´ ‹#Œ ¸ ‰ ´  ' )(. ,<..4,, * !>! ?> , / 7%$ 32( ,.72 ,

/ %$7 /.,10(45]$o4 ,( ,3 32( 54 ,.72j, *( 0 4 96+0),* 45 ^,<2(,. ./ %$

,.4 2.,.! f),A,.>9<= 1 ,* 4 0, 2(,. ./ + , / b %$7 32( K* ,3),A,.

,.*j 32( %$ ),* ..4 ,, -,* 40 b 4 v),* /.,104 ,*/32%$7/ ^ 32(

b >

P ^ €5PgY#‡qb›Y\p™L9–’•^`L9u™d¯rNP*W`P ^ Y\uNr+•P!^YULXWlrNP!uNd\^›P*^›LNP*uNP!’‘‡qLž€5dq[Nb`L™d¯dAr

d\”kY\uy_3L

\

QŸ­®Ps!dquNWX^`b`[Ns ^—Y9bXP?rN[™s!^›’•dqu&Y\W”Gdq•‘d‚–W?Ÿ P!^g[NWgO9Y\}P…Y9s!dqpy_Fd\”

Y\uNr~rNP?u™d\^›P…–’¦^›L“YpNb`’•O9P^›LNP°s!dqpN’‘P!W3Lr™P?uNd\^`P?W§^›L™PQs!dqpy_ d\”ML“”Gdqb§Y\uy_'L

\

Y\uNrb

r™P?uNd\^`P?W(^›L™P#WXP!^*d\”8s?dqp™’‘P?W(d}”8^`LNP—P?‘P!O9P!uy^›W(d\”/Y\uy_W`P ^b g

=

rNP!uNd\^`P?W

^›LNP—W`P ^gd}”cY\•C^›LNP#s?d\pN’‘P!W@ Ÿ P!^J€5PY}u˜P!‘P?OQP?uy^*uNd\^*’•u vuNdqb’•u Ÿ(md\uNW`’•rNP?b*Y

uNP!^œ– dqb`F–LNP?bXP^›L™P…uNdArNP?WgY\b`P J

i Ii Y\uNr –LNP!b`P^`LNP…uNP!’‘‡qLž€5dq[Nb`L™d¯dAr Y\bXP

^›LNP§”Gdq•‘d‚–’‘uN‡1

STUJW

STULXW /L

iYUVLXW

”Gd\bY\uy_ L \

S#UVL

W /L

i UVLXW3”GdqbY\uy_jL

\

B¯P?P# /’•‡q[Nb`P#ƒ„”Gdqb(Y\uFP ]™Y\OQpN‘P}Ÿ

BA[NstLY©r†Y}^›Y®Wš^›b`[™s!^›[™b`P s?Y\u%PZY\WX’‘¦_ €¤P˜s?dqOQpN[™^`P?r ’•u p5dq•_AuNdqOQ’¨Y}^›’•O9P}Ÿ­®P

s?¨Y}’‘O^`L†Y}^*^›LNPY\uNWX– P?b*^›d+^`LNPo˜[N•^`’‘p5dq’‘uy^¢UP?¨YZ_Facb`d\€N‘P!O”Gdqb*^`LNPu™d¯rNP^Jd\” ^`LNP

s?dqOQpN[™^`P?ruNP ^œ–d\b`9–’¦^›L+^`LNPT’•už^`P?‡qP!b 9 ’•W ¥}Y\‘’•rQ”Gd\b ^›L™P§gdqOQ’‘u†YJ^›’‘u™‡B¯P ^acb`dq€™‘P?O

”Gdqb ^`LNP§s?dqu™W`’‘r™P?b`P!r‡qb›Y\p™L–’•^›L9^›LNP§W`Y\O9Pg’‘uy^›P!‡qP?b 9 Ÿchº^(’‘W3WX[Qs?’•P?uy^(^›d°pNbXd‚¥qPT^›LNY}^

Y\uy_&O…[N¦^›’‘p5dq’•už^gb`P!¨YZ_&W`P ^^b ”GdqbT^›L™P9uNP ^œ–d\b`&’‘W—Y}W`W`dAs?’‘Y}^›P!r“^`d£YrNd\O9’•u†Y}^›’•uN‡WXP!^

d\”8^`LNP—‡qb›Y}pNL–’•^›LW`Y\O9P—s?Y\b`rN’•u†Y\•’•^œ_qŸCb ’‘WY°WX[N€NW`P ^gd}”]STUJHW 9Ÿ ­®P#WXLNd‚– ^›LNY}^

b ’•^`W`P!•”’‘WY'rNdqOQ’‘u†YJ^›’‘u™‡W`P ^gd}”8^`LNP—‡qb›Y\p™L ŸlmdquNWX’‘rNP!b¢Y°uNdArNP7L

\

Y}uNr˜’¦^›W(s!dqpy_

L

Ÿk±gW b ’‘WY°O…[N¦^›’‘p5dq’•už^ b`P!¨YZ_WXP!^?¡ L

’•W3^›LNP§u™P?’‘‡\LA€5dq[Nb(d}”/WXdqO9P§u™d¯rNP7`

\

b3Ÿc±gW

STU!`AW H`

i U!`AW €y_&rNP NuN’•^`’‘dqu ¡ – PQL†YZ¥qPQP?’•^`LNP?b7L ` d\bfL \

YU!`AW ¡dqb

Pž[N’•¥}Y\•P?uy^›¦_q¡LI`…dqb]L

\

YU!`AW Ÿ/K(LN’‘WkOQPZY\uNWc^`L†Y}^]L’•Wc’‘u b®dqbk’‘W ^`LNPuNP?’•‡qLž€¤d\[Nb

d\”/WXdqOQP§uNd¯r™P#’•ub(ŸMb ’‘W3^`LA[™WgY°rNdqOQ’‘uNY}^›’•uN‡…W`P ^¢Y\uNr^›LNP—pNbXd¯d\”’•WY\stLN’‘P ¥qP!r Ÿ

yzy|{M}~?€‚

(9)

c d e f g a b

a b

c e

d f

a’ b’ c’ d’ e’ f’ g’

s

(i) (ii)

g

 /’‘‡q[™b`P—ƒ =

’@ ± ‡\b›Y\pNLCŸ

=

’‘’@lK(LNP—uNP ^œ–dqbXdq€™^›Y\’‘uNP!r€y_^›LNP—bXP?rN[Ns ^›’•dqu Ÿ

#

’•W

Y'r™dqO9’•u†Y}^`’‘uN‡…WXP!^*’‘u

=

’@Y\u™r£Y°O…[N¦^›’•p¤dq’•uy^3b`P!¨YZ_WXP!^”GdqbZJ’•u

=

’‘’@ Ÿ

)# !" )*Q 2./! ?

± rNb`Y}”Ù^*€y_ ²§YZ_ž_A[NO E|.G’•uF^`LNP…oF±T«gjcK ‡qbXdq[Np˜d\”/^`LNPhijcK  p™b`dqp5dqW`P!WTY'LNP?[™b`’‘Wi¬

^›’‘s'”Gdqb#s?dqOQpN[™^`’‘uN‡£Y£O…[N¦^›’•p¤dq’•uy^—b`P?‘YZ_~W`P!^”GdqbY˜‡q’•¥\P?u uNdArNP}Ÿ˜­®P+pNbXd„¥\P+^›LNY}^…’¦^

s?dqOQpN[™^`P?W°Y O…[™•^›’•p¤d\’‘uy^#b`P?‘YZ_©WXP!^'d\”s?Y\b`rN’•u†Y\•’•^œ_®Y}^…OQdqWš^ ^`’‘OQP?W#^›LNPd\p™^›’½¬

O+Y\ O[N•^`’‘p5dq’‘uy^(bXP?¨YZ_už[NO…€5P?b–LNP!b`P“’‘W^›L™Puž[NO…€5P?b*d\”cu™d¯rNP!Wg’•uF^`LNPu™P!^œ– dqb`Ÿ

o˜dqb`P!d‚¥qP?b!¡¤– P‡\’•¥qPY9WXPž[NP!uNs?P…d}”cP ]™Y\OQpN‘P!W(”Gdqb*–L™’‘stL˜^`LNP#LNP?[NbX’‘Wš^›’‘s#s?d\O9pN[¯^›P?W¢Y

W`P!^–LN’‘stL ’‘WjU t/W [F^`’‘OQP?W€N’‘‡q‡\P?b^`L†Y\u ^`LNPdqp™^`’‘O9Y\ Ÿ P!^°[™W †b`Wš^°b`P!sZY\•

–L†Y}^^›LNP—LNP![Nb`’•WX^`’‘s§’‘W!Ÿ

"!$#%#'&( )*+&-,/.10324.15

P ^7J€5PYQuNdArNP}¡ S ’•^`W¢uNP!’‘‡qLž€5dq[NbXW?¡5Y\uNr S*S €5P#^`LNPu™d¯rNP!WTY}^^œ– dLNdqp™W*”GbXdqO JqŸ

mdqO9p™[™^›P—Y'O…[N¦^›’‘p5dq’•už^ b`P!¨YZ_WXP!^b Y\W(”Gdq•‘d‚–W7E|.G

RqŸ BA^tY}bX^(–’•^`LFY\uP?OQp™^œ_W`P ^7b d\”8O…[N•^`’‘p5dq’‘uy^b`P?‘YZ_¯W!Ÿ

ƒ¯Ÿ* /’‘bXWX^9W`P!‘P?s ^FY\W+O[N•^`’‘p5dq’‘uy^Qb`P!¨YZ_AW+^›LNd\W`P?Wu™d¯rNP!W’‘u S ^›L†YJ^Y\b`PF^›LNP£dquN•_

uN’‘‡\LA€5dq[Nbd\” WXdqO9P§u™d¯rNPf`

\

S*S®Ÿ

x™Ÿ(­LN’‘•P+^›L™P?b`PWš^›’‘•P]™’•WX^`WQWXdqOQPFu™d¯rNP’•u S*S –L™’‘stL ’‘W°uNd\^Qs!d„¥\P?bXP?r €y_ ^`LNP

O…[N¦^›’‘p5dq’•už^ b`P!¨YZ_WXP!^

=

!>! ?>

uNd STUVLXW–’¦^›L L \ b s?dquy^tY}’‘uNW’•^@

?-@zyA?-B

(10)

 †dqb#PZY\stLTL

\ S tIb3¡/s!dqO9p™[™^›P'^›LNP9už[NO…€5P?bd}”([NuNs!d„¥\P?bXP?r uNdArNP?W…’•u

S*S2^›L†Y}^*Y}b`P§’‘u S#UVLXW Ÿ

±(rNr˜’‘u b Y°uNdArNP§”Gdqb3–LN’•stLF^`LN’‘W(už[NO…€5P?b’•W(O+YJ]¯’•O+Y\ Ÿ

«*d\^›’•s?P#^›L†Y}^g^›LNP…Wš^›P!p©ƒ+WXP?•P?s!^`Wg^›L™P°uNd¯r™P?Wg–LN’•stL&O…[NWš^§€¤P’‘u&Y\uy_£O…[N•^`’‘p5dq’‘uy^

b`P?‘YZ_…W`P ^k”GdqbMJqŸ/­©P †bXWX^lW`L™d„– ^›L†YJ^l^`LN’‘WcL™P?[NbX’‘WX^`’‘s(s!dqO9p™[™^›P!W Y§WXP!^k–LN’‘stLQ’‘W –’¦^›LN’•u

Yg”ÆY\s!^`dqb ”GbXdqO^`LNP(dqp™^`’‘O9Y\¯Y\uNr°–P(–’•‘¯^`LNP?uQW`L™d„– ^›L†YJ^ ^›LN’•Wc€¤d\[NuNr'’‘W WXL†Y\bXp Ÿ

, &

­©P(‡q’¦¥qP(YTp™b`dAd\”5rN’‘bXP?s!^`•_’‘uNWXpN’‘bXP?r…”GbXdqO Ex GN–L™’‘stL'’‘W ’¦^›WXP?•”’•uNW`p™’‘b`P!r…”Gb`dqO YT‡qP!uNP?b`Y\

pNb`dAd\”k€y_“mLy¥}^tY\9E¦R G Ÿ#K(LNP †b`Wš^—pNb`dAd\”lY\€5dq[™^§Y\u~Y\u†Y\•dq‡qdq[™WgLNP![Nb`’•WX^›’•s–Y\W—‡q’•¥\P?u

’‘u E G Ÿ

P ^3b'€5P£^›L™P£uNdArNP?W9W`P!‘P?s ^›P!r%’‘u Wš^tY\‡qP ƒ“d\”§^`LNP Y}€¤d‚¥qP Y\‘‡qd\b`’•^`LNO Y}uNr ‘P ^

L

L€¤P^`LNPuNdArNP?W°WXP?•P?s!^`P?r ’‘u Wš^tY\‡qPx

=

L’‘W^`LNP ^`L Y\rNrNP!r u™d¯rNPo@ Ÿ P ^

b°€5PY&W`dq•[™^›’•dqu©–’•^`L O9’•uN’‘O9Y\ksZY\bXrN’‘uNY\‘’¦^œ_qŸ“ /’‘bXWX^°u™d\^›’•s?P^›LNY}^ b g b°W`’‘u™s?P

Y\uy_uNd¯r™P#’•u'b3’‘W(^›L™P#d\uN•_u™P?’‘‡\LA€5dq[Nbd\”/W`dqOQP—uNdArNP’•u S*S®Ÿ­®P#–’•‘ WXLNd‚– ^›LNY}^

b t b

!

bDt b

–LN’•stL’‘OQpN‘’•P?W ^›LNY}^^`LNPTs?d\O9pN[¯^›P?r+W`dq•[™^›’•dqu9’•W –’¦^›LN’•uY

”ÆY\s!^`dqb' &”GbXdqO,^`LNP—dqp™^›’•O+Y} Ÿ

P ^RS*S"c€¤P*^›LNP¢W`P!^3d\” uNdArNP?W ’‘u S*S ^›LNY}^(Y\bXP¢u™P?’‘‡\LA€5dq[NbXWd\” W`dqOQPguNdArNP*’‘ujb Ÿ

­©P¢W`P!^RS*S S*S t S*S" ¡ b b t bt¡*b b t blY\uNrjS ULXW STUVLXW# S*S

”GdqbcPZY\stLQuNdArNPxL

\

S®Ÿ/­®P*Y\W`WXd¯s!’¨Y}^`PY—s!dqWX^

k^`d#P?Y\stL+u™d¯rNPR`

\

S*S ٟ  d\b PZY\stL L

stLNdqW`P!u€y_'^›LNP¢Y\‘‡\dqb`’¦^›LNOF¡qY#[NuN’•^ls!dqWX^ ’‘WkPž[†Y}‘•_…rN’•¥A’‘r™P?r9Y\O9d\uN‡#^`LNP*uNd¯r™P?WluNP!–¦_

s?d‚¥qP!b`P?r&’•u S*S“Ÿ¢o˜dqb`P#”Gdqb`O9Y\••_7k’•”lL8’‘W^›L™P †bXWX^¢uNP?’•‡qLž€¤dq[™b¢d\”C`Y\rNrNP!r ’‘u b €y_

^›LNP#Y\•‡qdqb`’¦^›LNOF¡y^›LNP!uF– P#W`P!^3

k

$$S

UVLVWlt i

&%

')(

S UL

' W $$

K(LNP®s!dqWX^`W Y\bXP~‘’‘u™\P?r –’•^`L ^`LNP®s?Y\b`r™’‘u†Y\•’•^œ_ d\”^`LNP®s!dqOQpN[™^›P!r WXdq‘[™^`’‘dqu’•u^`LNP

”Gdq‘•d„–’•uN‡–YZ_ b *

kmonn+

k

­®P#Y\b`P—‡qd\’‘uN‡…^`dQWXLNd‚– ^`L†Y}^”GdqbY\uy_uNdArNP-,'’‘ub

¡N– P—L†YZ¥qP

*

kmon+-p/. s k S

U0, W =

RH@

yzy|{M}~?€‚

(11)

{

«¢d\^`’‘s?P †bXWX^¢^›L†Y}^*^›L™’‘W*’‘OQpN‘’•P?W3’•O9OQP?rN’‘Y}^›¦_9^`LNPbXP?WX[N•^?Ÿ(±guy_FuNd¯r™P ` \

S*S ’•W^`LNP

uNP?’•‡qLž€¤dq[™b¢d\”kWXdqOQP0L

\ b =

b`P!O9P!O…€¤P!b*^`L†Y}^guNduNdArNP’•u b’•WgY9uNP?’•‡qLž€¤dq[™b¢d\”+`

€ž_rNP †u™’•^›’•dqu ­©P§s?Y\uF^`Lž[NW¢r™P?rN[Ns!P

b *

kmonn+

k

*

.5mow + *

kmon+ep. s k

*

.5mow

+

S

U0,W

b

­®PWX^`’‘‘™L†YZ¥qP^›d#pNb`d‚¥qP*hœuNP A[NY}^›’•dqu˜R^›d#s?d\uNs?•[NrNP\Ÿ

P ^ ,€5P¢Y—uNdArNP’‘u b Y\uNr

‘P!^

$$S

U0, Wlt i ' ( S UL

' W $$

¡™”Gdqb(PZY}stL 9 =

S U, W @

€¤P—^›LNP#už[NO…€5P?b*d\”cuNP!’‘‡qLž€5dq[NbXW¢d\” ,+’‘uS*S ¤–LN’‘stL£Y\b`P#WX^`’‘• uNd\^¢s?d‚¥qP!b`P!r~Y}”Ù^›P!b^`LNP

stLNdq’‘s!Pd\”RL

A

LºŸ P!^€¤P^`LNP†b`Wš^'’‘uNr™P ]©W`[NstL ^›L†YJ^ ™Ÿ~­LNP!u L’•W

stLNdqW`P!u ¡

&%at

u™P?’‘‡\LA€5dq[NbXW(d\” ,+Y\bXPT^›LNP!u˜s!d„¥\P?bXP?r Ÿl­©P—sZY}uF^`LA[™W*r™P?rN[Ns!P

*

km/n +ep. s

k *( U %lt

W

$$S

UVLWlt i

&%

' ( S UVL

' W $$

­®P§^›LNP!u˜u™d\^›’•s?PT^›LNY}^^›LNP—stLNdq’•s?P§d\”aL€ž_+^›LNP#Y\•‡qdqb`’¦^›LNO ’•O9pN•’‘P!W3

$$S UVL!Wlt i %

')(

S UVL

' W $

$

$$S U0, WCt i

&%

' ( S UL

' W $$

&%

?-@zyA?-B

(12)

K(LN’‘W(’•O9pN•’‘P!W3

*

kmon +p/. s k *( U %lt

W

&%

S U , W

K(L™P§[NpNp5P?b3€5dq[NuNrd\u^›LNP§Y}pNpNb`dZ]¯’‘O9Y}^`’‘dquQ”ÆY\s!^`dqb ”Gdq•‘d‚–W?Ÿ «¢d\^`’‘s?P*^›L†YJ^3–P§s?Y\u

‡qP!^ YgW`L†Y}b`p5P?b/€5dq[Nu™r°dqu^›L™P3Y\pNp™b`dZ]¯’‘O9Y}^›’•dqu#”ÆY}s!^›d\b3’•^8’‘W8€¤dq[™uNrNP?r°€y_ –LNP!b`P

’‘W3^`LNP—O+YJ]¯’•O…[NO uA[™O…€¤P!bd\”8^œ– dQLNdqpFuNd¯r™P?W*Y°dquNP—LNdqpuNdArNP—O+YZ_+s?d‚¥qP!b?Ÿ

, &

­©PuNd‚–%‡q’¦¥qP*Y—W`Pž[NP!uNs?PU/W

9

—d\” P]NY}O9pN•P?Wc”Gdqb –LN’•stL9^`LNPgY}‘‡qdqbX’•^`LNO ‡q’•¥\P?W

Y\pNpNbXd‚]¯’•O+Y}^`’‘dquQ”ÆY\s!^`dqb`W(‡\b`PZYJ^›P?b3^`L†Y\u

%

ŸcK(LN’‘W3–’•‘¤pNb`d‚¥qP§^`L†Y}^(^›L™P—Y\‘‡qd\b`’•^`LNO

sZY\uNu™d\^'Y\stLN’•P!¥\PY\u Y\pNp™b`dZ]¯’‘O9Y}^›’•dqu®”ÆY\s ^›dqb€¤P ^`^`P?b^`L†Y\ujU W’‘u©W`dqOQP+s?Y\W`P!W?Ÿ

K(LNP—[NpNp5P?b€¤d\[NuNrF’‘W3^›Lž[NW*W`LNY\b`p Ÿ

’‘W°r™P †uNP!r Y\W…”Gd\‘‘d‚–W!Ÿ hº^›W'uNdArNP?WQY\bXPFY~WXdq[NbXs?P Jy¡

9

&dqu™PFL™dqp uNd¯r™P?W

A

Y}uNr U 9 HWW`P!^`WfS*S!

A S*S!

Y\uNr S*S#"

A S*S#"

d\”c^œ– dLNdqp

uNd¯r™P?W

=

Y\• ^›LN’•W#WXP!^`W…Y\bXPQr™’‘WGeXd\’‘uy^›W5@ ŸjkY\stLTS*S

.

–’¦^›L 9 Y\uNr , \

’‘WO+Y}rNPd}”

P?‘P!O9P!uy^›W?Ÿ(K(L™PuNP!’‘‡qLž€5dq[Nb`W*d\”cP?Y\stL uNdArNPY\b`P#rNP †uNP?r Y\Wg’•uNrN’‘s?Y}^›P!r

yzy|{M}~?€‚

(13)

R

€¤P!‘‘d‚–—Ÿ

S#U"JHW

# #

S#U

W J i S*S

i/i S*S

STU

W J i S*S

"

iHi S*S

"

”Gdqb 9

S#U

VW J i S*S

iS*S

"

”Gdqb 9 Y\uNr L \

S*S

S#UVLXW

”Gdqb 9

Y\uNr3L

\

S*S

"

S#UVLXW

#

 /’‘‡q[™b`P§x\‡\’•¥qP!W*Y\u’‘‘•[NWX^`b›Y}^`’‘dqu9d\” ‚Ÿ

K(L™P?b`P#’‘W(d\uN•_+dquNP—dqp™^`’‘O9Y\CO…[™•^›’•p¤d\’‘uy^3bXP?¨YZ_+W`P!^”Gdqb J—’‘u#k’•^(’•W

#

Ÿ ­®P

W`LNd‚– ^›L†YJ^…^›LNP+Y\‘‡\dqb`’¦^›LNO2d\”¢²§YZ_ž_A[NO bXP!^`[Nb`uNW

A 3

Y\W…Y£W`dq•[™^›’•dqu ¡^›Lž[NW

_¯’•P?•rN’‘uN‡…Y\u£Y}pNpNb`dZ]¯’‘O9Y}^`’‘dqu+”ÆY}s!^›d\b(‡qb`P?Y}^›P!b^`L†Y\u

Ÿ

 8’‘b`Wš^3uNd}^›’‘s!P¢^`L†Y}^(Y}‘†^›LNP¢^œ–d°LNd\puNdArNP?W3LNY‚¥\P§O9d\b`P¢^`L†Y\ud\uNPTdquNP¢LNdqpuNP!’‘‡qLA¬

€¤dq[™b?Ÿ°K(Lž[NW^BA^tY\‡\P+ƒd\” ^›LNPQY\•‡qdqbX’•^›L™O r™d¯P!W#u™d\^#W`P?•P?s ^Y}už_ uNd¯r™P\Ÿ°­©P'WXLNd‚–v€y_

b`P?s![Nb`WX’‘dqudqu ^`L†Y}^3^`LNP—Y\•‡qdqb`’¦^›LNOvWXP?•P?s!^`W

’•u^›L†YJ^dqb`rNP!b?ŸcK(LN’•W3’‘W(d\€ž¥A’½¬

dq[NW`¦_^›b`[™P”Gdqb 9 =

uNduNdArNP…WXP?‘P!s!^`P?r @Ÿ0B¯[NpNp5dqW`P’•^T’•W*^`b`[NP”Gdqb

q¡CY\uNr

^›LNPY\‘‡\dqb`’¦^›LNO L†Y\W*stLNdqWXP?u b

Ÿ K(LNP[Nu™s?d‚¥qP!b`P?r ^œ– d+L™dqp£uNdArNP!WgY\bXP

^›LNdqWXP’•u S*S!

&%

S*S#

Y\uNr'S*S#"

&%

A

S*S!"

Ÿ P!^*[NW*s?dq[™už^*^›L™P#už[NO€¤P!b

%ULXW

d\”/[™uNs?d‚¥qP!b`P!r uNdArNP?Ws?d‚¥\P?b`P!r£€y_PZY}stL˜dqu™P#L™dqp˜u™d¯rNPKL£uNd\^*’‘u b

%U

W

S*S

&%

S*S

%

*

t

U W

S*S

"

&%

S*S

"

&%

t

”Gdqb

U ' W

S*S

'

S*S

"

' '

K(LNP'Y\‘‡\dqb`’¦^›LNO^`LA[™W§stLNd¯d\W`P?W

&%gY\uNr ^`LNP°pNbXdqp¤P!bX^œ_˜’‘WTWš^›’•‘^›b`[™P”Gdqb at qŸ—hº^§’•W

^›b`[™P—”Gd\bY\uy_

'€y_bXP?s?[™b`W`’•dqu ŸcK(LNP#sZY}W`P °’‘W3–L†Y}^– P#–Y\uy^›P!rF^`d9WXLNd‚–—Ÿ

­®PLNY‚¥\PWXLNd‚–u£^›L†YJ^¢^`LNP…Y\p™pNb`dZ]¯’‘O9Y}^›’•dqu”ÆY\s ^›dqb*’‘W‡qbXPZY}^`P?b^›L†Y\u

Ÿ P!^¢[NW

P ]¯pNb`P!W`WQ^›LN’•WQ€5dq[Nu™r ’•u ^`LNP˜už[NO€¤P!b d\”*^œ–d®LNdqp uNd¯r™P?W'”Gb`dqO JF’‘u Fd\b'^`LNP

?-@zyA?-B

(14)

s

a

b

c

c

c

c M

M M

M M

M M

M

a

a

a

a b

b

b

b

0 0

0

1 1

1

2 2

2

3 3

3

 /’‘‡q[™b`PQx1 ZŸK(L™P²§YZ_A_A[NO LNP![Nb`’•WX^`’‘s°–’‘•cpNb`dArN[Ns!P

b›YJ^›LNP!b#^`L†Y\u

#

dqu! ‚Ÿ

O+YJ]¯’•O…[NO rNP?‡qbXP?P Ÿc­©P—L†YZ¥qP

S*S

S*S

S*S

"

S*S

"

t"

U

/W*

­©PQL†YZ¥qPQ^›Lž[NWp™b`d‚¥qP!r©^›L†YJ^#^`LNP9Y\pNpNbXd‚]¯’•O+Y}^`’‘dqu&”ÆY\s ^›dqb—’•W#‡\b`PZYJ^›P?b#dqb—Pž[†Y}l^`L†Y\u

p s%

Ÿ

yzy|{M}~?€‚

(15)

R„ƒ

hº^*’•W*p5dqWXW`’‘€™‘PT^›dQ‡qP!uNP?b`Y\‘’•ª?P§^›LNP#Y\€5d‚¥qP#W`P A[™P?uNs!P…d\” P]™Y\O9p™‘P?W(^`d9‡qP ^¢Y\pNpNbXd‚]¯’½¬

O+Y}^`’‘dquF”ÆY\s!^`dqb`Wgs?‘d\W`P…^›d €y_ ’•uNs?bXPZY\WX’‘uN‡^`LNP…W`’•ª?P°d\”k^`LNP…dqp™^`’‘O9Y\O…[N•^`’‘p5dq’‘uy^

b`P?‘YZ_'W`P!^

=

!>! ?> €y_'^tY\ž’‘u™‡~uNd¯r™P?Wl’‘uNWš^›PZY}r+d\”

Y}uNr

Y\uNr+s?[™^X^›’‘u™‡#^`LNPguNP!’‘‡qLž€5dq[Nbi¬

LNd¯dArd\”8PZY\stL

’•u WXP!^`W*’•uNWX^`PZY\rFd\” @ Ÿ

("}3.8

E¦R?GŸ5mLy¥ J^tY\ Ÿ ± ‡qb`P!P?r™_LNP?[NbX’‘Wš^›’‘sT”Gd\b^›LNP§WXP!^š¬œs?d‚¥\P?b`’•uN‡QpNb`dq€™‘P?OFŸ & ,.),* ,."

%$a* 4,.!" / ,.45 , ¡N|

=

x@ ΃\xqx™ƒ}xyz¯¡5R‚ ™Ÿ

E΃G#jcKZB¯h^B¯K*ml¬œU¢jMB¤R mdqO9OQ’•^X^›P!P\ŸFU¢Y}rN’‘d˜Pž[N’‘p™O9P!už^…Y\uNr Wš_¯Wš^›P!O9W LN’•‡qL©p5P?bi¬

”Gd\b`O9Y\uNs?P'b`Y\rN’‘d•d¯s?Y\cY\bXPZYuNP!^œ– dqb`&^œ_Ap5P˜Rq¡”G[NuNs ^›’•dqu†Y\cWXp¤P!s?’NsZY}^`’‘dquNW!Ÿ+jcKZB

x zqƒ¯¡N¢P?s?P!O…€5P?b#R‚qyz¯Ÿ

Ex G—K§Ÿ™mdqbXO9P!u ¡NmgŸ P?’•W`P!b`W`d\u ¡NY\u™rU—Ÿ¯U’•¥\P?Wš^ZŸ

*!4532. !"3%.(4-,(^

¡žstL†Y}p¯¬

^`P?b*x Ÿx™Ÿlo˜hœK acb`P?WXW?¡R‚q ¯Ÿ

E|.Go“Ÿ B¤Ÿ mdqb`WXdqu ¡ B5Ÿ a/Y\p†Y}rNP?OQP!^`b`’‘d\[ ¡ a/Y\p†Y}rNdqp5dq[N‘d\W?¡Ÿ a/Y\bX¡ Y\uNr

±Ÿ ²§YZ_ž_A[NO˜ŸF±gu©’‘uy^›P!b`uNP ^…O9Y\uNP!^P?u™sZY\pNWX[N¨YJ^›’‘d\u©pNbXd\^›dAs?dq

=

’‘OQP?p @§W`p5P?s!’†s?YJ¬

^`’‘dquCŸchœuž^`P?bXuNP!^ggb›YJ”Ù^Z¡NrNb`Y}”Ù^X¬º’‘P ^`”¨¬œO9Y\uNP ^X¬º’‘OQP?p¯¬ºW`p5P?s ¬ ™RqŸ^X]A^Z¡y±¢[N‡q[NWš^§ƒARQRZqq{™Ÿ

EÎzGo“ŸCU—Ÿ#Y}b`P!_FY}uNr ŸXB¤Ÿ\dqL™uNW`dquCŸ 45,.72 *!4,( , 0?!#6R, ..(%2( %

),* 0),* 4 %$K6 Ÿ/­ ŸfŸ †b`P!P?O9Y\u ¡ RZ ™Ÿ

E G B5ŸzB™Y\LNuN’kY}uNr®K§Ÿ§dquNª?Y\‘P!W?Ÿ'a/¬œs!dqOQpN‘P ^›P°Y\p™pNb`dZ]¯’‘O9Y}^›’•dqu&pNbXdq€N‘P!O9W!Ÿ 4,.

%$f),* a&

¡ƒ\x1Îzqzqz¯z z¯¡¤R‚ Ÿ

?-@zyA?-B

(16)

Unit´e de recherche INRIA Lorraine, Technop ˆole de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS L `ES NANCY

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex

Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

Editeur´

INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)

ISSN 0249-6399

Références

Documents relatifs

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT