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Algorithms Seminar, 2000-2001

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(1)Algorithms Seminar, 2000-2001 Frédéric Chyzak. To cite this version: Frédéric Chyzak. Algorithms Seminar, 2000-2001. [Research Report] RR-4406, INRIA. 2002. �inria00072182�. HAL Id: inria-00072182 https://hal.inria.fr/inria-00072182 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

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(117) u }~  ƒ‚ € 47-Wƒ „‡†;-Wƒ ˆ +„4‰,Š,‹+ŸŒ„ X ‹ŽC Š4.„ Y ?. Ji„ Žz‘!K ƒi’ À À À. 47+-. “4”J•–•˜—š™›]œ›žeŸš•– *¡ £¢”¥¤§¦¨Ÿ©”P›©Ÿ©”Wª£«¬¤.—”­6®1—E•z¯4° ± F²³yL1MW´µ³ ¶h·W¸h¹!º¸C»½¼½¹µ¾¨¿¹µ¾=ÀJÁW¼½»½¸`ÂÄÃJÀJ¸C»E¼L¿¹kÅ#¸»Æ»½ÇW»L¹!Èi¹!ÇJÉ6õ‡¹É6õ¾,Ê1Ë£É6¿ÈÇ=»Æ¸¿ÌP¸C¾W¸È6¹µ»Æ¼½ÍC¸4É6·W¸h¿¹µ¾=ÀJÁW¼½»Æ¸ÎÈ6ÇW»Æ¸¿,Ï*ÃPÈ Ìµ¸¾W¸Cȹ!»©ÌµÈ¹!Á=·=¿CÊi¶h·W¼L¿hÂÄÃEÀW¸C»·=¹P¿hº¥¸¸C¾§¼Æ¾šÉÈ6ÃJÀJÇ=ÅC¸ÀкšÑÐÒιµÓ©ÔW¶1¹µ¾WÌ=ÔJ¹!¾¥ÀzÕ]¼½¸¿¸C¾JÏ*¸»½À×ÖÆØÙ¼½¾qØÚµÛEÜEÊ ÝÞ·¥¹!ÈkÖ Ú.Ù¿%·=Ãßà¸À¨É6·=¹É`É·W¸m¿%¸CÉ`Ã!ÏÈ6¸Å#Ç=ÈÈ6¸C¾šÉ`Å#ÃP¾Já=ÌPÇWÈ6¹!ɼ½Ãµ¾=¿iõÏÉ·W¼L¿`¹µÇJÉÃP‡¹ÉÃP¾Ä·=¹µ¿5É·W¸m¿É6ÈÇ=Å#ÉÇWÈ6¸ Ã!Ïi¹âá=¾W¼Æɸ¬¹!º¸C»½¼L¹!¾§ÌµÈ6õÇWÁ,Ê ËH¾§É·W¼L¿hÉ6¹µ»ÆÓ©Ô¥ßà¸ãÀJ¸¿ÅCȼ½º¥¸¬¿¸CäP¸Cȹ!»,¹µ»ÆÌPõÈ6¼åÉ6·W‡¿`É6ÃeÀJ¸#É6¸CÈ6ÂļƾW¸¬É·W¸ã¼LÀJ¸C¾šÉ6¼åÉѧ¼Æ¾§É6·W¸¬ÌµÈ6õÇ=Á,Ê5¶h·W¼L¿ ¸C»½¸CÂĸC¾šÉzÁ=ȸ¿%¸¾PÉ¿‡Ï*ȹµÅ#É6¹!»k¹µ¿Á¥¸ÅÉ6¿eÉ·=¹!ÉÐßθ˜¹!È6¸¾Wõɧ¹!º=»Æ¸–ÑP¸#ÉeÉ6Ãæ¸#çJÁW»L¹!¼½¾,Êè¶h·W¸¿%¸q¹!»½ÌµÃµÈ6¼ÆÉ·W‡¿ ¹!»½»ÆÃß]Ç¥¿`É6ÃļƾšÉÈ6ÃJÀJÇ=ÅC¸kÈ6¸C»L¹É6¼ÆÃP¾=¿%·=¼ÆÁ=¿àº¸#ÉßθC¸C¾zÉ·W¸ã¿6¹!¾=ÀWÁW¼Æ»½¸m̵È6õÇWÁz¹µ¾=ÀÐß฻ƻÆé

(118) ÓE¾WÃ߾й!»½Ìµ¸Cº=È6¹µ¼½ÅÃPÈ Å#ÃP¨ºW¼½¾=¹É6õÈ6¼½¹µ»©ÃµºJê¸ÅÉ6¿Ê ; 1„ +  ‹*ƒ*ëãì  ˜ ˆ i„ÄíJ-WŒŠ,îq‹ïŠ +#‰,„ Ž „ 4.„!Š,+›‚- Ž#ð ë-Jíàñeòp´- Ž ™‹ ƒ ; ò ñr-Wë#ë#‹*Šˆƒ  Š,î ­ mò f  ƒ óˆ ? K B ƒ  ,Š î ; òñl-WëCë#‹*Š  ?‘ ’ Bò R  ‰ „‡† †i„ Ž ë ? ‘ ƒ ‘ À B ƒ+#‰„e;- - ð ? ’ B ƒ  Š,î9+#‰„(+#‰„!ë#‹*ë ? ‘ ~ B EŽ  „ 0:- -îS‹*Š,+ Ž -î,Œ47+#‹3-WŠ,ë+-˜ë  Š *ƒ „!ë!ò. ‘ ò¨ô l¥6µõšö,_X³7: 698

(119) ö¥l. Q3Q3Q. ÷ 1„ +×ø = Xù  Š8-WŠ2°v- Ž ‹ „!Š +„!î  Š,î47-WŠ,Š,„47+„!î Jrsxlnm ¢ f)iAj¾upz ‚¨‹3+#‰ ù =üû ‘ H H#ú Y i„ H ”hý ‹3+#ë–ë„1+-Jí Ž   Ž  Ž Ž Ž óJ„ +#‹54.„!ë Šî ú 눩ì–쁄1+ ‹54 ”ÿþ•” ì + ‹œ ‚¨‰8-W넧„!Š,+ ˆ E ¯ † ‹*ëE+#‰„{Š¥Œìzi„ -Jíބ!î80J„!ëG‚¨‹5+#‰S„!Š,î,†f-W‹ïŠ,+#ë ‚-ƒ*„ò —+ ‹*ë  ëCë#Œ,쁄!î +#‰  + íJ- Žã Š¥ˆŒ‚-ƒ E E = À ë2-‹+#‰  ++#‰„Ä짌,ƒ3+#‹°v0 ŽC †1‰˜‰  ë Š8-{ƒ3- -J†ëò ‹Ž „ŸŒ„!Š +#ƒ ˆƒ×ø ‹*ë  ¯ 0 ŽC †d ‰ ƒ  Š,î扄!Š4.„ E ¯ † ‹*ëã„!‹3+#‰,„ Ž À - Žz‘ ò R ‰,„Æt¥e±f)ikee‹-JíàóJ„ Ž +„hœ ‚k‹*ŠSø ‹*ë E = “ †Â¤ « E ¯ † ò š ìЌ,ƒ5+#‹°v0 ŽC †‰ ‹*G ë ikvvleˆtt‹* í -WŠ „ -Jíh‹3+#ëãóJ„ Ž +#‹54.„!Q3ë¨Q3‹*Q ë¨î,‹*ë+#‹*Š0WŒ,‹*ë#‰„!îdƒ=‹3+¬‹*ël4  ƒïƒ „!îA+#‰„¡z

(120) mog   Š,îæ‹*ë㊍Œ,ìАi„ Ž „!î ” ò š tZvg!¡@frsikjlnm™v#g =  óJ„47+- Ž -JíkŠ8-WŠs°%Š„10  +#‹ óJ„{‹*Š +„10J„ Ž ëòŽ—6Š +#‰„‹47-WŠ +„hœQ+ X  H H ¤ Y §

(121) ¤ -Jí ø ‹*ë  Ž C Ž  E  -J í +#‰„–ë Š,î†1‹*ƒ „{ì -î,„!™ƒ ƒ +#‰„óJ„ +#5‹ 4.„!> ë -Jœ í +#‰,„ 0 †‰ Ž  „ 4.„!ƒ*ƒïë ƒ  Š,î +#‰„–ŠŒ,ìАi„ Ž E ì  ˆ i„–‹*Š,+„ Ž † Ž „1+„!î  Á ë +#‰„‰„!3‹ 0W ‰ +Á-Jí  †‹*ƒ*> „ -J í 0 Ž ‹*Š,E ë -Jíkë  Šî 2ë +  Š,î,‹*8Š 0׋*Š 4.„!ƒ*X ƒ ‚#„ ò —Š +#‰„ Ž „!ë +Á-J í +#‰‹*E ë +  ƒ ð ƒ +#‰,„{ŠŒ,ìАi„ Ž -Jí 0 ŽC ‹*Š먋*Š +#‰,„Ðë#‹*Š ð ‹*ë 8Š -:+r+ Eð „!Š ‹* Š +-  4147-WŒ,Š +ò R ‰¥Œ,r ë +‚-@47-WŠ <0WŒ ŽC +#3‹ -WŠ ë  Š, î B‚¨‰,H‹ 4C‰Sî‹ h„ Ž -WŠ,ƒ ˆž‹*Š †;-Wë#3‹ +#3‹ -WŠ ” EŽ ‹ „ 47-WŠë#‹*î„ Ž „!î  먄 ŸŒ  ƒ ‚m4 „ ‚ Ž 3‹ + „ = ׋* í E =  E í - Žâ ƒ*ƒ ‘ Z ‚ƒ ” ò R ‰,‹*Ÿ ë + Ž Š,ë#ƒ  +„!ë +#‰„ { í 47+l+#‰  +l+#‰„Ðë#‹*Š ð 47-Wƒ*ƒ „ 47+#ë  ƒ*b C Ž   ƒ 0 ‹*Š, ë -Jíë Š,A î 0J1„ ++#‹ï8Š 0-W8Œ +l-Jª í +#‰„eë#ˆë +„!ìæò š l±v¾u*uuxmog_f -J í +#‰„{óJ„ Ž +h„ œ^‚-ƒ ‘ Z ‚/ƒ ” ƒ5‹*Š 47-WŠ <0WŒ Ž +#3‹ -W Š $47-WŠ,ë#‹*2ë +#뇋*Š î„ 4 Ž „  ë#‹*Š 0@+#‰,„{ŠŒ,ìАi„ Ž -Jí 0 ŽC ‹*Šë ‹*Š +#‰,‹*ë óJ„ Ž +h„ œ ¥ˆS5‹ +#ëâî,1„ 0 Ž „O „ ‚ ‰,‹*ƒ  „ +#‰,„{ŠŒ,ìАi„ Ž -J í +#‰ -Wë‹ „ -Jík„  4C‰ -JíÞ5‹ +#ëâŠ,„!3‹ 0W‰=;  -WŒ Ž Š ë „×‹*Š 4 Ž „  ë„!ë R   Ž      ˆ ò ‰‹*ë׋*ëׄ Ÿ¥Œ‹ ó ƒ*„!,Š +A+- +#‰„ î,î3‹ +#3‹ -WŠ +- -J6 í +#‰„]óJ„ 47+ë#Œ 4C‰ +#‰ + X E±YE =  E  Š,î ª X E Y † E ¯ † = E ¯ † í - Ž „=  ‚#ò R ‰„eŠ -:+  +#3‹ -WŠ  얄  Š,Ÿë +#‰  + ‹*ë -J +  ‹*Š,„!E îží Ž -W  ì S‰ ˆ l±v¾u*uuxmog_f óJ„ Ž +h„ œMƒ  ª  ®  Ž  ‘   R C Ž  ë - +#‰ +6+#‰„ „˜h„ œ‹*ë +#ë Š  Œ 4C= ‰ +#‰ + =. ‰,

(122) „ + Š,ëC3‹ +#‹ óJ„ 45ƒ -Wë#Œ Ž „ -JŸ í +#‰„ +-J††ƒ*‹*8Š 0 Z ‚\ƒ ” ë# Eò C Ž   Ž  -J†i„ +#3 ‹ -WŠ ‹ïëzî„!8Š -:+„!î . ª ]‹ í ]‹*>ë -J + ‹*Š„!î í -W! ì ¥ˆ ë„ ŸŒ„!Š 4.

(123) „ -JŸ í +-J††ƒ*‹*8Š 0Wë!ò š Š  ª   ª     ‹ Ž ‘  j ¨#j#xjgutzse‹*ë ë„ ŸŒ„!Š 4. „ -J½ í +-J†,†ƒï‹*8Š 0Wë X ë„„ 3‹ 0WŒ „!ë ƒ1’ ŠV î ~ Yò R ‰„ ë  Š,î,†‹*ƒ „ ì -î,„!ƒ ‰  ëאi„„!Š ‹* Š + Ž -©î, Œ 4.„!î ¥ˆ ­ Eð ƒ Rà Š 02ƒ  Š,î ž ‹ „!Š,ë„!Š,í „!ƒ*î ? ‘ B ‹*Š #‘ "%$'& ò —Š  Ž „ 4.„! Š +k; Ž *1EŽ „!ëY í - Ž  - - ð ƒ ­ Eð ? ’ B 0W‹ óJ„!ë 

(124) Š -PóJ„ Ž ó©‹ „ ‚ -Jí4ì  Š¥ˆ†‰¥ˆëC5‹ 4  ƒ† Ž -J1ƒ „!ì–)ë ( „ EŽ +#D‰ Ÿ¥Œ Eð „!ë  Š,îqë -Wƒ E+.

(125) ^. $± )! %¾ . $ . m~.

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