Lower Bounds and Algorithms for Dominating Sets in Web Graphs
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Lower Bounds and Algorithms for Dominating Sets in Web Graphs Colin Cooper — Ralf Klasing — Michele Zito. N° 5529 Mars 2005. ISSN 0249-6399. ISRN INRIA/RR--5529--FR+ENG. Thème COM. apport de recherche.
(3)
(4)
(5)
(6)
(7) "!#$% &'(!# ) * ,+-. / 0 132 4" ∗ 57698;:=<>576?69@BADC † EGFH 8;IKJL8 HNM :=<O ‡ EQP :;R%SAT8;AVU.:;WX6 § YZ\[^]7_a`cbed'fhgjilknmo[^]7_pkcqrD]7]as\t\uvq*wxtDmk y{zorx|}_md~wxkqrxmnm_ w\Nrxzoml_z_^qZ\_^zqZ\_t9 GxDf,d~wxzkxxD)fh7wxx_pk ^*%\xx tLmZ\uvkKwN_*zeB_k}mslimoZ_7kou^_arx{lrx]u tw%mutko_mk^¡¢wt~moZ\_^uzex_^t\_*zw£ u ^w%murDtk^¡ u t~m}.rDzwx\Z~\zrlq_pkoko_^kKcZ\uvqZVwz_(cuvl_^£iLsko_^~mor]rj\_*£Gwxko_pq¤mkKrxQmoZ_a.rxz£ l¥=cuvl_(._*¦?§ t mZ\_^ko_¨zorlq_pkoko_^k{_^wDqZ-t\_*ª©D_*zomo_*«q*rxt\t\_pq¤mkQmr
(8) moZ_K_*«juvknmou t\(Dzwx\Z-¦jiwaqrDtk}mwtTmBtjs\]a¦_^z^¡ m ¡Trx _plx_pk*§ YZ\_{m_*z]utwx£D©D_*zomouvq_pk¬rmoZ\_pkn_B_^lD_^kwz_{qZ\rDko_*tst\u®rDzo]£ i
(9) w%mzwxtlrx]¯rxz¦ji)\zo_*®_*z_*tTmouvw£ wmnmwxqZ\]_*tTmK\_*N_*tlu t\rxtLmZ\_(\zrlq_pkok^§c°V_7knZr%±mZw%m)w£ ]rDknmKw£ £kosqZLDzwx\ZL\zrjq*_^kkn_pkcZwX©D_ ]u t\u ]wx£Qlrx]u tw%mutko_mk)£u t\_^wxz)utVmoZ\_-kou^_r{mZ\_xzw\Z²wtxu ©x_7¦rDs\t\ke®rDzemoZ\uvk
(10) kou^_7wDkew ®stq¤murDtrx m §Q°²_
(11) rD¦lmwxutmoZ\_(s\\N_*z¦Nrxs\t\k®zrx]³kou ]7£_
(12) rDtl¥´£u t\_w£ xrDzoumoZ\]-k.®rDzlrx]u tw%mut\ ko_mk*§YZ\_c£ r%B_^z¦rDs\t\kQwxzo_rD¦lmwxut\_p(¦jia\zor%©ju t\KmoZwmGmoZ\_£ _«luvqrDxzw\Z\uvq*wx££ ieµzk}m{ko_mGr¢w)xu ©x_^t kou *_)u k.mZ\_]rDknm£u ¶x_^£i·mor-lrx]u tw%m_x§ ¸L¹Tº9»¼(½ %¾x wxtlrx]¿lijtw]uvqt\_m}.rxz¶lk*¡ lrx]u tw%mut\Lko_mk*¡B_^¦xzw\Zk^¡¬zwt\rx]'Dzwx\Z zorlq_pkoko_^k^§ À?ÁpÂÃ%Ä;ÅÇÆ ÆÉÈ9ÊcËÌpÆ;ÁXʤÄ;Å?È¢Â}Ä;ÂQÍ%ËÄ®Æ;ÎÏËÐÑÐ ÒKÅ;ÌXÍXÍʤĮÆ=Â}ÓÔÒ¨Æ;ÁXÂQÕ¬ÊoÒ*ËÐ\Ö*Ê^×}ÎÑÂØÆÉÒÙÄ;ËÚÆGÛ9Ö^ÛNÜ9ÝÞÜLßnà¤áâ¤âXã äÂ}Í%ËÄ®Æ;åcÂnÚÆ¢Êæjç9ʤåcÍXÌpÆ;ÂnÄ9Ö^×}ÎÑÂ}ÚX×}ÂèpéÎÑÚXêXë Å9ç¢Ê¤ÐÑÐÑÂ}ê¤Â¬ìjʤÚXÓpʤÚTèpìTʤÚXÓXʤÚTèpíé.èÂ}åËÎÑÐî¤îï¤ïoð*ñòóô^î¤õTö ÷pîøjöÉù*îTö ú¤÷%ã ʤÅ=ÌXÐÑÂnÍXÅnÍè ʤû)¢Ä®Æ=ÜüÂ}Ó¨ÖXç¬Ô*pèÒBýÀ?Æ=ÁpÀ?¤ÂÛ~à ÛÌp¤ÍXáÄ;ÄÇʤÊÍþ;Ö^Â}Ânʤ×ØËÍpÆnÚKÁXèNÎÏÍpÿË(ÖÄ;Êü þÇ®Â}Ú*ç ×ØÆ;Æ¢ÎÑÍÕÿÇʤÖ^ÖpÐÑÀ ÎÑÝÅcÿ ç9ÛÂ}Õ¬ÓXÀ"ÿÉ üGç¢ÝÕ?í %Û9ÚpÄ;ÖXÎ Ëç¬ÚXÂn×}ÄÇç¬Å= ýÎØÆ è¬) Â}×}ÓXåʤ ÚËÆ=ÎÑ?Ä;Ð ËÎÑ*×n×ØÂùÆ?Çø Ö*ÚX\ʤÊXö ÍXã**ÁXÿÇøÖ^ÎÏù*Ë)À õ ®üá ÚÆ;pó*ÎÑßõÍxïoʤð ÐÑpßÎÑÅnnùèlá ö ´ã9âaÜ\ò nùËÕ¬lÄ®Æ=ʤö ÎÏÌpËòXÆ;ÐxÂKãÅ=ÌXÓpÜ\ÍpÂnÍxËÅBÄÇʤÆ;ìjÄ®ÎÏÆ¢ÌXËÐÑ×}ÔÐ Î ÒÒ Æ=ÁX %ÄÇÂ}ÚX×´Á(ç Õ ÖcüGÖKä?ÒpÚ%ËåcÊXã ö *äïÂ}ó*Í%î¤Ëõ¤ÄÇîTÆ;åcö®ø Â}Ú*\ÆöÉù*ÊîTæDö ç9ú¤Ê¤÷%åcã ÍXÌpÆ;Â}ÄÖ^×}ÎÑÂnÚp×n¤è*í?ÚXÎ Â}Ä;Å;Î ÆÉÒBÊæìTÎ ÂnÄÇÍÊ^ʤÐvè*ÜlÂnË×´ÁÖ*Æ=ÄÇÂ}ÂØÆoè*ìjÎ Â}ÄÇÍxÊ*ʤÐXìlà íé ´èÂ}åËÎÑÐ ∗. †. . ‡. . .
(13)
(14)
(15) . . . . '. +. § 2 43576. (. . . .
(16)
(17) . . !#".
(18) $%. . & ! %. .
(19)
(20)
(21) )
(22) *#. . . . .
(23) -,
(24) ./0. 879. Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65. 1.
(25) c $c c .
(26) (¤ !#c4V
(27) ³c ?! 1"c (!#
(28) a /4²
(29) c4"c/ 0 1 wtkaq_*mawzomouvq£ _x¡t\rDsk mslu rxtka£vw"mwxu£ £_l_^k(_*tko_*]a¦\£_pk(lrD]utwxtTmk^¡wxutkou
(30) Ts\_ £ _*szk- *t ^zwx£uvkw%mou rxtk*¡.\wtkl_*s\« \zrlq_^kkosk·l_~Dzwx\Z\_
(31) Ts\uKkorxtTm·£vwzx_^]7_^tTm·slmou £uvk ^k·Nrxsz·£ w ]rl ^£uvkowmou rxt~l_7lu ^zo_^tDm)wxko_pq¤mk¨ls~°²_*¦?§ wtkeq_pk\zrlq_^kkosk*¡NqZw
(32) Ts\_(t\rxs\©D_^wxs~knrD]7]_*mKkn_ q*rxt\t_^q¤m_7wxsVDzwx\Z\_a_«lu knmwxtTmewxz)s\tt\rx]a¦\zo_7qrDtk}mwtTm m wxz*mo_pk*§¬_^k)korx]]_mk¨m_*z]utwxsl« \_Kq_pkBwxz *mo_pkBkorxtTm.qZrxuvknuvk{korxum.l_e]-wt\u [*z_s\tu®rDzo]_e_mcw£ ^w%mrxu zo_korxum.\_K]-wtu[^zo_K\z * *z_*tTmou _*£ £ _ *N_*t\wxtTmo_ls"\zrjq*_^kknsk*§ rDsk]rxtTmozrxtk
(33) Ts\_)Nrxs\zczo_pk
(34) Ts\_KmorDslm\zrjq*_^kknsk.\_xzw\Z\_)l_q_ m}ijN_x¡u £l_«luvk}m_cl_^k_^tkn_^](¦\£ _^kG\rx]utwtTmk]utu]-ws\«(£u t pwu zo_ckos\z£ wKmwxu£ £ _\saDzwx\Z\_x¡%_*m{lrxtt\rxtk Nrxsz.q*_mnm_mwu ££ _e\_^kB£u ]um_^kB_*t·®rDtq¤murDtl_ m !§ ?_^kB¦rDzot\_pk{kos\ *zu_^s\zo_pk{korxtTmBrD¦lmo_^tjs\_^#k "
(35) wxznmuz \_7kou]\£ _-w£ xrDzoumoZ\]_pk¨Nrxs\z)_^tkn_^](¦\£ _^k)lrD]utwxtTmk^$§ ¬_pkK¦Nrxzt\_pkKu tl ^zou _*s\z_^k)korxtTm)rx¦lm_*tjs\_^ke_^t \rxt\twxtTm)£vw"\z_*s\©D%_
(36) Ts\_-£_-zo_^]7u _*z_^tko_*](¦£_-£ _«luvqr¥´xzwZ\u
(37) Ts\_7Nrxs\zs\t_-mwxu£ £_·\rxt\t ^_-_^knm
(38) £_ £skzorD¦w¦\£ _*]_^tDmc&£ _^tko_*](¦£_lrD]7u twxtDmp§ ( ² ^ko_^wxsl«²\u knmozu¦s ^k
(39) \iTtw]u
(40) Ts\_pk**¡ )Qtko_*]a¦\£_pk
(41) lrD]7u twxtDmk*¡ +ezwx\Z\_pk
(42) ls°V_^¦¬¡ ' ½ p » y{zorlq_pkokoskl_Dzwx\Z\_^kwx£ pw%morDuz_^k^§.
(43) .
(44)
(45) !"$#
(46) %
(47) '&()*
(48) ,+.-'/01324 6. 7. 5. a?- 98{
(49) . tLz_^q*_*tTmcix_pwzk.moZ\_
(50) .rxz£vcuvl_._*¦LZwDkcxzr%ct"lzwx]-w%mouvq*wx££ ix§ mk¨q*s\zzo_^tDmkou^_
(51) uvkc]_^wxkos\z_^u t ¦u£ £u rxtkBrx?wxx_^k*:Ñ<;´¡\wt·wxx_^k.wz_KwD\l_p·mraumc_*©x_^zoi-\wXiD§>=¨kBmoZ\uvkxzwZ@?®t\rll_pkcqrDzoz_^korDt mr._*¦VwD_^k¨wxt~_plx_pkmr£ ut¶jke¦N_m}._*_^t~wD_^k3Aq*rxtTmou tjs\_^k¨morxzr%¯um)¦_pqrx]_pkKu tq*zo_pwxkout\D£i u ]rDznmwtTmmor knmos\i ]-w%mZ\_*]-w%mu q^w£c]rll_*£vk-cZ\uvqZ q*w\mos\z_Lumk·knmozsq¤ms\zwx£\zorD_^znmu_pBk : D¡ C^ ;=§ glsqZ~]rll_*£vkKq^wt~¦N_7skn_pLmrl_pknu xt~_ E·qu _*tTmewx£DrxzumZ\]-kc®rxze._*¦wx\\£ u q^w%mou rxtkKwxtL]-wXi_^©x_^t stqr%©D_*zQs\t\®rxz_^ko_*_*t-zorD_^znmu_pkQr¢mZ\uvkBZjs\D__*©Drx£ ©jut\ak}mzosqmos\z_x§Ggj_^©x_*zw£]-w%moZ_*]-w%mu q^w£]rll_^£ k ®rDzwtwx£ilkout\moZ\_._*¦VZwX©x_a¦_^_*t²\zorDrTkn_pF ?®®rxz)utk}mwtq*9_ :Ñl¡¬l>¡ Cp< ;A¤§aYZB_ ?Ç_*©Drx£ slmou rxtVrQmZ\._ A ._*¦ Dzwx\Zu k(skosw£ £iV]rj\_*£ £_p ¦jiVG w ?Çzwxtlrx'] A)zorlq_pkok
(52) ut cZ\uvqZ t\_^ ©D_*zomouvq_pk
(53) w\N_^wxz®zrx] mu]_emoramou ]_x§{gjsqZ©x_^znmu q*_^kB]-wXi-¦_)£ ut\¶D_^zwxtlrx]£ i7mor7moZ\_e_«luvk}mut7knmozsq¤ms\z_¨mZ\zorDs\xZknrD]_ ®rDzo] r( 2(1)!""1
(54) H%IJK)LNM3O"
(55) P)Q_«luvk}mut
(56) ©x_^znmu q*_^kQcumoZ]wxtjiat\_^uDZj¦rDs\zk{wz_¨knrD]_*cZwm{]rxz_ £ u ¶x_*£ i-mr7¦N_£ ut\¶D_^mor7mZ\_t\_*qrD]_*zk*§ YZ_K]-wu t·®rlq*sk.r?z_^ko_^wxzqZ·kor(ÇwxzBZwxk.¦_^_*trxtq^wlms\zut\a_*]\u zouvq*wx££ i-rx¦kn_^zo©D_^B : 5\¡l< ;¢®_^wmos\z_^k rxcmoZ\_._*¦¬§ rVwmnm_*]lmZwxka¦_^_*t ]wDl_·mrVqZwxzwDq¤m_*zu ko_-xzw\Z\¥;mZ\_*rDzo_*mouvq-kns\¦\¥´knmozsq¤ms\z_^k
(57) rx kosqZVxzw\Zk^§(°²_utumu wmo_-knsqZVu tj©x_^knmou Dwmou rxt~¦ji~£ rjrx¶ju t\wmko_mk)r.©x_*zomouvq_pkmZw%mp¡?u t²wko_*tko_x¡ q*r%©x_^z¨w£ £ rxmoZ\_^zK©D_*zomouvq_pk*§¨d~rxz_
(58) ®rxz]-w£ £iD¡Nw©x_^znm_«"utwxzw\F Z R
(59) KS"¨w£ £ ©D_*zomouvq_pkcmoZw%mwz_ wDX|nwxq*_*tTm{mrauTm ?®wxzwx££ _*£N_^\x_^kBxu ©x_K](s\£mou \£ _lrx]u tw%murDPt A¤§ t·mZ\_knuzumrV U¨wzwziwt9 U¨wXijt\_^k :WCXK;=¡?wxt NK
(60) (%
(61) Z"®rxzwxzw\Z uvkwko_m S ⊆ V knsqZmZw%m_^wDqZ©D_*zomo_«~u t hY G = (V, E) u c k \ x r ] u t % w o m p _ " % w c m £ p _ x w n k m o m u 7 ] p _ c k j ¦ · i x © ^ _ n z m u * q ^ _ . k u t § ¬_m. l_^t\rm_)moZ\_kou^_)rmoZ\_ V ko]-\wS£ £_pk}m h¥ØlrD]7u twmou t\(ko_mk.u th G §GYZ\_
(62) P h Y SNK
(63) (γ%
(64) [=γ(G) zorD¦\£_^\ ] ?Çd h Hg ABwDkn¶lkQ®rxz wxt h¥Ølrx]u tw%mut\-ko_mcrGkou^_ γ § rx]u tw%mut)ko_mk\£vwXi
(65) wt7u]NrxzomwtTmGzrx£ _Bu ta]-wxtTi(\zwDq¤mu q^w£lw\£uvq*wmou rxtk^¡%_x§ § ut7moZ_q*rxtTmo_*«jmrx \u knmozu¦slmo_p
(66) qrD]7slmou t\¨rDz?]rx¦u£ _{wxl¥=Z\rlqGt_m}.rxz¶j]k : \R¡ Cpl¡XD. ;´§ YZ\_Bzo_pwxl_^z?uvk¬z_®_*zz_^emor :^CXj¡ C^< ; ®rDz
(67) wt²ut\¥´l_^lmoZ©ju _* r{mZ\_·kos\¦l|}_^qm^§YZ_am}ij\uvq*wx£®s\t\wx]_*tTmwx£GmwDkn¶~u t knsqZ²wx\\£ u q^w%mou rxtk)u k mrko_*£ _^qm
(68) wkos\¦ko_mr.t\rll_^keu tmZ\_t\_m}.rxz¶"moZw%mcu£ £ \zor%©juvl_ 9w"q_^znmwu tVko_*z©Tuvq_(morLw£ £GrmZ\_*z ©D_*zomouvq_pk*§ _\rxzemZ\u kmor"¦_7mu]_¥´"_ E·q*u_^tDmp¡?w£ £GrmZ\_*z©D_*zomouvq_pk¨]ask}m
(69) ¦N_-lu zo_pq¤mo£ iq*rxt\t\_pq¤m_^mormoZ\_ ko_*£ _^qmo_p~t\rll_pk*¡?wxtVut²rxzl_*zK®rxzum)mor"¦N_q*rDknmn¥´_ 9_^qmou ©x_x¡9mZ\_tjs\](¦N_*zrx.ko_*£ _^q¤m_^t\rll_^k)]ask}m ¦N_]u t\u ]wx£;§ t zo_^£ wmou rxtmr._*¦ xzwZk*¡w~lrD]utwmou t\Vko_ma]wXi²¦N_sko_^mrVl_^©ju ko_·"_ E·q*u_^tDm ._*¦ ko_^wxzqZ_^k^9§ _\rDz h > 1 wxt h¥Ølrx]u tw%mut\~ko_mq*wxt²¦N_q*rxtkouvl_*z_^wxk(wL]rxz_Çws\£mn¥=morD£_^zwxtDm knmozsqmos\z_x§ QsLmor h − 1 ©x_^znmu q*_^k¨rxz¨_plx_pkcÇwxu£=¡NmoZ\_7lrx]u tw%murDtL\zorD_^znm}iuvk¨knmou ££]-wu tDmwu t\_^ ?Çu;§ _x§umuvkck}mu£ £¬NrDkknu ¦\£ _emor-\zr%©ju l_KmoZ_
(70) kn_^zo©juvq._ A§ YZ_Vd h gª\zrx¦\£ _*]#uvk ¨yG¥=Zwz` :^C.5la¡ Ccb<;
(71) wxt¢¡]rxz_*r%©D_*zp¡um"u kt\rxm£ u¶D_*£ i mZw%m"um]-wXi ¦_ wx\\zrX«lu]-w%m_^ _ N_pq¤mu©D_*£ d i :^CXje¡ Ccb;=§ yrx£ ijt\rx]uvw£.mou ]_"wx£DrxzumZ\]-k7_«lu knm-rxt knN_^q*u wx£cq£vwxkkn_pkarx Dzwx\Z9k ?®_x§ f § : 4 C;A¤§ YZ\_G d C g\zrx¦\£ _*] ZwDk(¦N_*_^t k}mslu _^ w£vknr~u t zwtlrx]¿xzw\Zk*§ t moZ\_ -
(72) K IKgOcR" :Ñ X;¢moZ_e©%wx£s\_)rx q*wxt¦N_eutl¥´Nrxu tDm_^
(73) Ts\um_)\zo_pquvkn_^£iD¡l\zr%©Tuvl_p u k trm.morjrako]-w£ £9q*rxG(n, ]wxzop)_p7mor n § tzwxtlrD] γz_*Ds\£vwz{Dzwx\ZkBr \_*xz_*_ r ?Çko_*_K®rxz._«\w]\£ _Kz_^kos\p£mk u t~mZ\Z_ MhK
(74) ciVK 1K)IK
(75) j[.Ng:Ñ ;{wtLz_®_^zo_^tq_pkmZ\_*z_*u Pt A¨s\\N_*zwt~£ r%B_^z¨¦Nrxs\tkewxzo_a¶jt\r%ct rDt γ § h. h. 1. 1. Õ Õ Úk ¤á. *
(76) *
(77) . h.
(78) X. ]c324 ^3
(79)
(80) gS. t moZ\uvkwx_^z7._£rjrx¶ wm-kou ]7£_"wt _"E·qu _*tTm·w£ xrDzoumoZ]k(®rxz7¦\s\u £vlutko]-w£ £ h¥Ølrx]u tw%mut ko_mkut Dzwx\Zk^§ YZ_N_*zo®rxz]-wtq*_"xswxzwxtDm_*_pk(rxKmZ\_^ko_~w£ xrDzoumoZ\]-kwz_Lwtwx£ilko_^ s\tl_^z-moZ\_ wDkokos\]lmurDtmoZwm(moZ_u t\\slmuvk(w~zwtlrx]¿B_^¦ xzwZ¬§~°²_w£vkorwtwx£ilko_-moZ_·mou xZTmt\_^kk(rxcmoZ\_ N_*zo®rxz]-wtq*_^k7rkosqZ w£ xrxzumoZ\]-k^¡{¦ji \zr%©jut²q*rx](¦utwmorDzouvw££ r%B_^z-¦rDs\t\k-rDt γ ¡{®rDzwtji µ«j_p h ≥ 1 § gjsqZ ¦rDs\t\kkoZ\r%/moZw%m-]rDknm7rx¨moZ_mu]_moZ_"ko_mkzo_*mos\zt\_p ¦jimoZ\_"©%wzu rxsk wx£DrxzumZ\]-k¨wxzo_
(81) rDt\£ i~wqrDtknmwtTmKÇwxq¤mrxzewX.wXi®zorD]rxlmu]-w£=§ _utwx££ ix¡9._aq*rx]wz_
(82) mZ\_
(83) Dsw£ um}i rx?mZ\_
(84) knrD£s\mou rxtkz_ms\zot_^¦Ti ?Çkorx]_rh A.rDs\zw£ xrDzoumoZ]k.cumoZ"_*]\u zu q^w£¬wX©x_^zwxx_©%wx£s\_pkcr γ § YZ_a]-wu t~rDslmq*rx]_(rxQmoZ\uvkewN_*z)q*wxt~¦N_7knmwmo_^ut\®rxz]wx££ i"¦ji"kwXijut·moZw%mK._*¦Vxzw\ZkZwX©D_ Çwxuz£ i-£ wxzoD_elrD]utwmou t\-kn_*mk^>§ U_*tq_
(85) wqzwXc£_^zBcZr7wtTmk{mrskn_wlrD]7u twmou t\-kn_*m.mr_«l\£ rxz_ mZ\_._*¦cu ££t\_*_p7mor
(86) knmorxz_w)£vwzx_\zrxNrxzomou rxt7r9moZ\_¨cZ\rx£ _cxzw\Z?§ tTmo_^zo_pk}mut\D£iD¡mZ\_z_^kos\£mkGu t mZ\uvkGwN_*zQwx£ kores\tqr%©D_*zGHw ?Çt\rmZ\_*z Alu N_^zo_^tq_c¦N_m}._*_^ta]rll_*£vkGrxmoZ_cB_^¦7¦wxko_^arxt7\zo_*®_*z_*tTmouvw£ wmnmwxqZ\]_*tTm{wxt7]7rDzo_.mzwDlumurDtw£zwxtlrx] xzw\Z7]rj\_*£vk*§GYZ\_m_*tl_^tqiamor
(87) qZrTrTkn_t\_^uDZj¦rDs\zk rx Z\u xZLl_^xz_*_)w 9_^qmkmZ\_
(88) knu *_)rx?moZ_
(89) kn]-w£ £ _^knmc\rx]utw%mou t\-ko_mk^§ d~rDknm)rx.rDs\z
(90) w£ xrDzoumoZ]k)wxzo_
(91)
(92) ut²moZ_kn_^tko_amZw%mmZ\_·l_pquvknu rxtVmor~wD\Vw"wzomouvqs\£vwz)©x_^zn¥ m_«²mormoZ\_\rx]utw%mou t\ko_m7u k
(93) mwxY ¶x_*t cumoZrxslmamormw£Butl®rDzo]-wmou rxt w¦Nrxs\m
(94) moZ\_B_^¦ xzw\Zs\tl_*z q*rxtkouvl_*zw%murDt¬¡¢wt Khh3au tmZ\_·ko_*tko_(mZw%m(\_^quvkourDtk*¡¢rDtq_7mw¶x_^t¬¡¬wz_t\_*©D_*zqZwxt\x_p¢§aYZ\_ wx£DrxzumZ\]-kQwz_Kwx£ ko$r
(95) Ts\umo_K"_ E·qu _*tTcm QrDt\£i-waqrxtk}mwtTmBw]rDs\tTmBr¢mu]_Ku k.sko_^-N_*z.s\9\w%m_¨rx¬moZ\_ \rx]utw%mou t\kn_*m^§QgjsqZ"w£ xrDzoumoZ]k.wz_er?wxznmu q*s\£vwzutTm_*z_^knm.u tmZ\_qrDtDm_«jmr?._*¦xzw\Zk*§ =¨k mZ\_
(96) ._*¦"xzwZuvk_^©xrx£ ©jut¡lrxt_.wxtDmkmor·\_^quvl_
(97) cZ_moZ_*z¨wt\_^ ©x_*zomo_*«uvkcmor-¦N_(wD\l_^mr-moZ\_ wx£z_^wDli·_*«juvknmou t\·lrx]u tw%mut·kn_*mcumZ\rxslmKzo_pqrx]\s\mou t\moZ\_(_«lu knmou t\·lrD]utwmou t\·ko_m¨wtcumoZ ]u t\u ]wx£¬qrD]7slmwmou rxtwx£¢_ 9rxzom^§BbKtl¥´£u t\_)k}mzwmo_^xu _^kB®rxzmZ\_
(98) lrx]u tw%mutko_mc\zrx¦£_^]ZwX©x_)¦N_*_^t q*rxtkouvl_*z_^7u t-moZ\_Kwxkn$m :WCNC¡ C.;®rDz{x_^t\_*zw£xzw\Zk^§ U¨r%B_^©x_^zGmoZ\_KwslmZ\rxzk{wxzo_ct\rxm.wXwz_r?wtji z_^kos\£mcrxtrxtl¥´£u t\_w£ xrDzoumoZ]kB®rDzmoZ\uvkc\zrx¦\£ _*] ut"zwtlrD]xzwZk*§ bKsz.z_^kos\£mk.Z\rD£ <$2L%IMhK K O<] 1 ?;w\§ w\§ k*§ A¤¡lu=§ _x§cumZ"\zorD¦w¦u£ um}i·wx\\zrDwDqZ\ut
(99) rxt\_ wDk
(100) mZ\_"knu *_rmZ\_._*¦ xzw\Z Dzor%k
(101) moru tlµtum}iD§ YZ\_wx£DrxzumZ\]u qzo_pkns\£mk7wz_·¦wDkn_p rxt moZ\_ wxtw£ ilknuvkrxBw·tjs\]a¦_^z)r ?;d~wz¶xr%©juvwPt A.zwt\rx]\zrjq*_^kkn_pk*§ t_pwxqZq^wxko_
(102) mZ\_azorD_^znmu_pk¨rxQmoZ\_ zorlq_pkokKs\t\_*z
(103) qrDtkou l_^zwmou rxt~£ _^wDLmrmZ\_-l_µt\umurDtVrx.w ?Çl_*mo_^zo]u t\u knmouvcq A)q*rxtTmou tjs\rxsk®s\tq¤mou rxt mZw%mBu kG©D_*ziaq*£rTknT_ ?Çut-\zrx¦w¦\u £um}4i A mormZ\_©%w£ s\_pkGr9moZ_\zrjq*_^kk*¡DwxkmZ\_¨kou^_rNmoZ\_¨xzw\Zxzr%k^§ m"knZrxs\£v ¦_VrDutTm_^ rDslmwmmoZu kk}mwD_"moZwmmoZ_V\zrxNrDko_^ wxtw£ ilknuvk·]_mZ\rllrx£ rxDi uvk
(104) Ts\umo_ D_*t\_^zwx£;§K°V_w\£iLumemorwxtw£ ilkn_aZ\_*szouvk}mu q^k®rDz¨mZ\_d gLzorD¦\£_^]rxt\£ ix¡¢¦\s\meume.rxs£ ~wx££ r% mr·\zr%©x_)z_^kos\£mkw¦Nrxs\m¨rmZ\_*zDzwx\Zkcwzw]_m_*zkcknsqZLhwDkmoZ\_autl_*N_*t\_*tq*_
(105) rxzcmZ\_(qZ\zrx]-wmouvq tjs\]a¦_^z^§VYZ\_]_mZ\rl uvk(q£ rDko_*£ i²zo_^£ wmo_^²mr~mZ\_korq*wx££ _^d K $1"
(106) P)IKD H%I
(107) O"4. :Ñx< ;=§ tÇwDq¤m¨w©x_^zkou rxtrx mZ\_]-wu tLwtwx£iTmouvq*wx£9morjrx£¢zorDrTkn_p¦ji°²rxz]-w£vq^wt"¦N_
(108) wx\wxlmo_pmrBrDzo¶ ®rDzmZ\_zorlq_pkoko_^kqrDtknuvl_^zo_p utªmoZ\uvkwN_*zp§ U¨r%B_^©x_^z-moZ\_²]-wxqZ\u t\_^zoi _*]\£ r%ix_p ut : D. ;u k trm)t\_*_pl_^~morwxtw£ ilkn_(moZ\_7\zrlq_^kko_^keqrxtknuvl_*z_^~utmoZ\uvkewN_*zp§(bKs\zKz_^kos\£mk)wz_(rD¦lmwxut_^L¦ji zor%©ju t\eq*rxtq*_*tTmozw%murDtarxmZ\_©%wzu rxsk\zrlq_pkoko_^kr¢u tTmo_*z_^knmQwzrxs\t
(109) mZ\_*u z{]_^wxt-wt7¦jial_^©ju kout w]_mZ\rl~®rDzKD_mnmutq*£rTkn_7_^knmou ]-w%mo_pkerxt~mZ\_zo_^£_^©%wtTmK_*«jN_^qmwmou rxtk^§ t gj_^qmou rxt²._az_*©ju _* mZ\_
(110) l_*µt\umou rxtkcrx mZ\_
(111) ]rll_*£vkr._*¦"xzw\Zk.moZwmB_cu ££?skn_D§Q°²_
(112) w£vknr-knmw%m_rxs\zc]-wxut"z_^kos\£m u t-mZ\_eq*rxtTmo_*«jmBr¢mZ\_^ko_¨]rll_^£ k^¡Twxt-\z_^ko_*tTm.]7rDzo_Kl_mwu £_pqrD]]7_^tTmkQrDt·rxszBD_*t\_^zwx£wtwx£ilkou k ]_*moZ\rl¢§ tmoZ_{®rD££ r%cutKko_^q¤murDt
(113) ._BqrDtkou l_^z w¨©x_*ziknu ]\£ _Bw£ xrDzoumoZ] wxt
(114) wx\\£ iKmZ\_.\zorDrTkn_p h. 1. ÿ Õ¬ÿÉü .
(115) . .
(116)
(117) !"$#
(118) %
(119) '&()*
(120) ,+.-'/01324. ] _*moZ\rlmorrx¦lmwu tt\rDtl¥;mzou ©ju wx£s\\N_*z)¦rDs\t\k)rDt γ §ac_µt_^²w£ xrxzumoZ\]-kewz_au tTmozrj\sq_pwt wxtw£ ilkn_put"gj_^qmou rxt'X7wtl§ tgj_pq¤murDt
(121) ._Kluvkqskk{D_*t\_^zwx£uvkw%mou rxtkmor §YZ\_^t._ms\zot mr£ r%B_^z¦Nrxs\tk*§ tVgj_pq¤mou rxt -wt b·._
(122) \z_^ko_*tTmKrDs\zKwxzoDs\]_*tTm®rDz¨mZ\_(h£ r%>B_^1z¨¦Nrxs\t\kKknmwmo_p u tLgj_^qmou rxt\>§ _ u twx££ i·._K¦\zu _ i·q*rx]]_*tTm.rDtkorx]_e_*]\u zouvq*wx£N.rxz¶-q*wxzozu_p-rxs\mrDtw7kns¦l¥´q*£ wDkok rx?mZ\_xzw\Zk.q*rxtkouvl_*z_^utmZ\u kwN_*zp§ 1. -c*/
(123) ³c ¤ . . Y Z_B]rll_*£vksko_^u t(mZ\uvk wN_*zwz_{¦wDkn_p
(124) rxt
(125) moZ\_.rxz¶KrxP=£ ¦_^znmGwtwzw¦wDknuP: 5;=§ = c- K1324 ?;kn_^_.wx£ kor: K;Aq*wxt(¦N_l_µt\_^7wxkwt7_*©D_*zxzr%cutKknmozsq¤ms\zo_.u tacZ\uvqZ¬¡xwmG_^wxqZ7k}m_*¬¡xt\_* ©x_*zomouvq_pk*¡ t_* _^lD_^k{rxz{w
(126) q*rx]a¦\utw%mou rxt-rx9moZ\_pkn_eq*wt-¦N_KwD\l_p¢§ _^q*u kou rxtk{rxt-cZwm{morawx\7mr
(127) moZ__«luvk}mut\ Dzwx\ZVwxzo_]-wx\_·w%m
(128) zwtlrD]'¦wDkn_prDtVmZ\_-©%w£ s\_^k)rxw"tjs\]a¦_^zrcl_µt\utLwzw]_m_*zk*§-YZ\_ _*«lu knmo_^tq_rx9moZ\_pkn_wzw]_m_*zkG]-w¶x_pkmoZ_]rj\_*£N©x_*zi(D_*t\_^zwx£;V§ _\rDzQmoZ_\s\zrTkn_pkQrxNmZ\uvkQwx_^z^¡ mr7wX©Drxuv·q£ slmomo_*zu t\(mZ\_l_^kqzu lmou rxt·rx rDs\zBz_^kos\£mk^¡jB_e\z_®_*z.mra]-w¶D_ew7lzwxknmouvq¨kou]\£ uµq*wmou rxt¬§ °²_¨cu £ £qrDtkou l_^z{Dzwx\ZkGD_*t\_^zwmo_^wDq*qrDz\ut\emr
(129) m}.rzwmoZ\_^z{_«jmzo_^]7_¨\zorlq*_^ls\z_^k{l_*zu©D_^a®zrx] mZ\_D_*t\_^zwx£¨]rll_*£=§ t±_^wDqZªq*wxko_LmZ\_VD_*t\_^zwmou rxt \zrlq_^kk·uvkxr%©x_^zot_^ ¦ji w knu t\x£ _u tTmo_^x_*z wzw]_m_*z m §GYZ\_¨ko_^qrDtr¢mZ\_^ko_¨]u ]u q^kQmZ\_¨\z_®_^tTmouvw£wmnmwxqZ\]_*tTm{Z\_*t\rD]_*t\rDt¬§GYZ\_cµzk}m rDt\_x¡Tz_*£vw%mo_p7mor
(130) ]rDzo_cmzwDlumou rxtwx£zwtlrD]Dzwx\Z]rj\_*£vk*¡juvk{qrDtkou l_^zo_p]-wu t\£i7®rDzBqrD]wzu korxt?§ l %
(131) ½ ¹ ^ YZ\_Kutumu wx£Dzwx\Z G uvkBwaknu t\D£_©D_*zomo_*« v cumZ m £ rjrxkBw%momwxqZ_^7mor umpe§ _\rxz ¡£ _m ¦_
(132) moZ\_
(133) Dzwx\Z"x_^t\_*zw%m_^ut"mZ\_µzk}m knmo_*krmoZu k¨\zorlq*_^kk*¡ morl_µtt_ ≥G1 w·Gt\_* ©x_^znm_« v u k¨x_*t_*zw%mo_p¢¡wxt"umKuvk¨qrDt\t\t_p−q¤mo_p1mor G moZzorDs\xZ m ?®s\tluz_^qmo_pH Ac_plx_pk*§YZ\_(t\_*u xZj¦Nrxs\zkcr wxzo_
(134) qZ\rTkn_^t"s\t\u®rxz]7£ iLw%m¨zwtlrx] ?®swx3z AcumoZ v zo_^\£vwxq_^]_*tTm®zorD] {v , . . . , v } § ¹ w% mo½mwx qº Z_^ % mr
(135) ump§ _½ rx z ¹ ^ YZ\¡B_xuzt\wu\mouvZ w£Kxzw\Z uvkGl_µt\_^uv k·®w zorDkn] u t\D£_L©D_*zomo¦T_i « Dv_*t\_^czwumomoZ u t\mw£ rjt\rx_^ k ©x_^znm_« v ¡{wt qrxtt\_^tqmo≥u t\1 um(mr G G mZ\zrxs\DZ m ?Çs\tlu z_^Gq¤m_^H A(_^\x_^k^§ = ©D_*zomo_« u ∈ u keqrDt\t\_pq¤mo_p"mor v cumZ\zrx¦wx¦\u £um}i ?®cZ\_^zo_ {v , . . . , v } Γ(u) = {w : {u, w} ∈ A§ E(G )} °²_(cu £ £rxÉmo_^t~zo_*®_*z¨mor·mZ\_7swzKxzw\ZL\zrlq_pkokKwxkew·zwtlrD]³Dzwx\ZL\zrlq_^kk^§°V_7sko_
(136) mZ\_(.rxz q*rxji-wxkButTm_*zqZwt\D_^wx¦\£_¨cumZ\z_®_^zo_^tTmouvw£9w%mnmwxqZ\]_^tDmp§GYZ\_K_plx_pk{ZwX©x_Kwxtu tTmozutkouvq¨lu zo_pq¤murDt cZu qZB_)u xt\rDzo_D§ .rxmoZ~]rll_*£vk)wz_alijtwx]u qx¡¢cumZt\_^ ©x_^znmu q*_^kewt_^lD_^keqrxtTmutjs\rDskn£ i"utqz_^wxkou t\-moZ_7kou^_arx mZ\_cxzwZ¬§ U¨r%B_^©x_*z moZ_*i(z_*zo_pkn_^tDmm}.re_*«jmoz_*]_cq*wxko_^k^§ t7mZ\_cswzGDzwx\Z(zorlq_pkok mZ\_mo_^zo]u tw£ ©D_*zomo_*«·r wxt_plx_eu kcqZrDko_*tzwtlrx]£ ia®zrx]moZ\_ko_mcrwX©Xwxu£vw¦£_K©x_*zomouvq_pk*¡jcZ\_^zo_pwxk.u tmZ\_qrxji ]rll_^£{mZ\_·m_*z]7u twx£{©x_^znm_«rwt _^lD_·u kaqZ\rDko_*t zorDrDznmurDtw£Gmr~mZ\_q*s\zoz_*tTm(l_^xz_*_·rxcmoZ\_ ©%wxzou rxsk.©D_*zomouvq_^k^§ R,m t. R,m 0. R,m t−1. 0. R,m t−1. t. t. 0. t−1. C,m t C,m t−1. t. 0. t−1. C,m t−1. Õ Õ Úk ¤á. *
(137) *
(138) . t. C,m 0. C,m t−1. |Γ(u)| 2mt. 0.
(139) . ]c324 ^3
(140)
(141) gS. Y wx¦\£_[CRQ s\]_*zuvq*w£¢¦Nrxst\k.®rxzmZ\_]utu]as\] \rx]utw%mou t\-ko_mc\zrx¦\£ _*]"§ m. C. . 5 X. . . § 5D b ^§ C.bxN b ^§ CcXRbNb ^§ CXx ^§ CRC^D § DN b § R bxN b. \§Ñ \§ 5 CX \§ 5xTX \§ÑR 5KX \§ÑK 5R5D \§Ñ CRC^ \W§ C^R 5D. αR lo. \§ÑR5x \W§ C^Dx \§ x \§ DK XNX \§ K X5x \§ N 5Dx \§ N 5HCc5. αR up. αC lo. \§ 5N5N5R5 \§ÑK 5KXj \W§ C \W§ CXTD \W§ CNCKb \W§ C^xD \§ N bxT. αC up. _^q*wx££mZw%m γ (G ) uvkmoZ_~knu *_LrxKmZ\_L]ut\u ]-w£ h¥Ølrx]u tw%mut\ ko_m·u t G §>°²_"µt rTknumou ©x_ q*rxtknmwxtTmk ¡ knsqZ²mZw%m wx£]rDknm(kos\zo_^£iD§YZ_·s\\N_*za¦rDs\t\k q*rx]_®zrx] αmZ\_¨αkou^_rNmoZ_KlrD]7u αtwmo·u t\t
(142) ≤kn_*γmQzo(G _*mos\z)t\≤_pα¦Ti7kn·u ]t \£ _rxtl¥´£ ut\_wx£DrxzumZ\]-k*¡xcZ\_*z_^wDkmoZ\_ £ r%._*z¦rDs\t\kcq*rx]_)®zrx]\zrx¦w¦\u £uvknmouvqKwxzoDs\]_*tTmk^§ d~rxz_(\z_^q*u ko_*£ (i Q ¬_m ¡¢wt ¦N_-w·NrDkoumu©D_au tTmo_^x_*zpa§ _\rxze_^wDqZ mZ\_*z_a_*«lu knm NrDkoumou ©x_)zo_pw£¬q*rxtknmwxtDmMk α∈ {R, wt C}α ?Çl_^m_^tl_*tTm¨rDt M ¡ m wt h ¦\slmu tl_^h_^t≥l_*1tTm¨rx tA.cumoZ kosqZ-mZw%m w\§ w\§ k^V§ _rxz h = 1 ¡TmZ\_¨¦Nrxs\t\kwz_xu ©x_^t α <α <1 u tLY wx¦\£ _ CD§ BrDs\t\k.®rDz h α> 1· wxt zo≤_lγuvko(G q*skkn_p)u t~≤gjα_^qmou ·rxtt"\§ YZ_\zorjrxrx mZ\_(w£ xrDzoumoZ]7uvqezo_pkns£mkwz_)¦wxko_^rxtmZ\_ÇwxqmcmoZwm¨twmos\zw£¬_plx_*¥=_*«lrTkns\z_e]wxzn¥ mutDw£ _^kq^wt¦N_l_µt\_^rDtVmZ\_·Dzwx\ZVzorlq_pkoko_^k)stl_*zaqrxtknuvl_*zw%murDf t : < ;´§d~rxz_7zo_pquvkn_^£iD¡?u v u k w j t i D z x w \ Z m \ Z * _ D r o z * _ o m v u q ® s t ¤ q m u D r t ?Ç_x§ § moZ\_~kou^_rx¨mZ\_~lrx]u tw%mut²ko_m-z_moszot\_p ¦ji w f (G) wzomouvqs\£vwz
(143) wx£DrxzumZ\'] A¤¡ moZ\_zwxtlrx] \zrjq*_^kkl_µt\_^ ¦ji²kn_*mnmou t\ Z = E (f (G )) ¡Qwxt Z ?®®rxz Amr(¦N_moZ_K_*«jN_^qmwmou rxt-r qrxtlumurDt\_^rDt-moZ_ _«lrTknszo_ crx¢moZ\_ µzknm ii ∈_^l{1, D_^k. .u t. , momt} Z\_Dzwx\Z\zrlq_pkokuvkcw7]wxznmutDfw(G £ _x!§ ¨rmo)uvq_emZw%mcmZ\_
(144) knwxq_)rxwx££¢Dzwx\Zk.cZ\uvqZ q^wt-¦N_KD_*t\_^zwmo_p7wDq*q*rxzlut)mor
(145) mZ\_¨Du©D_*t·]rll_^£ G uvkQwxznmumurDt\_^-u tTmor7q£vwxkkn_pDk ?®rDz i Y -"WcM< A q*rxtTmwxutut\w£ £¢mZ\rDko_)xzw\Zk.cZ\uvqZ"qrxu tq*u l_§ zp§ mp§ mZ\_)µzk}m i _^lD_e_*«lrTkns\z_^k^§ t"mZ\_®rDznmZqrD]7u t\-ko_^q¤murDtkc._
(146) cu ££¬z_*N_^wmo_pl£isko_)moZ\_
(147) ®rx£ £r%cu t\·qrDtq_^tTmozw%mou rxtz_^kos\£m ?®®rxz¨w zorjrkn_^_x¡j®rDzcutknmwxtq_D¡ :^C;A§ h. lo. t. t. up. lo. M lo. M lo. M up. h. t. up. M up. M lo. h. M,m t. M up. M,m t. M,m t. 0. i. M,m t. .
(148) ¹j(½ H ¹"
(149) . 1}. .
(150) . ". -h aO%
(151) ^$>. c = Z 0 , . . . , Zn √ 2 [|Zn − c| > λ n] ≤ 2e−λ /2. |Zi+1 −Zi | ≤ 1. !"K$. i ∈ {0, . . . , n−. t w£ £rxs\z-wx\\£ u q^w%mou rxtk c = E (f (G )) ¡ n = mt wt λ = O(log t) § t rDz\_*z7morw\£i YZ_*rxz_*] \§^Crxt\_t\_*_p\k(mor²\zor%©D_·mZw%m |Z − Z | ≤ 1 § gjsqZ ut_
(152) Tsw£ um}iV®rD££ r%k(®zorD] moZ\_ ko]rjrmoZt\_^kker ?®u=§ _D§ u wt \u 9_*z§ z^§ m^§(moZ\_\z_^ko_*tq_7r.w"koutx£ _ _plx<_ Awt(moZ\_wxf¦\u £um}imor|fl(G) _^]rx−tknfmoz(H)| w%m_{mZ\≤_c1_«luvk}mG_*tq*_.rxNHwe]_^wDkns\z_.\zo_pkn_^zo©ju t\¨¦u |}_pq¤mou rxt¦N_m}._*_^t M,m t. i+1. i. ÿ Õ¬ÿÉü .
(153) . .
(154)
(155) !"$#
(156) %
(157) '&()*
(158) ,+.-'/01324. ¥´¦\£rlq¶lkeut wLkw]_ ¥=¦\£ rlq¶9§·YZ\u k
(159) u k)rD¦T©ju rxsk)u t²moZ\_-zwtlrx]'Dzwx\Z²\zorlq*_^kkwxk)_^\x_^kwxzo_ u tko_*zomo_put\_*N_*tl_^tTmo£ ix§ i t-moZ_Kq*wDkn_crx¢moZ\_eqrDTia]rj\_*£Num.u kBqrxtj©D_*t\u _*tTmGmorau \_*tTmou®ia_pwxqZ-Dzwx\Z cumoZ moZ\_·zorx|}_^q¤murDtr¨wLwxznmu q*s\£vwzm}ijN_·rM3
(160) .iN 1%I
(161) ¡u=§ _x§~wt rDz\_*z_^qrD££ _^qmou rxt r mt £vw¦N_*£ £ _^7wu zkr9rDutTm$k ?Çko_*_ : X;A§!¬_m wxt ¦_m}.rknsqZq*rxtlµxs\zw%murDtk moZwm{wz_.uvl_^tDmu q^w£ smrmZ\_ imZVwu z^§agls\\NrDko_amZ\_ i +C1¥Øk}m)wCu z)u k {a, b} u t C wt {a, c} ut C § C t\_^©x_*z skn_pk)rDutTm b wTwu tmZ\_*tmZ\_-u]-wxx_r C s\tl_^zmoZ_-]7_pwxkos\z_7zo_pkn_^zo©ju t\¦u |}_pq¤mou rxt cu £ £Q¦_·w q*rxtlµxs\zw%murDt u \_*tTmouvq*wx£lmor _«\q_^lmQmoZw%m{wxuz b} uvkQz_*\£vwxq*_^7¦ji(wxuz {a, c} § b u k{skn_p u t w"®rD££ r%cu t\LCwu 9z ?;kowXi {d, b}CAr C mZ\_*t C c{a, u £ £{ZwX©D_ u tknmo_^wDr {a, b} wt {d, c} u tknmo_pwxr {d, b} ¡lwtkor7rxt¬§{gju ]u£vwzq*rxtknmozsqmou rxtu k\z_^ko_*tT{a,m_^c}ut : ;´§ _u tw£ £ i·B_)cu ££?t\_*_pmZ\_)®rx£ £ r%cut\ B¹ ! Z O%
(162) ^eK
(163) $H"heR JMhK
(164) 4L
(165) H) c , c > 0 HMhT4 µ = E (Z ) ∈. i+1. 1. 2. 1. 2. 1. 1. 0. 1. 0. 1. >4
(166) Tn!"3NM3ai3
(167) HLSR ^ . [c1 n, c2 n] |(Zn )j − (µn )j | ≤ Knj−. j>0. H"1 R )]2H<1)2MhK
(168) 4L
(169) H.
(170) n. K. ½¬ ½ u kw]-wzomou t\Dwx£_D¡9moZ\_^t²¦jiYZ_*rxz_*] \§^CD¡ √ § £=§ r§ §._Zwxkkns]7_ λ = o(√n) §]_\zrx]/mZ\uvkcB_wx£ koraZwX©x_xµ¡j®rD−zc_^λwDqZn µ\≤«l_pZu tT≤mo_*Dµ_*z n. n. (µn )j (1 −. √ λ n j µn ). n. ≤ (Zn )j ≤ (µn )j (1 +. √ λ n j µn ). n. . HMh n 4K. √ +λ n j>0. ¡. cZ_*z_. §. YZ_zo_pkns\£mt\r%±®rx£ £r%k ?Ç\zr%©Tuvl_p K uvkqZ\rTkn_^t¦\u -_*trxs\DZPAknu tq*_x¡lmZ\_(wxkkos\]lmou rxtkrDt λ wt _*tTmwxu£9mZw%m (1 + ) u kcwm.]rDknm 1 + ¡lcZ\_*z_^wDk (1 − ) u kcwm£ _^wDk}m 1 − 2 µ √ λ n j µn. n. √ j2 λ n µn. √ λ n j µn. √ 2jλ n µn. +^! 4c¬
(171)
(172)
(173) !. YZ_~w£ xrxzumoZ\] zo_pkn_^tDm_^ ut mZ\uvk·kn_pq¤murDt uvk·w©D_*zi knu ]\£ _ µzknmw%mnm_*]lm Lkorx£ slmou rxt ®rDz-moZ\_ zorD¦\£_^]³wmZwt¬§ =¨£mZ\rxs\DZLutL]-wxtTiq*wDkn_pkumK\rT_pkt\rxm£_pwxmorw©x_*ziko]-w£ £?lrD]7u twmou t\·kn_*m^¡ umz_*\z_^ko_*tTmkw7twmos\zw£¢¦N_*tqZ]wxzo¶®rDz¨wtji-]rxz_)zo_*µt\_pZ\_^s\zu knmouvq§ ( ½
(174) Ç
(175) ._®rDzo_moZ\_¨µzknm.knmo_^·r¢mZ\_)w£ xrxzumoZ\]moZ\_Kxzw\Z·q*rxtkouvk}mkQr?waknu t\D£_©D_*zomo_*« v wxt S = {v } *§ =.meknmo_* t uGmoZ_(t\_^c£i"x_^t\_*zw%m_^"©D_*zomo_« v lrj_^kKt\rxm¨ZwX©x_awtjit_*u xZj¦rDs\zku t ?Çu;§ _x§ AQmZ\_*t u k¨wx\l_p·mr S § S Γ(v ) ∩ S = ∅ v tVmoZ_®rxzomoZq*rx]ut"\u kqskkourDt m uvkwµ\«l_^²rTknumou ©x_u tTmo_*D_*zp§ ¬_m X \_*t\rxmo_7moZ\_·kou *_r.moZ\_ \rx]utw%mou t\kn_*m S q*rx]\slm_^¦ji =£ xrDzoumoZ\] Ce¦N_®rDzo_ v uvkcwxl_^mor7moZ_
(176) qs\zzo_^tTm.Dzwx\Zwxt£ _m § _\rDzxzw\ZkKx_^t\_*zw%m_^Vwxq*q*rxzlu t\·mrmZ\_ [ ]rj\_*£=¡¢moZ\_\zrx¦w¦\u £um}i"moZw%m v µ = E (X ) ]uvkoko_^k.mZ\_
(177) lrD]7u twmou t\-kn_*muvk (1 − ) ]§ U_*tq_._)q^wtGczoumo_. 0. 0. t. t. t. t. t. t. t. Õ Õ Úk ¤á. *
(178) *
(179) . Xt m t. R,m t. t.
(180) ]c324 ^3
(181)
(182) gS b. µt+1 = µt + E [(1 −. ?_m ¦N_
(183) mZ\_(st\u
(184) Ts\_7knrD£s\mou rxt~rxmZ\_7_
(185) Tsw%murDt mZ\_©%xw=£ s\_^x(m) krx x ®rDzmoZ\_)µzknm®_*¯©Xwx£s_^kr m §. B¹ . . . 4KP!"K . K'
(186) . . Xt m t ) ]. §. x = (1 − x)m. u t. §¨Yw¦\£ _7Du©D_^k. (0, 1). . . M3
(187) HSK
(188) H 041' R IO
(189) fR- S 2H<% < M3
(190) H3L
(191) P ^ . 1 2. <ρ<1 t > 0 |µt − xt| ≤ Ctρ. ½¬½ °²_
(192) q£vwu ]/mZw%mcmZ\_
(193) lu N_^zo_^tq_ . |µt − xt|. C. 3 HM3. kw%mu knµ_^kcw7z_^q*s\zoz_*tq_ermoZ\_)®rDzo]. |µt+1 − x(t + 1)| ≤ |µt − xt| + O. q. log t t. . §. YZu kBq*wt-¦N_\zr%©x_p7¦Tiautlsq¤murDt·rxt § .i7\_µt\umou rxt ¡TZ\_*tq*_ §°V_Kw£vknr ZwX©x_ |µ − x(t + 1)| = |µ − xt + E t[(1 − ) ] − (1X− x)= 1| §GYZ\_Kl|µu9_*z−_*tx|q*_ =E1[(1− x− ) ] − q*wt-¦N_¨z_*czumnmo_^t·wxk − (µ − xt) + {E [(1 − ) ] − 1 + m − (1 − x) + 1 − mx} § (1 − x) U¨_*tq*_ § |µ − x(t + 1)| = |(1 − )(µ − xt) + {E [(1 − ) ] − 1 + m − (1 − x) + 1 − mx}| Y r-qrD]7£_*mo_emoZ\_zorjrt\rmu q*_)moZwm^¡\¦ji ?_*]]-w Cx¡ t+1 m. m t. m t. t+1. E [(1 −. wxt mZ\_~®s\tq¤mou rxt § z , z ∈ [0, 1] 1. 2. Xt m t ) ]. 1. Xt m t. t. Xt m t. t. f (z) = (1 − z)m − 1 + mz. µt t. Xt m t. t. − 1 + m µtt = (1 −. 1. m. µt m t ). µt t. − 1 + m µtt + O(. kowmouvk}µ_pk. m. Xt m t. m. q. log t t ). §. |f (z1 ) − f (z2 )| ≤ m|z1 − z2 |. ¡c®rxz 2. Yw¦\£ _
(194) Q!s]7_^zouvq*wx£¢©%w£ s\_^k\_µt\_pu t¬_^]7]-w·\§ m. C. . 5 X. . . § § 5RbD § 5HC § D § K XT4 C § D Cp § xN 5T. x. ÿ Õ¬ÿÉü .
(195) . .
(196)
(197) !"$#
(198) %
(199) '&()*
(200) ,+.-'/01324. YZ_-®rx£ £r%cu t\YZ_*rxz_*]¿uvk(w\uz_^qmaqrDtkn_
(201) Ts\_*tq*_·r ? _*]]-w~wxt²moZ\_qrDtq_^tDmzwmou rxtz_^kos\£m ]_^tDmurDt\_^u tmZ\_\z_*©ju rxskko_^q¤murDt¬§.
(202) ¹j½ ¹ . . 7. Xt ∼ xt. ^ . ! 4
(203) c
(204) 44·*!#) (
(205) (!
(206) 4 4O8.c?. =¨£mZ\rxsx[ Z =£ xrxzumoZ\] Ccuvk!
(207) Ts\umo_kou]\£ _x¡um{ko_*_*]-kQ\u E·qs£mGmre¦N_^wm^¡TwxkGwex£vwtq*_cw%m u t-Y wx¦\£ _*C xw taekoZ\r%k^§ YZ\uvku k_pknN_^q*u wx££ i)mozs\_.®rxz m = 1 cZ\_^zo_.t\r)u ]7zor%©D_*]_*tTmq*rxs\£va¦_αrx¦\mwu t\_p¢§V_rxz £vwzx_^zQ©%w£ s\_^k{r m ¡Tw(¦_*mnm_*z..wXiar¢µt\ut\7kn]-wx££Nlrx]u tw%mut\7kn_*mk{u kBrx¦lmwu t\_^¦jirlq*q^wxkourDtw£ £i wx££ r%cu t\ ©x_^znmu q*_^kmr ¦_²lzrx\N_^ ®zrx] S § mu kqrDtT©D_*t\u _*tTm-mr q£vwxkknu®i moZ\_V©x_*zomouvq_pk-u t moZ\_ \rx]utw%mou t\kn_*mwxk 24O
(208)
(209) H]?;kn_*m P ABwxt 1S2(^NM33N-^?Çko_m RA¤§QYZjsk S = P ∪ R § ?_m P wt \_*t\rxmo_emoZ\_
(210) kou *_^krxkosqZ"kn_*mkcwmcmou ]7_ t ?Çko_m P = 0 wxt R = 1A¤§ R ( ½
(211) Ç
(212) ._®rDzo_moZ\_¨µzknm.knmo_^·r¢mZ\_)w£ xrxzumoZ\]moZ\_Kxzw\Z·q*rxtkouvk}mkQr?waknu t\D£_©D_*zomo_*« v wxt § =Émo_*z uvkcqz_^wmo_^wtq*rxt\t_^q¤m_^mor t_*u xZj¦rDs\zk*¡ju ) ∩ P 6= ∅ moZ\_^t v ] uvk]r%R©x_p=-{v mor V} \ S §Bb¨mvZ\_*zcu ko_ v uvkcwxl_^mor R u Γ(vm) ∩ R = ∅ ¡lrmoZ_*zΓ(v cu ko_ v uvkcwx\\_^mor P wxtw£ £¢©x_*zomouvq_pkut Γ(v ) ∩ R wxzo_e]r%©x_^mor V \ S § YZ_e_*«l_pq¤mwmou rxtk π = E (P ) wt ρ = E (R ) kw%mouvkn®iQ R up. t. t. 1. 1. 0. 0. t. t. t. t. t. t. t. t. _µt\_. αR up. wDk. t. t. πt+1. = πt + E [(1 −. Pt m t ) ]. ρt+1. = ρt + E [(1 −. Rt t. p+r. ¡cZ_*z_. p = p(m) r. B¹ . . =. wt. −. − E [(1 −. Pt m t ) ]. Pt t. −. Rt m t ) ],. − mE [ Rtt (1 −. r = r(m). kowmouvk}®i. 1 2. <ρ<1. Pt m−1 ]. t ). (1−p)m −p 1+m(1−p)m−1 ,. p = (1 − p)m − (1 − p − r)m .. K[K
(213) HM3B4KP!" . t. . . M3
(214) H3L
(215) P 041' N $R- S 2H<% <9M3
(216) 4L
(217) PI ρ
(218) ( ρ ^ . t > 0 |πt − pt| ≤ C1 t. |ρt − rt| ≤ C2 t. wxzo D½¬s\½ ] _* tTmYuZ\tL_Lkorx\]zrj_rKl_uvmk^w¡.u _p£ k.koko®_*rDtTz mu wx£§]£ ix=.¡mcw²moZx__*tu_*tz\ws£ u q¤kmw%um©Du_)rDt knmo_^rJeQ moZw%mrx ?_*]]-w . πt. Õ Õ Úk ¤á. *
(219) *
(220) . C1.
(221) . C2. l§ª°²_L\z_^ko_*tTm-moZ\_.
(222) Cp. ]c324 ^3
(223)
(224) gS. |πt+1 − p(t + 1)| = |πt − pt + E [(1 − Ptt )m ] − (1 − p)m − {E [(1 − Ptt + Rtt )m ] − (1 − p − r)m}|. §. YZ_\zrTrx¢u kBqrx]\£ _m_^-¦jil_^q*rx]NrDkout\)mZ\_K\u 9_*z_*tq*_^k E [(1 − ) ] − (1 − p) wxt E [(1 − u twzomk.mZw%m¨wxzo_)\zrxNrxzomou rxtwx£mr-_*umoZ\_^z P − π rDz π − pt § − ) ] − (1 − p − r) YZ_ez_^kos\£m¨w¦Nrxslm ρ uvk\zr%©x_^knu ]u£vwz£ iw%Ém_*zct\rxmouvqu t\7moZwm r kw%mu knµ_^kcQ § r = (1 − p − r) − mr(1 − p) Rt m t. Pt t. Pt m t. m. m. t. t. t. t. m. m−1. ?_m l_*t\rxmo_BmoZ\_ckou^_BrxmZ\_.\rx]utw%mou t\)kn_*mz_ms\zt\_^(¦TiT=¨£DrxzumZ\] ¨cZ\_*t7zost(rxt7w¨zwtlrx] Dzwx\XZzorlq_pkokstDmu£¬mou ]7_ t §. 2. 2 t.
(225) ¹j½ ¹ 0 ^ ½¬½ YZ_*rxz_*]l§WC
(226) u ]7£u _^kmoZwmmoZ\_7kns] z_^kos\£m®rx£ £r%k^§ . . YZ_e©%wx£s\_pkcr. 7. Xt2 ∼ (p + r)t. ®rxz p+r. p+r. uvk¨w\§ w\§ k^§B©D_*ziq£ rDko_mr. wz_Kz_*Nrxzomo_pu tmoZ\_
(227) q*rx£ s\]t£vw¦N_*£ £_p m≤7. αR up. |S| t. =. Pt t. +. Rt t. §YZ\_ 2. rGY wx¦\£_[CD§. ! 4
(228) c 44 · *!#) *
(229) ¿4" B8 4,4O8.c?. =¨£DrxzumZ\] Cx¡el_^kqzu ¦_pªu t¯gj_pq¤mou rxt 5\¡)q*wxtª¦_ wxtw£ ilkn_p ut±moZ_²q*rxji ]7rll_^£
(230) wxkB_^££=§ YZ\_ *_ «l_pq¤mo_pqZwxt\x_KutmoZ\_e©%wzu wx¦\£ _ X q*wxt¦N_eq*rx]\slm_^·¦ji-¶D_*_*ut\(mozwxq¶7r¬mZ\_emorxmw£9l_^xz_*_¨rx mZ\_
(231) lrD]7u twmou t\-kn_*m^¡ D § twzomouvqs\£vwz.moZ_)®rx£ £r%cu t\zo_^£ wmou rxtkoZ\u k.Z\rx£v t. t. E (Xt+1 ). = E (Xt ) + E [(1 −. E (Dt+1 ). = (1 +. 1 2t )E. Dt m 2mt ) ],. (Dt ) + mE [(1 −. Dt m 2mt ) ].. ¨rmGknszo\zuvknu t\x£ iwt7wtwx£ilkou kknu ]u£vwzmr¨mZ\_rxt\_l_^kqzu ¦_p
(232) u t(mZ\_\zo_^©jurDsk ko_^qmou rxtku ]\£u _^kmZw%m kosqZLw£ xrDzoumoZ]zo_*mos\ztkclrD]utwmou t\·ko_mkrGknu *_ xt u t G § Ur%._*©x_^z^¡ju t"moZ\_aqrxji·]rll_^£¬B_ q^wt-u ]\zor%©D_rxtmoZu kB¦jia\sknZ\u t\
(233) ZuDZ·l_*Dzo_^_c©x_*zomouvq_pkGu t-moZ\_Klrx]u tw%mut\akn_*m^§GYZ\_twmos\zw£.wXi mrewDq*q*rx]\£ u koZ
(234) mZ\u k.rxs£ (¦_D¡®rxzGwxtTit_*c£ ix_*t_*zw%mo_p
(235) s\tqr%©x_^zo_p©x_^znm_«¢¡Xmor)ko_*£ _^q¤mQwKt\_*u xZj¦Nrxs\z rx{]-w%«lu ](s\]'l_^xz_*_7wt~wD\~um)mor § ¨tl®rxzomos\tw%mo_^£i~kosqZwx£DrxzumZ\]u ket\rxme_pwxkoimor"wtwx£ilko_ ¦N_^q^wsko_7utVmoZ\_-Dzwx\Zzorlq_pkoko_^k¨moZw%SmB_-q*rxtkou \_*zKmoZ\_^zo_]-wXiL¦_®_^ ©D_*zomouvq_^kerxBzw%moZ_*ze£ wxzoD_ \_*xz_*_ ?®moZ\uvkGuvkQwqrDtkn_
(236) Ts\_*tq*_rNmoZ\_cNr%B_^zG£ wX luvk}mzou ¦\slmurDt7rxN©D_*zomo_*«al_*Dzo_^_^0k : 5\N¡ b;A§ U¨r%B_^©x_^z . C,m t. ÿ Õ¬ÿÉü .
(237) .
(238)
(239) !"$#
(240) %
(241) '&()*
(242) ,+.-'/01324. CRC. w)lzwxq*rxt\uvwta©x_^zkourDt(rxmZ\u kQZ\_*s\zuvk}mu qq*wxt¦_wxtw£ ilkn_p¢§ YZ\_.®rD££ r%cut)wx£DrxzumZ\] mwx¶x_pkGwxku t\\slm wxtwD\lumou rxtwx£¢utTm_*x_^zcwzw]_*mo_*z k > 0 § ( ½
(243) Ç
(244) B_*®rxz_emoZ\_µzknm¨k}m_*"rmoZ_(w£ xrDzoumoZ\]moZ\_
(245) Dzwx\Z"qrxtknuvk}mkcrGw-kout\D£_uvkorx£vw%mo_p ©D_*zomo_*« v wxt S = {v } V§ =Émo_^z v uvkBq*zo_pw%mo_p-wt-q*rxt\t_^q¤m_^7mr m t_*u xZj¦rDs\zk*¡D£_*m Z ¦N_mZ\_Kko_m rxwx££¢t_*u xZj¦rDs\zkr u t rGl_*Dzo_^_ + 1 § Z 6= ∅ mZ\_*t~wx££¢©D_*zomouvq_^ku t Z wz_wx\l_p mr S §~b¨moZ\_^zocuvkn_u vv uvkt\Vrxm(\ Slrx]u tw%m_^¦jkm iVkorx]__*£ _*]_*tTm(rx S mZ\_*t w"©D_*zomo_*«r]-w%«lu]as\] \_*xz_*_eu t Γ(v ) uvkcwx\\_^mor7moZ_
(246) lrx]u tw%mut\-ko_mp§ u kD_*t\_^zwmo_^wtqrDt\t\_^qmo_p·mr ¡\w£ £N©D_*zomouvq_pkBcZrDko_el_^xz_*_eZwxk ¨rmu q*_KmZw%mwÉmo_*z_pwxqZ ¦N_^q*rx]_£ wxzoD_*zemoZwt kmv wz_]7r%©D_^utkouvl_ S §YZ\_-wxtw£ ilGknuvk)rBmoZ\_-_^©xrx£ slmurDtVrx |S| uvk¦wDkn_p wxDwxut"rxt~moZ\_7l_µtumurDt~rQw·zwtlrx]³\zorlq*_^kkcmoZw%me\_^kqzu¦N_^kmoZ\_7w£ xrxzumoZ\]lijtw]uvq*kKwt"rDt mZ\_\zrjr?mZw%m¨kosqZ\zrjq*_^kk.¦_^ZwX©x_pkBu tLw7\zo_pluvq¤mwx¦\£ _.wXi®rDzc£ wxzoD_ t § ?_m § _rxz._^wDqZ i ∈ {0, . . . , n − 1} wt t > 0 ¡jl_µt\_ Y = |V \ S| u t G V +1 ? u nkGmo=Z\_¨(kko_−mB1)m r 9 x © * _ o z o m v u q _pkGr¢l_^xz_*_ A¦_*®rxz_ u kQwD\l_p7mor)moZ_xzw\Z?§ ?_m l_^t\rm_moZ\_mormw£ \V_*xz_*_u tkou \_ S ?Çu;§ _x§ Y = P i|Γ(v)|AKwtv X mZ\_akou^_
(247) rmZ\_alrD]utwmoYu t\ko_me¦_*®rxz_ v u k wD\l_p"mormoZ\_7xzwZ¬§eYZ_7knmwmo_(rxQmoZ_7koilk}m_*]"¡¢w%mK_pwxqZknmo_^ ¡¢uvk¨]rll_^££ _^¦TimZ\_ ?ÇzwxtlrDO] A ©D_^qmorxz (Y , . . . , Y , X ) § rmu q*_(mZw%mp¡9®rxze_^wDqZ t > 0 ¡NmoZ_a©%wtzuvw%mou rxtutV_^wxqZrGmoZ_a©%wzuvw¦\£ _^k uvkw%m]rDknm m O§ =¨£ kor¡ Y + P (m + i)Y = 2mt ¡?wxt¢¡?wm)_pwxqZk}m_* t ≥ 1 ¡?cZ\_*t v u k q*zo_pw%m_^mZ\_)\zorD¦w¦u£ um}i7mZw%mcum.Z\umkcwa©x_^znm_«-rl_*Dzo_^_ ¡D®rDz ?®z_^ko¬§ mZ\_¨lrD]utwmou t\
(248) ko_3m AGut·wxtTi(rxt\_rxmZ\_ m mozu wx£ kQwX©%wxu£vw¦\£ _.mmor+uim{uvk{wi\∈zorX{0, «lu ]-. w%. mo. _^, £(k9i ?®−rx]1)m} umnmut\ o(1) ÇwDq¤mrxzk*¡¢®rDz t £vwzx<_ AK_
(249) Tsw£Qmor P = ?ÇcZ\_*z_ u wxt²*_^zor δ =1 i=n rxmoZ\_^zocuvko._ A¤§ _rxz ?®z_^ko¬§ A¤¡T£ _m l_*t\rxmo_cmZ\__^©x_*tTm v ]7uvkkn_p S wtamoZ_]-w%«lu]as\] d ∈ {0, . . . , n − 1} d= n E \_*xz_*_But ) u k m+d*?Çzo_pkn¬§ v \u at\rmG]uvkok S A¤§GYZ\_._«lN_^q¤m_^qZwxt\x_{mor Y ¡xqrDtlumou rxt\_p mr-moZ_
(250) \zrjΓ(v q*_^kkcZ\uvk}mrxzis\~mormou ]_ t q*wxtL¦_aqrx]\s\mo_^~¦Ti®s\zomoZ\_^zKqrDtlumou rxt\u t\·rDtmoZ\_(Çw]u£ i rx _^©x_*tTmk (E ) §G°²_
(251) q*wxtczumo_knsqZ
Documents relatifs
cosinus (en bleu à gauche) et d’une droite (en noir à droite) sur les données simulées pour 4 années à une température de 45°C, avec un bruit de fond interne réduit de 100,
Florence CARRÉ ~ Réoccupations funéraires de sépultures collectives néolithiques dans la boucle du
2 Vue plus en détails dans le chapitre 1... 2 phénomène d’agrégation des protéines amyloïdes en fibres et d’autres facteurs impor- tants doivent être pris en compte comme
In this chapter, we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for
Revue française d’héraldique et de sigillographie – Études en ligne – 2020-10 © Société française d’héraldique et de sigillographie, Paris,
Dans ce cas, comme les coulées du Bras de Sainte-Suzanne appartiennent au massif de La Montagne daté à plus de 2 Ma (McDougall, 1971), l’ensemble des coulées pahoehoe
In infantile Pompe disease patients, the glycogen storage diffusely affects brainstem motor and sensory neurons, and the whole spinal cord sensory neurons, interneurons, and
De ce point de vue, il est possible d’affirmer que la mission est accomplie : les projets de renouvellement urbain (GPV et ORU) élaborés depuis 1998 dans une centaine de villes