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Web","\u002Fsnt-2nde\u002Fsb-web\u002Fhistoire-web","0.snt-2nde\u002Fsb-web\u002F1.histoire-web",{"title":47,"path":48,"stem":49},"URL et hypertexte","\u002Fsnt-2nde\u002Fsb-web\u002Furl-hypertexte","0.snt-2nde\u002Fsb-web\u002F2.url-hypertexte",{"title":51,"path":52,"stem":53},"HTML et CSS — le contenu et la présentation","\u002Fsnt-2nde\u002Fsb-web\u002Fhtml-css","0.snt-2nde\u002Fsb-web\u002F3.html-css",{"title":55,"path":56,"stem":57},"Client-serveur et requêtes HTTP","\u002Fsnt-2nde\u002Fsb-web\u002Fclient-serveur-http","0.snt-2nde\u002Fsb-web\u002F4.client-serveur-http",{"title":59,"path":60,"stem":61},"Moteurs de recherche","\u002Fsnt-2nde\u002Fsb-web\u002Fmoteurs-recherche","0.snt-2nde\u002Fsb-web\u002F5.moteurs-recherche",{"title":63,"path":64,"stem":65},"Sécurité du navigateur et notions juridiques","\u002Fsnt-2nde\u002Fsb-web\u002Fsecurite-droits","0.snt-2nde\u002Fsb-web\u002F6.securite-droits",{"title":67,"path":68,"stem":69,"children":70},"Les réseaux 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tables","\u002Fsnt-2nde\u002Fsd-donnees-structurees\u002Ftraitement-tables","0.snt-2nde\u002Fsd-donnees-structurees\u002F4.traitement-tables",{"title":115,"path":116,"stem":117},"Le cloud","\u002Fsnt-2nde\u002Fsd-donnees-structurees\u002Fcloud","0.snt-2nde\u002Fsd-donnees-structurees\u002F5.cloud",{"title":119,"path":120,"stem":121,"children":122},"Localisation, cartographie et mobilité","\u002Fsnt-2nde\u002Fse-localisation-cartographie","0.snt-2nde\u002Fse-localisation-cartographie\u002F0.index",[123,124,128,132,136,140],{"title":119,"path":120,"stem":121},{"title":125,"path":126,"stem":127},"GPS et Galileo — se situer grâce aux satellites","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fgps-galileo","0.snt-2nde\u002Fse-localisation-cartographie\u002F1.gps-galileo",{"title":129,"path":130,"stem":131},"Décoder une trame NMEA","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Ftrame-nmea","0.snt-2nde\u002Fse-localisation-cartographie\u002F2.trame-nmea",{"title":133,"path":134,"stem":135},"Cartes numériques — GeoPortail et OpenStreetMap","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fcartes-numeriques","0.snt-2nde\u002Fse-localisation-cartographie\u002F3.cartes-numeriques",{"title":137,"path":138,"stem":139},"Itinéraires et graphes","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fitineraires-graphes","0.snt-2nde\u002Fse-localisation-cartographie\u002F4.itineraires-graphes",{"title":141,"path":142,"stem":143},"Confidentialité de la position","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fconfidentialite-position","0.snt-2nde\u002Fse-localisation-cartographie\u002F5.confidentialite-position",{"title":145,"path":146,"stem":147,"children":148},"Informatique embarquée et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee","0.snt-2nde\u002Fsf-info-embarquee\u002F0.index",[149,150,154,158],{"title":145,"path":146,"stem":147},{"title":151,"path":152,"stem":153},"Systèmes informatiques embarqués","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fsystemes-embarques","0.snt-2nde\u002Fsf-info-embarquee\u002F1.systemes-embarques",{"title":155,"path":156,"stem":157},"Capteurs et actionneurs","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fcapteurs-actionneurs","0.snt-2nde\u002Fsf-info-embarquee\u002F2.capteurs-actionneurs",{"title":159,"path":160,"stem":161},"IHM et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fihm-objet-connecte","0.snt-2nde\u002Fsf-info-embarquee\u002F3.ihm-objet-connecte",{"title":163,"path":164,"stem":165,"children":166},"Photographie numérique","\u002Fsnt-2nde\u002Fsg-photo-numerique","0.snt-2nde\u002Fsg-photo-numerique\u002F0.index",[167,168,172,176,180],{"title":163,"path":164,"stem":165},{"title":169,"path":170,"stem":171},"Photosites, pixels et résolution","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fpixels-resolution","0.snt-2nde\u002Fsg-photo-numerique\u002F1.pixels-resolution",{"title":173,"path":174,"stem":175},"Les algorithmes de l'appareil photo","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Falgorithmes-photo","0.snt-2nde\u002Fsg-photo-numerique\u002F2.algorithmes-photo",{"title":177,"path":178,"stem":179},"Les métadonnées EXIF","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fmetadonnees-exif","0.snt-2nde\u002Fsg-photo-numerique\u002F3.metadonnees-exif",{"title":181,"path":182,"stem":183},"Traiter une image en Python","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Ftraitement-rgb","0.snt-2nde\u002Fsg-photo-numerique\u002F4.traitement-rgb",{"title":185,"path":186,"stem":187,"children":188},"Notions transversales de programmation","\u002Fsnt-2nde\u002Fsh-prog-notions","0.snt-2nde\u002Fsh-prog-notions\u002F0.index",[189,190,194,198,202],{"title":185,"path":186,"stem":187},{"title":191,"path":192,"stem":193},"Séquences d'instructions et variables","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fsequences-affectations","0.snt-2nde\u002Fsh-prog-notions\u002F1.sequences-affectations",{"title":195,"path":196,"stem":197},"Instructions conditionnelles","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fconditionnelles","0.snt-2nde\u002Fsh-prog-notions\u002F2.conditionnelles",{"title":199,"path":200,"stem":201},"Boucles bornées et non bornées","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fboucles","0.snt-2nde\u002Fsh-prog-notions\u002F3.boucles",{"title":203,"path":204,"stem":205},"Définitions et appels de fonctions","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Ffonctions","0.snt-2nde\u002Fsh-prog-notions\u002F4.fonctions",{"title":207,"path":208,"stem":209,"children":210},"Démo des widgets MDC","\u002Fsnt-2nde\u002Fzz-demo-widgets","0.snt-2nde\u002Fzz-demo-widgets\u002F0.index",[211,212,216,220],{"title":207,"path":208,"stem":209},{"title":213,"path":214,"stem":215},"Démo statique — callouts, code, BO, illustration","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fstatique","0.snt-2nde\u002Fzz-demo-widgets\u002F1.statique",{"title":217,"path":218,"stem":219},"Démo interactive — self-eval et python-snippet","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Finteractif","0.snt-2nde\u002Fzz-demo-widgets\u002F2.interactif",{"title":221,"path":222,"stem":223},"Démo · 14 widgets de manipulation NSI","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fwidgets-manipulation","0.snt-2nde\u002Fzz-demo-widgets\u002F3.widgets-manipulation",{"title":225,"path":226,"stem":227,"children":228},"NSI — Première","\u002Fnsi-1ere","1.nsi-1ere\u002F0.index",[229,230,234,256,282,304,328,354,380,410],{"title":225,"path":226,"stem":227},{"title":231,"path":232,"stem":233},"Relecture pédagogique — 8 chapitres NSI 1ère (PA → PH)","\u002Fnsi-1ere\u002Freview","1.nsi-1ere\u002FREVIEW",{"title":235,"path":236,"stem":237,"children":238},"Histoire de l'informatique","\u002Fnsi-1ere\u002Fpa-histoire-info","1.nsi-1ere\u002Fpa-histoire-info\u002F0.index",[239,240,244,248,252],{"title":235,"path":236,"stem":237},{"title":241,"path":242,"stem":243},"De la préantiquité aux machines à calculer","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fpreantiquite-aux-machines","1.nsi-1ere\u002Fpa-histoire-info\u002F1.preantiquite-aux-machines",{"title":245,"path":246,"stem":247},"La machine universelle (1930-1950)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fmachine-universelle","1.nsi-1ere\u002Fpa-histoire-info\u002F2.machine-universelle",{"title":249,"path":250,"stem":251},"Les ordinateurs modernes (1947-1990)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fordinateurs-modernes","1.nsi-1ere\u002Fpa-histoire-info\u002F3.ordinateurs-modernes",{"title":253,"path":254,"stem":255},"Réseaux, Web et IA (1969 à aujourd'hui)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Freseaux-et-web","1.nsi-1ere\u002Fpa-histoire-info\u002F4.reseaux-et-web",{"title":257,"path":258,"stem":259,"children":260},"Représentation des données — types et valeurs","\u002Fnsi-1ere\u002Fpb-types-valeurs","1.nsi-1ere\u002Fpb-types-valeurs\u002F0.index",[261,262,266,270,274,278],{"title":257,"path":258,"stem":259},{"title":263,"path":264,"stem":265},"Bases de numération et entiers positifs","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbases-entiers-positifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs",{"title":267,"path":268,"stem":269},"Entiers relatifs et complément à deux","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs",{"title":271,"path":272,"stem":273},"Nombres flottants et virgule flottante","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants","1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants",{"title":275,"path":276,"stem":277},"Textes et encodages — ASCII, ISO-8859-1, Unicode","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Ftextes-encodages","1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages",{"title":279,"path":280,"stem":281},"Valeurs booléennes et opérateurs logiques","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbooleens","1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens",{"title":283,"path":284,"stem":285,"children":286},"Représentation des données : types construits","\u002Fnsi-1ere\u002Fpc-types-construits","1.nsi-1ere\u002Fpc-types-construits\u002F0.index",[287,288,292,296,300],{"title":283,"path":284,"stem":285},{"title":289,"path":290,"stem":291},"Tableaux et listes","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftableaux-listes","1.nsi-1ere\u002Fpc-types-construits\u002F1.tableaux-listes",{"title":293,"path":294,"stem":295},"Listes en compréhension","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fcomprehension","1.nsi-1ere\u002Fpc-types-construits\u002F2.comprehension",{"title":297,"path":298,"stem":299},"Tuples et p-uplets nommés","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftuples","1.nsi-1ere\u002Fpc-types-construits\u002F3.tuples",{"title":301,"path":302,"stem":303},"Dictionnaires","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fdictionnaires","1.nsi-1ere\u002Fpc-types-construits\u002F4.dictionnaires",{"title":305,"path":306,"stem":307,"children":308},"Traitement de données en tables","\u002Fnsi-1ere\u002Fpd-tables-donnees","1.nsi-1ere\u002Fpd-tables-donnees\u002F0.index",[309,310,314,318,322],{"title":305,"path":306,"stem":307},{"title":311,"path":312,"stem":313},"Tables et CSV","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftables-csv","1.nsi-1ere\u002Fpd-tables-donnees\u002F1.tables-csv",{"title":315,"path":316,"stem":317},"Recherche et filtrage","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Frecherche","1.nsi-1ere\u002Fpd-tables-donnees\u002F2.recherche",{"title":319,"path":320,"stem":321},"Trier une table","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftri","1.nsi-1ere\u002Fpd-tables-donnees\u002F3.tri",{"title":323,"path":324,"stem":325,"children":326},"Fusion et indexation","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ffusion-indexation","1.nsi-1ere\u002Fpd-tables-donnees\u002F4.fusion-indexation",[327],{"title":323,"path":324,"stem":325},{"title":329,"path":330,"stem":331,"children":332},"IHM sur le Web","\u002Fnsi-1ere\u002Fpe-ihm-web","1.nsi-1ere\u002Fpe-ihm-web\u002F0.index",[333,334,338,342,346,350],{"title":329,"path":330,"stem":331},{"title":335,"path":336,"stem":337},"HTML et CSS — rappel de structure","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fhtml-css-bases","1.nsi-1ere\u002Fpe-ihm-web\u002F1.html-css-bases",{"title":339,"path":340,"stem":341},"Formulaires HTML","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fformulaires","1.nsi-1ere\u002Fpe-ihm-web\u002F2.formulaires",{"title":343,"path":344,"stem":345},"Événements JavaScript","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fevenements-js","1.nsi-1ere\u002Fpe-ihm-web\u002F3.evenements-js",{"title":347,"path":348,"stem":349},"Manipulation du DOM","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fdom-interaction","1.nsi-1ere\u002Fpe-ihm-web\u002F4.dom-interaction",{"title":351,"path":352,"stem":353},"Client, serveur et HTTP","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fclient-serveur-http","1.nsi-1ere\u002Fpe-ihm-web\u002F5.client-serveur-http",{"title":355,"path":356,"stem":357,"children":358},"Architectures matérielles et systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os","1.nsi-1ere\u002Fpf-archi-os\u002F0.index",[359,360,364,368,372,376],{"title":355,"path":356,"stem":357},{"title":361,"path":362,"stem":363},"Le modèle de von Neumann","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fvon-neumann","1.nsi-1ere\u002Fpf-archi-os\u002F1.von-neumann",{"title":365,"path":366,"stem":367},"Périphériques et interface homme-machine","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fperipheriques-ihm","1.nsi-1ere\u002Fpf-archi-os\u002F2.peripheriques-ihm",{"title":369,"path":370,"stem":371},"Les systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fsystemes-exploitation","1.nsi-1ere\u002Fpf-archi-os\u002F3.systemes-exploitation",{"title":373,"path":374,"stem":375},"Linux et le shell","\u002Fnsi-1ere\u002Fpf-archi-os\u002Flinux-shell","1.nsi-1ere\u002Fpf-archi-os\u002F4.linux-shell",{"title":377,"path":378,"stem":379},"Réseaux et protocoles","\u002Fnsi-1ere\u002Fpf-archi-os\u002Freseaux","1.nsi-1ere\u002Fpf-archi-os\u002F5.reseaux",{"title":381,"path":382,"stem":383,"children":384},"Langages et programmation","\u002Fnsi-1ere\u002Fpg-langages-prog","1.nsi-1ere\u002Fpg-langages-prog\u002F0.index",[385,386,390,394,398,402,406],{"title":381,"path":382,"stem":383},{"title":387,"path":388,"stem":389},"Diversité et unité des langages de programmation","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fdiversite-langages","1.nsi-1ere\u002Fpg-langages-prog\u002F1.diversite-langages",{"title":391,"path":392,"stem":393},"Constructions élémentaires","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fconstructions-elementaires","1.nsi-1ere\u002Fpg-langages-prog\u002F2.constructions-elementaires",{"title":395,"path":396,"stem":397},"Fonctions","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Ffonctions","1.nsi-1ere\u002Fpg-langages-prog\u002F3.fonctions",{"title":399,"path":400,"stem":401},"Spécification d'une fonction","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fspecification","1.nsi-1ere\u002Fpg-langages-prog\u002F4.specification",{"title":403,"path":404,"stem":405},"Mise au point des programmes","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fmise-au-point","1.nsi-1ere\u002Fpg-langages-prog\u002F5.mise-au-point",{"title":407,"path":408,"stem":409},"Utilisation de bibliothèques","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fbibliotheques","1.nsi-1ere\u002Fpg-langages-prog\u002F6.bibliotheques",{"title":411,"path":412,"stem":413,"children":414},"Algorithmique","\u002Fnsi-1ere\u002Fph-algorithmique","1.nsi-1ere\u002Fph-algorithmique\u002F0.index",[415,416,420,424,428,432,436],{"title":411,"path":412,"stem":413},{"title":417,"path":418,"stem":419},"Parcours séquentiel d'un tableau","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fparcours-sequentiel","1.nsi-1ere\u002Fph-algorithmique\u002F1.parcours-sequentiel",{"title":421,"path":422,"stem":423},"Bases de la complexité","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fcomplexite-bases","1.nsi-1ere\u002Fph-algorithmique\u002F2.complexite-bases",{"title":425,"path":426,"stem":427},"Tris par sélection et par insertion","\u002Fnsi-1ere\u002Fph-algorithmique\u002Ftris","1.nsi-1ere\u002Fph-algorithmique\u002F3.tris",{"title":429,"path":430,"stem":431},"Recherche dichotomique","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fdichotomie","1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie",{"title":433,"path":434,"stem":435},"Algorithmes gloutons","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fgloutons","1.nsi-1ere\u002Fph-algorithmique\u002F5.gloutons",{"title":437,"path":438,"stem":439},"Les k plus proches voisins (k-NN)","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fknn","1.nsi-1ere\u002Fph-algorithmique\u002F6.knn",{"title":441,"path":442,"stem":443,"children":444},"NSI — Terminale","\u002Fnsi-tale","2.nsi-tale\u002F0.index",[445,446,450,476,498,536,560],{"title":441,"path":442,"stem":443},{"title":447,"path":448,"stem":449},"Review pédagogique — chapitres NSI Terminale (TA, TB, TC, TD, TE)","\u002Fnsi-tale\u002Freview","2.nsi-tale\u002FREVIEW",{"title":451,"path":452,"stem":453,"children":454},"Structures de données","\u002Fnsi-tale\u002Fta-structures-donnees","2.nsi-tale\u002Fta-structures-donnees\u002F0.index",[455,456,460,464,468,472],{"title":451,"path":452,"stem":453},{"title":457,"path":458,"stem":459},"Objet, classe, interface","\u002Fnsi-tale\u002Fta-structures-donnees\u002Finterface-implementation","2.nsi-tale\u002Fta-structures-donnees\u002F1.interface-implementation",{"title":461,"path":462,"stem":463},"Implémenter une structure — l'exemple de la file FIFO","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fimplementer-une-structure","2.nsi-tale\u002Fta-structures-donnees\u002F2.implementer-une-structure",{"title":465,"path":466,"stem":467},"Listes, piles, files, dictionnaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Flistes-piles-files-dict","2.nsi-tale\u002Fta-structures-donnees\u002F3.listes-piles-files-dict",{"title":469,"path":470,"stem":471},"Arbres binaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Farbres-binaires","2.nsi-tale\u002Fta-structures-donnees\u002F4.arbres-binaires",{"title":473,"path":474,"stem":475},"Graphes","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fgraphes","2.nsi-tale\u002Fta-structures-donnees\u002F5.graphes",{"title":477,"path":478,"stem":479,"children":480},"Bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees","2.nsi-tale\u002Ftb-bases-de-donnees\u002F0.index",[481,482,486,490,494],{"title":477,"path":478,"stem":479},{"title":483,"path":484,"stem":485},"Le modèle relationnel","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fmodele-relationnel","2.nsi-tale\u002Ftb-bases-de-donnees\u002F1.modele-relationnel",{"title":487,"path":488,"stem":489},"Bases de données relationnelles et anomalies de schéma","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fbase-relationnelle","2.nsi-tale\u002Ftb-bases-de-donnees\u002F2.base-relationnelle",{"title":491,"path":492,"stem":493},"Le langage SQL","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsql","2.nsi-tale\u002Ftb-bases-de-donnees\u002F3.sql",{"title":495,"path":496,"stem":497},"Les systèmes de gestion de bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsgbd","2.nsi-tale\u002Ftb-bases-de-donnees\u002F4.sgbd",{"title":499,"path":500,"stem":501,"children":502},"Architectures matérielles, systèmes d'exploitation et réseaux","\u002Fnsi-tale\u002Ftc-archi-os-reseaux","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F0.index",[503,504,508,512,516,520,524,528,532],{"title":499,"path":500,"stem":501},{"title":505,"path":506,"stem":507},"Le système sur puce (SoC)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fsysteme-sur-puce","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F1.systeme-sur-puce",{"title":509,"path":510,"stem":511},"Processus et ordonnancement par l'OS","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprocessus-os","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F2.processus-os",{"title":513,"path":514,"stem":515},"L'interblocage (deadlock)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Finterblocage","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F3.interblocage",{"title":517,"path":518,"stem":519},"Principes du routage","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Froutage-principes","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F4.routage-principes",{"title":521,"path":522,"stem":523},"Protocoles de routage RIP et OSPF","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprotocoles-rip-ospf","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F5.protocoles-rip-ospf",{"title":525,"path":526,"stem":527},"Chiffrement symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-symetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F6.chiffrement-symetrique",{"title":529,"path":530,"stem":531},"Chiffrement asymétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-asymetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F7.chiffrement-asymetrique",{"title":533,"path":534,"stem":535},"HTTPS — l'échange de clé symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fhttps-tls","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F8.https-tls",{"title":381,"path":537,"stem":538,"children":539},"\u002Fnsi-tale\u002Ftd-langages-prog","2.nsi-tale\u002Ftd-langages-prog\u002F0.index",[540,541,544,548,552,556],{"title":381,"path":537,"stem":538},{"title":403,"path":542,"stem":543},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmise-au-point","2.nsi-tale\u002Ftd-langages-prog\u002F1.mise-au-point",{"title":545,"path":546,"stem":547},"Modularité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmodularite","2.nsi-tale\u002Ftd-langages-prog\u002F2.modularite",{"title":549,"path":550,"stem":551},"Récursivité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Frecursivite","2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite",{"title":553,"path":554,"stem":555},"Paradigmes de programmation","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fparadigmes","2.nsi-tale\u002Ftd-langages-prog\u002F4.paradigmes",{"title":557,"path":558,"stem":559},"Programme comme donnée — calculabilité et indécidabilité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fprogramme-comme-donnee","2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee",{"title":411,"path":561,"stem":562,"children":563},"\u002Fnsi-tale\u002Fte-algorithmique","2.nsi-tale\u002Fte-algorithmique\u002F0.index",[564,565,569,573,577,581],{"title":411,"path":561,"stem":562},{"title":566,"path":567,"stem":568},"Algorithmes sur les arbres binaires","\u002Fnsi-tale\u002Fte-algorithmique\u002Farbres-binaires-algos","2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos",{"title":570,"path":571,"stem":572},"Parcours de graphes — DFS, BFS, cycles, chemins","\u002Fnsi-tale\u002Fte-algorithmique\u002Fgraphes-parcours","2.nsi-tale\u002Fte-algorithmique\u002F2.graphes-parcours",{"title":574,"path":575,"stem":576},"Diviser pour régner","\u002Fnsi-tale\u002Fte-algorithmique\u002Fdiviser-pour-regner","2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner",{"title":578,"path":579,"stem":580},"Programmation dynamique","\u002Fnsi-tale\u002Fte-algorithmique\u002Fprogrammation-dynamique","2.nsi-tale\u002Fte-algorithmique\u002F4.programmation-dynamique",{"title":582,"path":583,"stem":584},"Recherche textuelle — l'algorithme de Boyer-Moore","\u002Fnsi-tale\u002Fte-algorithmique\u002Frecherche-textuelle","2.nsi-tale\u002Fte-algorithmique\u002F5.recherche-textuelle",{"id":586,"title":574,"bo":587,"body":589,"description":5885,"eval":5886,"extension":5887,"labo":5886,"meta":5888,"navigation":4025,"path":575,"quizzes":5889,"readingTime":3873,"seo":5896,"stem":576,"tp":5886,"__hash__":5897},"cours\u002F2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner.md",[588],"TE03",{"type":590,"value":591,"toc":5874},"minimark",[592,597,630,802,805,809,812,838,971,1142,1146,1431,1916,1923,2374,2378,2719,3836,3843,3847,3860,4153,4785,5046,5053,5076,5080,5134,5138,5273,5287,5530,5534,5537,5552,5571,5575,5870],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,604,605,609,610,613,614,617,618,621,622,625,626,629],"p",{},[601,602,603],"strong",{},"« Diviser pour régner »"," (",[606,607,608],"em",{},"divide and conquer",") est une stratégie\nalgorithmique générale qui structure beaucoup d'algorithmes efficaces : on\n",[601,611,612],{},"découpe"," un problème en sous-problèmes plus petits du ",[601,615,616],{},"même type",", on\nles ",[601,619,620],{},"résout récursivement",", puis on ",[601,623,624],{},"combine"," les solutions pour\nconstruire celle du problème initial. Cette méthode est l'",[601,627,628],{},"inverse\nnaturelle de la récursion"," : ce qui descend en récursion se reconstruit en\nremontant.",[598,631,632,633,636,637,683,684,687,688,801],{},"La formule latine — ",[606,634,635],{},"divide et impera"," — date de l'Antiquité : Jules César\nappliquait à la conquête des Gaules la même logique que l'algorithmicien\napplique à un tableau de ",[638,639,642,664],"span",{"className":640},[641],"katex",[638,643,646],{"className":644},[645],"katex-mathml",[647,648,650],"math",{"xmlns":649},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[651,652,653,660],"semantics",{},[654,655,656],"mrow",{},[657,658,659],"mi",{},"n",[661,662,659],"annotation",{"encoding":663},"application\u002Fx-tex",[638,665,669],{"className":666,"ariaHidden":668},[667],"katex-html","true",[638,670,673,678],{"className":671},[672],"base",[638,674],{"className":675,"style":677},[676],"strut","height:0.4306em;",[638,679,659],{"className":680},[681,682],"mord","mathnormal"," éléments. Un ",[601,685,686],{},"tournoi de tennis à élimination\ndirecte"," suit le même schéma : à chaque ronde, le nombre de joueurs encore\nen lice est divisé par deux, jusqu'au vainqueur — un arbre binaire à\n",[638,689,691,721],{"className":690},[641],[638,692,694],{"className":693},[645],[647,695,696],{"xmlns":649},[651,697,698,718],{},[654,699,700,716],{},[701,702,703,712],"msub",{},[654,704,705,708],{},[657,706,707],{},"log",[709,710,711],"mo",{},"⁡",[713,714,715],"mn",{},"2",[657,717,659],{},[661,719,720],{"encoding":663},"\\log_2 n",[638,722,724],{"className":723,"ariaHidden":668},[667],[638,725,727,731,793,798],{"className":726},[672],[638,728],{"className":729,"style":730},[676],"height:0.9386em;vertical-align:-0.2441em;",[638,732,735,743],{"className":733},[734],"mop",[638,736,738,739],{"className":737},[734],"lo",[638,740,742],{"style":741},"margin-right:0.0139em;","g",[638,744,747],{"className":745},[746],"msupsub",[638,748,752,784],{"className":749},[750,751],"vlist-t","vlist-t2",[638,753,756,779],{"className":754},[755],"vlist-r",[638,757,761],{"className":758,"style":760},[759],"vlist","height:0.207em;",[638,762,764,769],{"style":763},"top:-2.4559em;margin-right:0.05em;",[638,765],{"className":766,"style":768},[767],"pstrut","height:2.7em;",[638,770,776],{"className":771},[772,773,774,775],"sizing","reset-size6","size3","mtight",[638,777,715],{"className":778},[681,775],[638,780,783],{"className":781},[782],"vlist-s","​",[638,785,787],{"className":786},[755],[638,788,791],{"className":789,"style":790},[759],"height:0.2441em;",[638,792],{},[638,794],{"className":795,"style":797},[796],"mspace","margin-right:0.1667em;",[638,799,659],{"className":800},[681,682]," niveaux.",[803,804],"bo-ref",{"code":588},[593,806,808],{"id":807},"le-schéma-général","Le schéma général",[598,810,811],{},"Tout algorithme « diviser pour régner » suit trois étapes :",[813,814,815,822,832],"ol",{},[816,817,818,821],"li",{},[601,819,820],{},"Diviser"," — partitionner l'entrée en sous-entrées plus petites.",[816,823,824,827,828,831],{},[601,825,826],{},"Régner"," — résoudre chaque sous-entrée par appel récursif (ou\ndirectement si elle est suffisamment petite : c'est le ",[601,829,830],{},"cas de base",").",[816,833,834,837],{},[601,835,836],{},"Combiner"," — assembler les sous-solutions en une solution globale.",[598,839,840,841,844,845,874,875,918,919,970],{},"Le coût d'un tel algorithme s'exprime par une ",[601,842,843],{},"relation de récurrence"," :\nsi l'on découpe en ",[638,846,848,862],{"className":847},[641],[638,849,851],{"className":850},[645],[647,852,853],{"xmlns":649},[651,854,855,860],{},[654,856,857],{},[657,858,859],{},"a",[661,861,859],{"encoding":663},[638,863,865],{"className":864,"ariaHidden":668},[667],[638,866,868,871],{"className":867},[672],[638,869],{"className":870,"style":677},[676],[638,872,859],{"className":873},[681,682]," sous-problèmes de taille ",[638,876,878,899],{"className":877},[641],[638,879,881],{"className":880},[645],[647,882,883],{"xmlns":649},[651,884,885,896],{},[654,886,887,889,893],{},[657,888,659],{},[657,890,892],{"mathvariant":891},"normal","\u002F",[657,894,895],{},"b",[661,897,898],{"encoding":663},"n\u002Fb",[638,900,902],{"className":901,"ariaHidden":668},[667],[638,903,905,909,912,915],{"className":904},[672],[638,906],{"className":907,"style":908},[676],"height:1em;vertical-align:-0.25em;",[638,910,659],{"className":911},[681,682],[638,913,892],{"className":914},[681],[638,916,895],{"className":917},[681,682],", et que les étapes\ndiviser + combiner coûtent ",[638,920,922,946],{"className":921},[641],[638,923,925],{"className":924},[645],[647,926,927],{"xmlns":649},[651,928,929,943],{},[654,930,931,934,938,940],{},[657,932,933],{},"f",[709,935,937],{"stretchy":936},"false","(",[657,939,659],{},[709,941,942],{"stretchy":936},")",[661,944,945],{"encoding":663},"f(n)",[638,947,949],{"className":948,"ariaHidden":668},[667],[638,950,952,955,959,963,966],{"className":951},[672],[638,953],{"className":954,"style":908},[676],[638,956,933],{"className":957,"style":958},[681,682],"margin-right:0.1076em;",[638,960,937],{"className":961},[962],"mopen",[638,964,659],{"className":965},[681,682],[638,967,942],{"className":968},[969],"mclose",", alors",[638,972,975],{"className":973},[974],"katex-display",[638,976,978,1034],{"className":977},[641],[638,979,981],{"className":980},[645],[647,982,984],{"xmlns":649,"display":983},"block",[651,985,986,1031],{},[654,987,988,991,993,995,997,1000,1002,1005,1007,1009,1011,1013,1015,1017,1020,1022,1024,1026,1028],{},[657,989,990],{},"T",[709,992,937],{"stretchy":936},[657,994,659],{},[709,996,942],{"stretchy":936},[709,998,999],{},"=",[657,1001,859],{},[709,1003,1004],{},"⋅",[657,1006,990],{},[709,1008,937],{"stretchy":936},[657,1010,659],{},[657,1012,892],{"mathvariant":891},[657,1014,895],{},[709,1016,942],{"stretchy":936},[709,1018,1019],{},"+",[657,1021,933],{},[709,1023,937],{"stretchy":936},[657,1025,659],{},[709,1027,942],{"stretchy":936},[657,1029,1030],{"mathvariant":891},".",[661,1032,1033],{"encoding":663},"T(n) = a \\cdot T(n\u002Fb) + f(n).",[638,1035,1037,1067,1088,1121],{"className":1036,"ariaHidden":668},[667],[638,1038,1040,1043,1047,1050,1053,1056,1060,1064],{"className":1039},[672],[638,1041],{"className":1042,"style":908},[676],[638,1044,990],{"className":1045,"style":1046},[681,682],"margin-right:0.1389em;",[638,1048,937],{"className":1049},[962],[638,1051,659],{"className":1052},[681,682],[638,1054,942],{"className":1055},[969],[638,1057],{"className":1058,"style":1059},[796],"margin-right:0.2778em;",[638,1061,999],{"className":1062},[1063],"mrel",[638,1065],{"className":1066,"style":1059},[796],[638,1068,1070,1074,1077,1081,1085],{"className":1069},[672],[638,1071],{"className":1072,"style":1073},[676],"height:0.4445em;",[638,1075,859],{"className":1076},[681,682],[638,1078],{"className":1079,"style":1080},[796],"margin-right:0.2222em;",[638,1082,1004],{"className":1083},[1084],"mbin",[638,1086],{"className":1087,"style":1080},[796],[638,1089,1091,1094,1097,1100,1103,1106,1109,1112,1115,1118],{"className":1090},[672],[638,1092],{"className":1093,"style":908},[676],[638,1095,990],{"className":1096,"style":1046},[681,682],[638,1098,937],{"className":1099},[962],[638,1101,659],{"className":1102},[681,682],[638,1104,892],{"className":1105},[681],[638,1107,895],{"className":1108},[681,682],[638,1110,942],{"className":1111},[969],[638,1113],{"className":1114,"style":1080},[796],[638,1116,1019],{"className":1117},[1084],[638,1119],{"className":1120,"style":1080},[796],[638,1122,1124,1127,1130,1133,1136,1139],{"className":1123},[672],[638,1125],{"className":1126,"style":908},[676],[638,1128,933],{"className":1129,"style":958},[681,682],[638,1131,937],{"className":1132},[962],[638,1134,659],{"className":1135},[681,682],[638,1137,942],{"className":1138},[969],[638,1140,1030],{"className":1141},[681],[593,1143,1145],{"id":1144},"lintuition-par-larbre-des-appels","L'intuition par l'arbre des appels",[598,1147,1148,1149,1152,1153,1295,1296,1324,1325,1361,1362,1400,1401,1030],{},"Avant de poser le théorème maître, regardons ",[601,1150,1151],{},"ce qui se passe à l'exécution",".\nLa récurrence ",[638,1154,1156,1208],{"className":1155},[641],[638,1157,1159],{"className":1158},[645],[647,1160,1161],{"xmlns":649},[651,1162,1163,1205],{},[654,1164,1165,1167,1169,1171,1173,1175,1177,1181,1183,1185,1187,1189,1191,1193,1195,1199,1201,1203],{},[657,1166,990],{},[709,1168,937],{"stretchy":936},[657,1170,659],{},[709,1172,942],{"stretchy":936},[709,1174,999],{},[713,1176,715],{},[1178,1179,1180],"mtext",{}," ",[657,1182,990],{},[709,1184,937],{"stretchy":936},[657,1186,659],{},[657,1188,892],{"mathvariant":891},[713,1190,715],{},[709,1192,942],{"stretchy":936},[709,1194,1019],{},[657,1196,1198],{"mathvariant":1197},"script","O",[709,1200,937],{"stretchy":936},[657,1202,659],{},[709,1204,942],{"stretchy":936},[661,1206,1207],{"encoding":663},"T(n) = 2\\,T(n\u002F2) + \\mathcal{O}(n)",[638,1209,1211,1238,1275],{"className":1210,"ariaHidden":668},[667],[638,1212,1214,1217,1220,1223,1226,1229,1232,1235],{"className":1213},[672],[638,1215],{"className":1216,"style":908},[676],[638,1218,990],{"className":1219,"style":1046},[681,682],[638,1221,937],{"className":1222},[962],[638,1224,659],{"className":1225},[681,682],[638,1227,942],{"className":1228},[969],[638,1230],{"className":1231,"style":1059},[796],[638,1233,999],{"className":1234},[1063],[638,1236],{"className":1237,"style":1059},[796],[638,1239,1241,1244,1247,1250,1253,1256,1259,1263,1266,1269,1272],{"className":1240},[672],[638,1242],{"className":1243,"style":908},[676],[638,1245,715],{"className":1246},[681],[638,1248],{"className":1249,"style":797},[796],[638,1251,990],{"className":1252,"style":1046},[681,682],[638,1254,937],{"className":1255},[962],[638,1257,659],{"className":1258},[681,682],[638,1260,1262],{"className":1261},[681],"\u002F2",[638,1264,942],{"className":1265},[969],[638,1267],{"className":1268,"style":1080},[796],[638,1270,1019],{"className":1271},[1084],[638,1273],{"className":1274,"style":1080},[796],[638,1276,1278,1281,1286,1289,1292],{"className":1277},[672],[638,1279],{"className":1280,"style":908},[676],[638,1282,1198],{"className":1283,"style":1285},[681,1284],"mathcal","margin-right:0.0278em;",[638,1287,937],{"className":1288},[962],[638,1290,659],{"className":1291},[681,682],[638,1293,942],{"className":1294},[969]," du tri fusion engendre un\narbre d'appels : à la racine, on traite un tableau de taille ",[638,1297,1299,1312],{"className":1298},[641],[638,1300,1302],{"className":1301},[645],[647,1303,1304],{"xmlns":649},[651,1305,1306,1310],{},[654,1307,1308],{},[657,1309,659],{},[661,1311,659],{"encoding":663},[638,1313,1315],{"className":1314,"ariaHidden":668},[667],[638,1316,1318,1321],{"className":1317},[672],[638,1319],{"className":1320,"style":677},[676],[638,1322,659],{"className":1323},[681,682]," ; au niveau\nsuivant, deux tableaux de taille ",[638,1326,1328,1346],{"className":1327},[641],[638,1329,1331],{"className":1330},[645],[647,1332,1333],{"xmlns":649},[651,1334,1335,1343],{},[654,1336,1337,1339,1341],{},[657,1338,659],{},[657,1340,892],{"mathvariant":891},[713,1342,715],{},[661,1344,1345],{"encoding":663},"n\u002F2",[638,1347,1349],{"className":1348,"ariaHidden":668},[667],[638,1350,1352,1355,1358],{"className":1351},[672],[638,1353],{"className":1354,"style":908},[676],[638,1356,659],{"className":1357},[681,682],[638,1359,1262],{"className":1360},[681]," ; puis quatre de taille ",[638,1363,1365,1384],{"className":1364},[641],[638,1366,1368],{"className":1367},[645],[647,1369,1370],{"xmlns":649},[651,1371,1372,1381],{},[654,1373,1374,1376,1378],{},[657,1375,659],{},[657,1377,892],{"mathvariant":891},[713,1379,1380],{},"4",[661,1382,1383],{"encoding":663},"n\u002F4",[638,1385,1387],{"className":1386,"ariaHidden":668},[667],[638,1388,1390,1393,1396],{"className":1389},[672],[638,1391],{"className":1392,"style":908},[676],[638,1394,659],{"className":1395},[681,682],[638,1397,1399],{"className":1398},[681],"\u002F4"," ;\njusqu'aux feuilles de taille ",[638,1402,1404,1418],{"className":1403},[641],[638,1405,1407],{"className":1406},[645],[647,1408,1409],{"xmlns":649},[651,1410,1411,1416],{},[654,1412,1413],{},[713,1414,1415],{},"1",[661,1417,1415],{"encoding":663},[638,1419,1421],{"className":1420,"ariaHidden":668},[667],[638,1422,1424,1428],{"className":1423},[672],[638,1425],{"className":1426,"style":1427},[676],"height:0.6444em;",[638,1429,1415],{"className":1430},[681],[1432,1433,1434,1590,1705],"ul",{},[816,1435,1436,1437,1440,1441,1485,1486,1537,1538,1589],{},"À chaque niveau, le ",[601,1438,1439],{},"travail total"," de fusion vaut ",[638,1442,1444,1464],{"className":1443},[641],[638,1445,1447],{"className":1446},[645],[647,1448,1449],{"xmlns":649},[651,1450,1451,1461],{},[654,1452,1453,1455,1457,1459],{},[657,1454,1198],{"mathvariant":1197},[709,1456,937],{"stretchy":936},[657,1458,659],{},[709,1460,942],{"stretchy":936},[661,1462,1463],{"encoding":663},"\\mathcal{O}(n)",[638,1465,1467],{"className":1466,"ariaHidden":668},[667],[638,1468,1470,1473,1476,1479,1482],{"className":1469},[672],[638,1471],{"className":1472,"style":908},[676],[638,1474,1198],{"className":1475,"style":1285},[681,1284],[638,1477,937],{"className":1478},[962],[638,1480,659],{"className":1481},[681,682],[638,1483,942],{"className":1484},[969]," —\ndeux moitiés à ",[638,1487,1489,1513],{"className":1488},[641],[638,1490,1492],{"className":1491},[645],[647,1493,1494],{"xmlns":649},[651,1495,1496,1510],{},[654,1497,1498,1500,1502,1504,1506,1508],{},[657,1499,1198],{"mathvariant":1197},[709,1501,937],{"stretchy":936},[657,1503,659],{},[657,1505,892],{"mathvariant":891},[713,1507,715],{},[709,1509,942],{"stretchy":936},[661,1511,1512],{"encoding":663},"\\mathcal{O}(n\u002F2)",[638,1514,1516],{"className":1515,"ariaHidden":668},[667],[638,1517,1519,1522,1525,1528,1531,1534],{"className":1518},[672],[638,1520],{"className":1521,"style":908},[676],[638,1523,1198],{"className":1524,"style":1285},[681,1284],[638,1526,937],{"className":1527},[962],[638,1529,659],{"className":1530},[681,682],[638,1532,1262],{"className":1533},[681],[638,1535,942],{"className":1536},[969],", quatre quarts à ",[638,1539,1541,1565],{"className":1540},[641],[638,1542,1544],{"className":1543},[645],[647,1545,1546],{"xmlns":649},[651,1547,1548,1562],{},[654,1549,1550,1552,1554,1556,1558,1560],{},[657,1551,1198],{"mathvariant":1197},[709,1553,937],{"stretchy":936},[657,1555,659],{},[657,1557,892],{"mathvariant":891},[713,1559,1380],{},[709,1561,942],{"stretchy":936},[661,1563,1564],{"encoding":663},"\\mathcal{O}(n\u002F4)",[638,1566,1568],{"className":1567,"ariaHidden":668},[667],[638,1569,1571,1574,1577,1580,1583,1586],{"className":1570},[672],[638,1572],{"className":1573,"style":908},[676],[638,1575,1198],{"className":1576,"style":1285},[681,1284],[638,1578,937],{"className":1579},[962],[638,1581,659],{"className":1582},[681,682],[638,1584,1399],{"className":1585},[681],[638,1587,942],{"className":1588},[969],", etc.,\nsomme stable.",[816,1591,1592,1593,1676,1677,1030],{},"Le nombre de niveaux est ",[638,1594,1596,1619],{"className":1595},[641],[638,1597,1599],{"className":1598},[645],[647,1600,1601],{"xmlns":649},[651,1602,1603,1617],{},[654,1604,1605,1615],{},[701,1606,1607,1613],{},[654,1608,1609,1611],{},[657,1610,707],{},[709,1612,711],{},[713,1614,715],{},[657,1616,659],{},[661,1618,720],{"encoding":663},[638,1620,1622],{"className":1621,"ariaHidden":668},[667],[638,1623,1625,1628,1670,1673],{"className":1624},[672],[638,1626],{"className":1627,"style":730},[676],[638,1629,1631,1636],{"className":1630},[734],[638,1632,738,1634],{"className":1633},[734],[638,1635,742],{"style":741},[638,1637,1639],{"className":1638},[746],[638,1640,1642,1662],{"className":1641},[750,751],[638,1643,1645,1659],{"className":1644},[755],[638,1646,1648],{"className":1647,"style":760},[759],[638,1649,1650,1653],{"style":763},[638,1651],{"className":1652,"style":768},[767],[638,1654,1656],{"className":1655},[772,773,774,775],[638,1657,715],{"className":1658},[681,775],[638,1660,783],{"className":1661},[782],[638,1663,1665],{"className":1664},[755],[638,1666,1668],{"className":1667,"style":790},[759],[638,1669],{},[638,1671],{"className":1672,"style":797},[796],[638,1674,659],{"className":1675},[681,682],", puisqu'à chaque descente on divise la\ntaille par ",[638,1678,1680,1693],{"className":1679},[641],[638,1681,1683],{"className":1682},[645],[647,1684,1685],{"xmlns":649},[651,1686,1687,1691],{},[654,1688,1689],{},[713,1690,715],{},[661,1692,715],{"encoding":663},[638,1694,1696],{"className":1695,"ariaHidden":668},[667],[638,1697,1699,1702],{"className":1698},[672],[638,1700],{"className":1701,"style":1427},[676],[638,1703,715],{"className":1704},[681],[816,1706,1707,1710,1711,1794,1795,1851,1852,1030],{},[601,1708,1709],{},"Coût total"," : ",[638,1712,1714,1737],{"className":1713},[641],[638,1715,1717],{"className":1716},[645],[647,1718,1719],{"xmlns":649},[651,1720,1721,1735],{},[654,1722,1723,1733],{},[701,1724,1725,1731],{},[654,1726,1727,1729],{},[657,1728,707],{},[709,1730,711],{},[713,1732,715],{},[657,1734,659],{},[661,1736,720],{"encoding":663},[638,1738,1740],{"className":1739,"ariaHidden":668},[667],[638,1741,1743,1746,1788,1791],{"className":1742},[672],[638,1744],{"className":1745,"style":730},[676],[638,1747,1749,1754],{"className":1748},[734],[638,1750,738,1752],{"className":1751},[734],[638,1753,742],{"style":741},[638,1755,1757],{"className":1756},[746],[638,1758,1760,1780],{"className":1759},[750,751],[638,1761,1763,1777],{"className":1762},[755],[638,1764,1766],{"className":1765,"style":760},[759],[638,1767,1768,1771],{"style":763},[638,1769],{"className":1770,"style":768},[767],[638,1772,1774],{"className":1773},[772,773,774,775],[638,1775,715],{"className":1776},[681,775],[638,1778,783],{"className":1779},[782],[638,1781,1783],{"className":1782},[755],[638,1784,1786],{"className":1785,"style":790},[759],[638,1787],{},[638,1789],{"className":1790,"style":797},[796],[638,1792,659],{"className":1793},[681,682]," niveaux ",[638,1796,1798,1824],{"className":1797},[641],[638,1799,1801],{"className":1800},[645],[647,1802,1803],{"xmlns":649},[651,1804,1805,1821],{},[654,1806,1807,1810,1813,1815,1817,1819],{},[709,1808,1809],{},"×",[1178,1811,1812],{}," ",[657,1814,1198],{"mathvariant":1197},[709,1816,937],{"stretchy":936},[657,1818,659],{},[709,1820,942],{"stretchy":936},[661,1822,1823],{"encoding":663},"\\times\\ \\mathcal{O}(n)",[638,1825,1827],{"className":1826,"ariaHidden":668},[667],[638,1828,1830,1833,1836,1839,1842,1845,1848],{"className":1829},[672],[638,1831],{"className":1832,"style":908},[676],[638,1834,1809],{"className":1835},[681],[638,1837,1812],{"className":1838},[796],[638,1840,1198],{"className":1841,"style":1285},[681,1284],[638,1843,937],{"className":1844},[962],[638,1846,659],{"className":1847},[681,682],[638,1849,942],{"className":1850},[969]," par niveau, soit\n",[638,1853,1855,1881],{"className":1854},[641],[638,1856,1858],{"className":1857},[645],[647,1859,1860],{"xmlns":649},[651,1861,1862,1878],{},[654,1863,1864,1866,1868,1870,1872,1874,1876],{},[657,1865,1198],{"mathvariant":1197},[709,1867,937],{"stretchy":936},[657,1869,659],{},[657,1871,707],{},[709,1873,711],{},[657,1875,659],{},[709,1877,942],{"stretchy":936},[661,1879,1880],{"encoding":663},"\\mathcal{O}(n \\log n)",[638,1882,1884],{"className":1883,"ariaHidden":668},[667],[638,1885,1887,1890,1893,1896,1899,1902,1907,1910,1913],{"className":1886},[672],[638,1888],{"className":1889,"style":908},[676],[638,1891,1198],{"className":1892,"style":1285},[681,1284],[638,1894,937],{"className":1895},[962],[638,1897,659],{"className":1898},[681,682],[638,1900],{"className":1901,"style":797},[796],[638,1903,738,1905],{"className":1904},[734],[638,1906,742],{"style":741},[638,1908],{"className":1909,"style":797},[796],[638,1911,659],{"className":1912},[681,682],[638,1914,942],{"className":1915},[969],[598,1917,1918,1919,1922],{},"Cette intuition est ",[601,1920,1921],{},"plus importante à retenir que la table abstraite ci-dessous"," :\nelle donne la bonne lecture mentale, le théorème maître en est la version\nmécanique.",[1924,1925,1926,1929,2306],"note",{},[598,1927,1928],{},"Trois notations asymptotiques apparaissent dans la suite :",[1432,1930,1931,1995,2058],{},[816,1932,1933,1990,1991,1994],{},[638,1934,1936,1962],{"className":1935},[641],[638,1937,1939],{"className":1938},[645],[647,1940,1941],{"xmlns":649},[651,1942,1943,1959],{},[654,1944,1945,1947,1949,1951,1953,1955,1957],{},[657,1946,1198],{"mathvariant":1197},[709,1948,937],{"stretchy":936},[657,1950,933],{},[709,1952,937],{"stretchy":936},[657,1954,659],{},[709,1956,942],{"stretchy":936},[709,1958,942],{"stretchy":936},[661,1960,1961],{"encoding":663},"\\mathcal{O}(f(n))",[638,1963,1965],{"className":1964,"ariaHidden":668},[667],[638,1966,1968,1971,1974,1977,1980,1983,1986],{"className":1967},[672],[638,1969],{"className":1970,"style":908},[676],[638,1972,1198],{"className":1973,"style":1285},[681,1284],[638,1975,937],{"className":1976},[962],[638,1978,933],{"className":1979,"style":958},[681,682],[638,1981,937],{"className":1982},[962],[638,1984,659],{"className":1985},[681,682],[638,1987,1989],{"className":1988},[969],"))"," — ",[601,1992,1993],{},"majoration"," (au plus), déjà connue de Première.",[816,1996,1997,1990,2054,2057],{},[638,1998,2000,2027],{"className":1999},[641],[638,2001,2003],{"className":2002},[645],[647,2004,2005],{"xmlns":649},[651,2006,2007,2024],{},[654,2008,2009,2012,2014,2016,2018,2020,2022],{},[657,2010,2011],{"mathvariant":891},"Ω",[709,2013,937],{"stretchy":936},[657,2015,933],{},[709,2017,937],{"stretchy":936},[657,2019,659],{},[709,2021,942],{"stretchy":936},[709,2023,942],{"stretchy":936},[661,2025,2026],{"encoding":663},"\\Omega(f(n))",[638,2028,2030],{"className":2029,"ariaHidden":668},[667],[638,2031,2033,2036,2039,2042,2045,2048,2051],{"className":2032},[672],[638,2034],{"className":2035,"style":908},[676],[638,2037,2011],{"className":2038},[681],[638,2040,937],{"className":2041},[962],[638,2043,933],{"className":2044,"style":958},[681,682],[638,2046,937],{"className":2047},[962],[638,2049,659],{"className":2050},[681,682],[638,2052,1989],{"className":2053},[969],[601,2055,2056],{},"minoration"," (au moins).",[816,2059,2060,1990,2117,2120,2121,2215,2216,831],{},[638,2061,2063,2090],{"className":2062},[641],[638,2064,2066],{"className":2065},[645],[647,2067,2068],{"xmlns":649},[651,2069,2070,2087],{},[654,2071,2072,2075,2077,2079,2081,2083,2085],{},[657,2073,2074],{"mathvariant":891},"Θ",[709,2076,937],{"stretchy":936},[657,2078,933],{},[709,2080,937],{"stretchy":936},[657,2082,659],{},[709,2084,942],{"stretchy":936},[709,2086,942],{"stretchy":936},[661,2088,2089],{"encoding":663},"\\Theta(f(n))",[638,2091,2093],{"className":2092,"ariaHidden":668},[667],[638,2094,2096,2099,2102,2105,2108,2111,2114],{"className":2095},[672],[638,2097],{"className":2098,"style":908},[676],[638,2100,2074],{"className":2101},[681],[638,2103,937],{"className":2104},[962],[638,2106,933],{"className":2107,"style":958},[681,682],[638,2109,937],{"className":2110},[962],[638,2112,659],{"className":2113},[681,682],[638,2115,1989],{"className":2116},[969],[601,2118,2119],{},"encadrement strict"," (entre ",[638,2122,2124,2151],{"className":2123},[641],[638,2125,2127],{"className":2126},[645],[647,2128,2129],{"xmlns":649},[651,2130,2131,2148],{},[654,2132,2133,2140,2142,2144,2146],{},[701,2134,2135,2138],{},[657,2136,2137],{},"c",[713,2139,1415],{},[657,2141,933],{},[709,2143,937],{"stretchy":936},[657,2145,659],{},[709,2147,942],{"stretchy":936},[661,2149,2150],{"encoding":663},"c_1 f(n)",[638,2152,2154],{"className":2153,"ariaHidden":668},[667],[638,2155,2157,2160,2203,2206,2209,2212],{"className":2156},[672],[638,2158],{"className":2159,"style":908},[676],[638,2161,2163,2166],{"className":2162},[681],[638,2164,2137],{"className":2165},[681,682],[638,2167,2169],{"className":2168},[746],[638,2170,2172,2194],{"className":2171},[750,751],[638,2173,2175,2191],{"className":2174},[755],[638,2176,2179],{"className":2177,"style":2178},[759],"height:0.3011em;",[638,2180,2182,2185],{"style":2181},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[638,2183],{"className":2184,"style":768},[767],[638,2186,2188],{"className":2187},[772,773,774,775],[638,2189,1415],{"className":2190},[681,775],[638,2192,783],{"className":2193},[782],[638,2195,2197],{"className":2196},[755],[638,2198,2201],{"className":2199,"style":2200},[759],"height:0.15em;",[638,2202],{},[638,2204,933],{"className":2205,"style":958},[681,682],[638,2207,937],{"className":2208},[962],[638,2210,659],{"className":2211},[681,682],[638,2213,942],{"className":2214},[969]," et ",[638,2217,2219,2245],{"className":2218},[641],[638,2220,2222],{"className":2221},[645],[647,2223,2224],{"xmlns":649},[651,2225,2226,2242],{},[654,2227,2228,2234,2236,2238,2240],{},[701,2229,2230,2232],{},[657,2231,2137],{},[713,2233,715],{},[657,2235,933],{},[709,2237,937],{"stretchy":936},[657,2239,659],{},[709,2241,942],{"stretchy":936},[661,2243,2244],{"encoding":663},"c_2 f(n)",[638,2246,2248],{"className":2247,"ariaHidden":668},[667],[638,2249,2251,2254,2294,2297,2300,2303],{"className":2250},[672],[638,2252],{"className":2253,"style":908},[676],[638,2255,2257,2260],{"className":2256},[681],[638,2258,2137],{"className":2259},[681,682],[638,2261,2263],{"className":2262},[746],[638,2264,2266,2286],{"className":2265},[750,751],[638,2267,2269,2283],{"className":2268},[755],[638,2270,2272],{"className":2271,"style":2178},[759],[638,2273,2274,2277],{"style":2181},[638,2275],{"className":2276,"style":768},[767],[638,2278,2280],{"className":2279},[772,773,774,775],[638,2281,715],{"className":2282},[681,775],[638,2284,783],{"className":2285},[782],[638,2287,2289],{"className":2288},[755],[638,2290,2292],{"className":2291,"style":2200},[759],[638,2293],{},[638,2295,933],{"className":2296,"style":958},[681,682],[638,2298,937],{"className":2299},[962],[638,2301,659],{"className":2302},[681,682],[638,2304,942],{"className":2305},[969],[598,2307,2308,2309,2373],{},"En pratique, dans ce chapitre, ",[638,2310,2312,2338],{"className":2311},[641],[638,2313,2315],{"className":2314},[645],[647,2316,2317],{"xmlns":649},[651,2318,2319,2335],{},[654,2320,2321,2323,2325,2327,2329,2331,2333],{},[657,2322,2074],{"mathvariant":891},[709,2324,937],{"stretchy":936},[657,2326,659],{},[657,2328,707],{},[709,2330,711],{},[657,2332,659],{},[709,2334,942],{"stretchy":936},[661,2336,2337],{"encoding":663},"\\Theta(n \\log n)",[638,2339,2341],{"className":2340,"ariaHidden":668},[667],[638,2342,2344,2347,2350,2353,2356,2359,2364,2367,2370],{"className":2343},[672],[638,2345],{"className":2346,"style":908},[676],[638,2348,2074],{"className":2349},[681],[638,2351,937],{"className":2352},[962],[638,2354,659],{"className":2355},[681,682],[638,2357],{"className":2358,"style":797},[796],[638,2360,738,2362],{"className":2361},[734],[638,2363,742],{"style":741},[638,2365],{"className":2366,"style":797},[796],[638,2368,659],{"className":2369},[681,682],[638,2371,942],{"className":2372},[969]," signifie « ni plus, ni\nmoins, en ordre de grandeur ».",[593,2375,2377],{"id":2376},"le-théorème-maître-forme-appliquée","Le théorème maître (forme appliquée)",[598,2379,2380,2381,2384,2385,2535,2536,2579,2580,2718],{},"Le ",[601,2382,2383],{},"théorème maître"," donne directement la complexité asymptotique de\n",[638,2386,2388,2436],{"className":2387},[641],[638,2389,2391],{"className":2390},[645],[647,2392,2393],{"xmlns":649},[651,2394,2395,2433],{},[654,2396,2397,2399,2401,2403,2405,2407,2409,2411,2413,2415,2417,2419,2421,2423,2425,2427,2429,2431],{},[657,2398,990],{},[709,2400,937],{"stretchy":936},[657,2402,659],{},[709,2404,942],{"stretchy":936},[709,2406,999],{},[657,2408,859],{},[709,2410,1004],{},[657,2412,990],{},[709,2414,937],{"stretchy":936},[657,2416,659],{},[657,2418,892],{"mathvariant":891},[657,2420,895],{},[709,2422,942],{"stretchy":936},[709,2424,1019],{},[657,2426,933],{},[709,2428,937],{"stretchy":936},[657,2430,659],{},[709,2432,942],{"stretchy":936},[661,2434,2435],{"encoding":663},"T(n) = a \\cdot T(n\u002Fb) + f(n)",[638,2437,2439,2466,2484,2517],{"className":2438,"ariaHidden":668},[667],[638,2440,2442,2445,2448,2451,2454,2457,2460,2463],{"className":2441},[672],[638,2443],{"className":2444,"style":908},[676],[638,2446,990],{"className":2447,"style":1046},[681,682],[638,2449,937],{"className":2450},[962],[638,2452,659],{"className":2453},[681,682],[638,2455,942],{"className":2456},[969],[638,2458],{"className":2459,"style":1059},[796],[638,2461,999],{"className":2462},[1063],[638,2464],{"className":2465,"style":1059},[796],[638,2467,2469,2472,2475,2478,2481],{"className":2468},[672],[638,2470],{"className":2471,"style":1073},[676],[638,2473,859],{"className":2474},[681,682],[638,2476],{"className":2477,"style":1080},[796],[638,2479,1004],{"className":2480},[1084],[638,2482],{"className":2483,"style":1080},[796],[638,2485,2487,2490,2493,2496,2499,2502,2505,2508,2511,2514],{"className":2486},[672],[638,2488],{"className":2489,"style":908},[676],[638,2491,990],{"className":2492,"style":1046},[681,682],[638,2494,937],{"className":2495},[962],[638,2497,659],{"className":2498},[681,682],[638,2500,892],{"className":2501},[681],[638,2503,895],{"className":2504},[681,682],[638,2506,942],{"className":2507},[969],[638,2509],{"className":2510,"style":1080},[796],[638,2512,1019],{"className":2513},[1084],[638,2515],{"className":2516,"style":1080},[796],[638,2518,2520,2523,2526,2529,2532],{"className":2519},[672],[638,2521],{"className":2522,"style":908},[676],[638,2524,933],{"className":2525,"style":958},[681,682],[638,2527,937],{"className":2528},[962],[638,2530,659],{"className":2531},[681,682],[638,2533,942],{"className":2534},[969]," en comparant ",[638,2537,2539,2558],{"className":2538},[641],[638,2540,2542],{"className":2541},[645],[647,2543,2544],{"xmlns":649},[651,2545,2546,2556],{},[654,2547,2548,2550,2552,2554],{},[657,2549,933],{},[709,2551,937],{"stretchy":936},[657,2553,659],{},[709,2555,942],{"stretchy":936},[661,2557,945],{"encoding":663},[638,2559,2561],{"className":2560,"ariaHidden":668},[667],[638,2562,2564,2567,2570,2573,2576],{"className":2563},[672],[638,2565],{"className":2566,"style":908},[676],[638,2568,933],{"className":2569,"style":958},[681,682],[638,2571,937],{"className":2572},[962],[638,2574,659],{"className":2575},[681,682],[638,2577,942],{"className":2578},[969]," à ",[638,2581,2583,2614],{"className":2582},[641],[638,2584,2586],{"className":2585},[645],[647,2587,2588],{"xmlns":649},[651,2589,2590,2611],{},[654,2591,2592],{},[2593,2594,2595,2597],"msup",{},[657,2596,659],{},[654,2598,2599,2609],{},[701,2600,2601,2607],{},[654,2602,2603,2605],{},[657,2604,707],{},[709,2606,711],{},[657,2608,895],{},[657,2610,859],{},[661,2612,2613],{"encoding":663},"n^{\\log_b a}",[638,2615,2617],{"className":2616,"ariaHidden":668},[667],[638,2618,2620,2624],{"className":2619},[672],[638,2621],{"className":2622,"style":2623},[676],"height:0.8491em;",[638,2625,2627,2630],{"className":2626},[681],[638,2628,659],{"className":2629},[681,682],[638,2631,2633],{"className":2632},[746],[638,2634,2636],{"className":2635},[750],[638,2637,2639],{"className":2638},[755],[638,2640,2642],{"className":2641,"style":2623},[759],[638,2643,2645,2648],{"style":2644},"top:-3.063em;margin-right:0.05em;",[638,2646],{"className":2647,"style":768},[767],[638,2649,2651],{"className":2650},[772,773,774,775],[638,2652,2654,2711,2715],{"className":2653},[681,775],[638,2655,2657,2671],{"className":2656},[734,775],[638,2658,2660,2664,2668],{"className":2659},[734,775],[638,2661,2663],{"className":2662},[775],"l",[638,2665,2667],{"className":2666},[775],"o",[638,2669,742],{"className":2670,"style":741},[775],[638,2672,2674],{"className":2673},[746],[638,2675,2677,2702],{"className":2676},[750,751],[638,2678,2680,2699],{"className":2679},[755],[638,2681,2684],{"className":2682,"style":2683},[759],"height:0.2302em;",[638,2685,2687,2691],{"style":2686},"top:-2.2341em;margin-right:0.0714em;",[638,2688],{"className":2689,"style":2690},[767],"height:2.5em;",[638,2692,2696],{"className":2693},[772,2694,2695,775],"reset-size3","size1",[638,2697,895],{"className":2698},[681,682,775],[638,2700,783],{"className":2701},[782],[638,2703,2705],{"className":2704},[755],[638,2706,2709],{"className":2707,"style":2708},[759],"height:0.2659em;",[638,2710],{},[638,2712],{"className":2713,"style":2714},[796,775],"margin-right:0.1952em;",[638,2716,859],{"className":2717},[681,682,775]," :",[2720,2721,2722,2781],"table",{},[2723,2724,2725],"thead",{},[2726,2727,2728,2732,2778],"tr",{},[2729,2730,2731],"th",{},"Cas",[2729,2733,2734,2735],{},"Condition sur ",[638,2736,2738,2757],{"className":2737},[641],[638,2739,2741],{"className":2740},[645],[647,2742,2743],{"xmlns":649},[651,2744,2745,2755],{},[654,2746,2747,2749,2751,2753],{},[657,2748,933],{},[709,2750,937],{"stretchy":936},[657,2752,659],{},[709,2754,942],{"stretchy":936},[661,2756,945],{"encoding":663},[638,2758,2760],{"className":2759,"ariaHidden":668},[667],[638,2761,2763,2766,2769,2772,2775],{"className":2762},[672],[638,2764],{"className":2765,"style":908},[676],[638,2767,933],{"className":2768,"style":958},[681,682],[638,2770,937],{"className":2771},[962],[638,2773,659],{"className":2774},[681,682],[638,2776,942],{"className":2777},[969],[2729,2779,2780],{},"Conclusion",[2782,2783,2784,3162,3546],"tbody",{},[2726,2785,2786,2789,2982],{},[2787,2788,1415],"td",{},[2787,2790,2791],{},[638,2792,2794,2846],{"className":2793},[641],[638,2795,2797],{"className":2796},[645],[647,2798,2799],{"xmlns":649},[651,2800,2801,2843],{},[654,2802,2803,2805,2807,2809,2811,2813,2815,2817,2841],{},[657,2804,933],{},[709,2806,937],{"stretchy":936},[657,2808,659],{},[709,2810,942],{"stretchy":936},[709,2812,999],{},[657,2814,1198],{},[709,2816,937],{"stretchy":936},[2593,2818,2819,2821],{},[657,2820,659],{},[654,2822,2823,2833,2835,2838],{},[701,2824,2825,2831],{},[654,2826,2827,2829],{},[657,2828,707],{},[709,2830,711],{},[657,2832,895],{},[657,2834,859],{},[709,2836,2837],{},"−",[657,2839,2840],{},"ε",[709,2842,942],{"stretchy":936},[661,2844,2845],{"encoding":663},"f(n) = O(n^{\\log_b a - \\varepsilon})",[638,2847,2849,2876],{"className":2848,"ariaHidden":668},[667],[638,2850,2852,2855,2858,2861,2864,2867,2870,2873],{"className":2851},[672],[638,2853],{"className":2854,"style":908},[676],[638,2856,933],{"className":2857,"style":958},[681,682],[638,2859,937],{"className":2860},[962],[638,2862,659],{"className":2863},[681,682],[638,2865,942],{"className":2866},[969],[638,2868],{"className":2869,"style":1059},[796],[638,2871,999],{"className":2872},[1063],[638,2874],{"className":2875,"style":1059},[796],[638,2877,2879,2883,2886,2889,2979],{"className":2878},[672],[638,2880],{"className":2881,"style":2882},[676],"height:1.0991em;vertical-align:-0.25em;",[638,2884,1198],{"className":2885,"style":1285},[681,682],[638,2887,937],{"className":2888},[962],[638,2890,2892,2895],{"className":2891},[681],[638,2893,659],{"className":2894},[681,682],[638,2896,2898],{"className":2897},[746],[638,2899,2901],{"className":2900},[750],[638,2902,2904],{"className":2903},[755],[638,2905,2907],{"className":2906,"style":2623},[759],[638,2908,2909,2912],{"style":2644},[638,2910],{"className":2911,"style":768},[767],[638,2913,2915],{"className":2914},[772,773,774,775],[638,2916,2918,2967,2970,2973,2976],{"className":2917},[681,775],[638,2919,2921,2933],{"className":2920},[734,775],[638,2922,2924,2927,2930],{"className":2923},[734,775],[638,2925,2663],{"className":2926},[775],[638,2928,2667],{"className":2929},[775],[638,2931,742],{"className":2932,"style":741},[775],[638,2934,2936],{"className":2935},[746],[638,2937,2939,2959],{"className":2938},[750,751],[638,2940,2942,2956],{"className":2941},[755],[638,2943,2945],{"className":2944,"style":2683},[759],[638,2946,2947,2950],{"style":2686},[638,2948],{"className":2949,"style":2690},[767],[638,2951,2953],{"className":2952},[772,2694,2695,775],[638,2954,895],{"className":2955},[681,682,775],[638,2957,783],{"className":2958},[782],[638,2960,2962],{"className":2961},[755],[638,2963,2965],{"className":2964,"style":2708},[759],[638,2966],{},[638,2968],{"className":2969,"style":2714},[796,775],[638,2971,859],{"className":2972},[681,682,775],[638,2974,2837],{"className":2975},[1084,775],[638,2977,2840],{"className":2978},[681,682,775],[638,2980,942],{"className":2981},[969],[2787,2983,2984],{},[638,2985,2987,3033],{"className":2986},[641],[638,2988,2990],{"className":2989},[645],[647,2991,2992],{"xmlns":649},[651,2993,2994,3030],{},[654,2995,2996,2998,3000,3002,3004,3006,3008,3010,3028],{},[657,2997,990],{},[709,2999,937],{"stretchy":936},[657,3001,659],{},[709,3003,942],{"stretchy":936},[709,3005,999],{},[657,3007,2074],{"mathvariant":891},[709,3009,937],{"stretchy":936},[2593,3011,3012,3014],{},[657,3013,659],{},[654,3015,3016,3026],{},[701,3017,3018,3024],{},[654,3019,3020,3022],{},[657,3021,707],{},[709,3023,711],{},[657,3025,895],{},[657,3027,859],{},[709,3029,942],{"stretchy":936},[661,3031,3032],{"encoding":663},"T(n) = \\Theta(n^{\\log_b a})",[638,3034,3036,3063],{"className":3035,"ariaHidden":668},[667],[638,3037,3039,3042,3045,3048,3051,3054,3057,3060],{"className":3038},[672],[638,3040],{"className":3041,"style":908},[676],[638,3043,990],{"className":3044,"style":1046},[681,682],[638,3046,937],{"className":3047},[962],[638,3049,659],{"className":3050},[681,682],[638,3052,942],{"className":3053},[969],[638,3055],{"className":3056,"style":1059},[796],[638,3058,999],{"className":3059},[1063],[638,3061],{"className":3062,"style":1059},[796],[638,3064,3066,3069,3072,3075,3159],{"className":3065},[672],[638,3067],{"className":3068,"style":2882},[676],[638,3070,2074],{"className":3071},[681],[638,3073,937],{"className":3074},[962],[638,3076,3078,3081],{"className":3077},[681],[638,3079,659],{"className":3080},[681,682],[638,3082,3084],{"className":3083},[746],[638,3085,3087],{"className":3086},[750],[638,3088,3090],{"className":3089},[755],[638,3091,3093],{"className":3092,"style":2623},[759],[638,3094,3095,3098],{"style":2644},[638,3096],{"className":3097,"style":768},[767],[638,3099,3101],{"className":3100},[772,773,774,775],[638,3102,3104,3153,3156],{"className":3103},[681,775],[638,3105,3107,3119],{"className":3106},[734,775],[638,3108,3110,3113,3116],{"className":3109},[734,775],[638,3111,2663],{"className":3112},[775],[638,3114,2667],{"className":3115},[775],[638,3117,742],{"className":3118,"style":741},[775],[638,3120,3122],{"className":3121},[746],[638,3123,3125,3145],{"className":3124},[750,751],[638,3126,3128,3142],{"className":3127},[755],[638,3129,3131],{"className":3130,"style":2683},[759],[638,3132,3133,3136],{"style":2686},[638,3134],{"className":3135,"style":2690},[767],[638,3137,3139],{"className":3138},[772,2694,2695,775],[638,3140,895],{"className":3141},[681,682,775],[638,3143,783],{"className":3144},[782],[638,3146,3148],{"className":3147},[755],[638,3149,3151],{"className":3150,"style":2708},[759],[638,3152],{},[638,3154],{"className":3155,"style":2714},[796,775],[638,3157,859],{"className":3158},[681,682,775],[638,3160,942],{"className":3161},[969],[2726,3163,3164,3166,3346],{},[2787,3165,715],{},[2787,3167,3168],{},[638,3169,3171,3217],{"className":3170},[641],[638,3172,3174],{"className":3173},[645],[647,3175,3176],{"xmlns":649},[651,3177,3178,3214],{},[654,3179,3180,3182,3184,3186,3188,3190,3192,3194,3212],{},[657,3181,933],{},[709,3183,937],{"stretchy":936},[657,3185,659],{},[709,3187,942],{"stretchy":936},[709,3189,999],{},[657,3191,2074],{"mathvariant":891},[709,3193,937],{"stretchy":936},[2593,3195,3196,3198],{},[657,3197,659],{},[654,3199,3200,3210],{},[701,3201,3202,3208],{},[654,3203,3204,3206],{},[657,3205,707],{},[709,3207,711],{},[657,3209,895],{},[657,3211,859],{},[709,3213,942],{"stretchy":936},[661,3215,3216],{"encoding":663},"f(n) = \\Theta(n^{\\log_b a})",[638,3218,3220,3247],{"className":3219,"ariaHidden":668},[667],[638,3221,3223,3226,3229,3232,3235,3238,3241,3244],{"className":3222},[672],[638,3224],{"className":3225,"style":908},[676],[638,3227,933],{"className":3228,"style":958},[681,682],[638,3230,937],{"className":3231},[962],[638,3233,659],{"className":3234},[681,682],[638,3236,942],{"className":3237},[969],[638,3239],{"className":3240,"style":1059},[796],[638,3242,999],{"className":3243},[1063],[638,3245],{"className":3246,"style":1059},[796],[638,3248,3250,3253,3256,3259,3343],{"className":3249},[672],[638,3251],{"className":3252,"style":2882},[676],[638,3254,2074],{"className":3255},[681],[638,3257,937],{"className":3258},[962],[638,3260,3262,3265],{"className":3261},[681],[638,3263,659],{"className":3264},[681,682],[638,3266,3268],{"className":3267},[746],[638,3269,3271],{"className":3270},[750],[638,3272,3274],{"className":3273},[755],[638,3275,3277],{"className":3276,"style":2623},[759],[638,3278,3279,3282],{"style":2644},[638,3280],{"className":3281,"style":768},[767],[638,3283,3285],{"className":3284},[772,773,774,775],[638,3286,3288,3337,3340],{"className":3287},[681,775],[638,3289,3291,3303],{"className":3290},[734,775],[638,3292,3294,3297,3300],{"className":3293},[734,775],[638,3295,2663],{"className":3296},[775],[638,3298,2667],{"className":3299},[775],[638,3301,742],{"className":3302,"style":741},[775],[638,3304,3306],{"className":3305},[746],[638,3307,3309,3329],{"className":3308},[750,751],[638,3310,3312,3326],{"className":3311},[755],[638,3313,3315],{"className":3314,"style":2683},[759],[638,3316,3317,3320],{"style":2686},[638,3318],{"className":3319,"style":2690},[767],[638,3321,3323],{"className":3322},[772,2694,2695,775],[638,3324,895],{"className":3325},[681,682,775],[638,3327,783],{"className":3328},[782],[638,3330,3332],{"className":3331},[755],[638,3333,3335],{"className":3334,"style":2708},[759],[638,3336],{},[638,3338],{"className":3339,"style":2714},[796,775],[638,3341,859],{"className":3342},[681,682,775],[638,3344,942],{"className":3345},[969],[2787,3347,3348],{},[638,3349,3351,3403],{"className":3350},[641],[638,3352,3354],{"className":3353},[645],[647,3355,3356],{"xmlns":649},[651,3357,3358,3400],{},[654,3359,3360,3362,3364,3366,3368,3370,3372,3374,3392,3394,3396,3398],{},[657,3361,990],{},[709,3363,937],{"stretchy":936},[657,3365,659],{},[709,3367,942],{"stretchy":936},[709,3369,999],{},[657,3371,2074],{"mathvariant":891},[709,3373,937],{"stretchy":936},[2593,3375,3376,3378],{},[657,3377,659],{},[654,3379,3380,3390],{},[701,3381,3382,3388],{},[654,3383,3384,3386],{},[657,3385,707],{},[709,3387,711],{},[657,3389,895],{},[657,3391,859],{},[657,3393,707],{},[709,3395,711],{},[657,3397,659],{},[709,3399,942],{"stretchy":936},[661,3401,3402],{"encoding":663},"T(n) = \\Theta(n^{\\log_b a} \\log n)",[638,3404,3406,3433],{"className":3405,"ariaHidden":668},[667],[638,3407,3409,3412,3415,3418,3421,3424,3427,3430],{"className":3408},[672],[638,3410],{"className":3411,"style":908},[676],[638,3413,990],{"className":3414,"style":1046},[681,682],[638,3416,937],{"className":3417},[962],[638,3419,659],{"className":3420},[681,682],[638,3422,942],{"className":3423},[969],[638,3425],{"className":3426,"style":1059},[796],[638,3428,999],{"className":3429},[1063],[638,3431],{"className":3432,"style":1059},[796],[638,3434,3436,3439,3442,3445,3529,3532,3537,3540,3543],{"className":3435},[672],[638,3437],{"className":3438,"style":2882},[676],[638,3440,2074],{"className":3441},[681],[638,3443,937],{"className":3444},[962],[638,3446,3448,3451],{"className":3447},[681],[638,3449,659],{"className":3450},[681,682],[638,3452,3454],{"className":3453},[746],[638,3455,3457],{"className":3456},[750],[638,3458,3460],{"className":3459},[755],[638,3461,3463],{"className":3462,"style":2623},[759],[638,3464,3465,3468],{"style":2644},[638,3466],{"className":3467,"style":768},[767],[638,3469,3471],{"className":3470},[772,773,774,775],[638,3472,3474,3523,3526],{"className":3473},[681,775],[638,3475,3477,3489],{"className":3476},[734,775],[638,3478,3480,3483,3486],{"className":3479},[734,775],[638,3481,2663],{"className":3482},[775],[638,3484,2667],{"className":3485},[775],[638,3487,742],{"className":3488,"style":741},[775],[638,3490,3492],{"className":3491},[746],[638,3493,3495,3515],{"className":3494},[750,751],[638,3496,3498,3512],{"className":3497},[755],[638,3499,3501],{"className":3500,"style":2683},[759],[638,3502,3503,3506],{"style":2686},[638,3504],{"className":3505,"style":2690},[767],[638,3507,3509],{"className":3508},[772,2694,2695,775],[638,3510,895],{"className":3511},[681,682,775],[638,3513,783],{"className":3514},[782],[638,3516,3518],{"className":3517},[755],[638,3519,3521],{"className":3520,"style":2708},[759],[638,3522],{},[638,3524],{"className":3525,"style":2714},[796,775],[638,3527,859],{"className":3528},[681,682,775],[638,3530],{"className":3531,"style":797},[796],[638,3533,738,3535],{"className":3534},[734],[638,3536,742],{"style":741},[638,3538],{"className":3539,"style":797},[796],[638,3541,659],{"className":3542},[681,682],[638,3544,942],{"className":3545},[969],[2726,3547,3548,3551,3741],{},[2787,3549,3550],{},"3",[2787,3552,3553],{},[638,3554,3556,3606],{"className":3555},[641],[638,3557,3559],{"className":3558},[645],[647,3560,3561],{"xmlns":649},[651,3562,3563,3603],{},[654,3564,3565,3567,3569,3571,3573,3575,3577,3579,3601],{},[657,3566,933],{},[709,3568,937],{"stretchy":936},[657,3570,659],{},[709,3572,942],{"stretchy":936},[709,3574,999],{},[657,3576,2011],{"mathvariant":891},[709,3578,937],{"stretchy":936},[2593,3580,3581,3583],{},[657,3582,659],{},[654,3584,3585,3595,3597,3599],{},[701,3586,3587,3593],{},[654,3588,3589,3591],{},[657,3590,707],{},[709,3592,711],{},[657,3594,895],{},[657,3596,859],{},[709,3598,1019],{},[657,3600,2840],{},[709,3602,942],{"stretchy":936},[661,3604,3605],{"encoding":663},"f(n) = \\Omega(n^{\\log_b a + \\varepsilon})",[638,3607,3609,3636],{"className":3608,"ariaHidden":668},[667],[638,3610,3612,3615,3618,3621,3624,3627,3630,3633],{"className":3611},[672],[638,3613],{"className":3614,"style":908},[676],[638,3616,933],{"className":3617,"style":958},[681,682],[638,3619,937],{"className":3620},[962],[638,3622,659],{"className":3623},[681,682],[638,3625,942],{"className":3626},[969],[638,3628],{"className":3629,"style":1059},[796],[638,3631,999],{"className":3632},[1063],[638,3634],{"className":3635,"style":1059},[796],[638,3637,3639,3642,3645,3648,3738],{"className":3638},[672],[638,3640],{"className":3641,"style":2882},[676],[638,3643,2011],{"className":3644},[681],[638,3646,937],{"className":3647},[962],[638,3649,3651,3654],{"className":3650},[681],[638,3652,659],{"className":3653},[681,682],[638,3655,3657],{"className":3656},[746],[638,3658,3660],{"className":3659},[750],[638,3661,3663],{"className":3662},[755],[638,3664,3666],{"className":3665,"style":2623},[759],[638,3667,3668,3671],{"style":2644},[638,3669],{"className":3670,"style":768},[767],[638,3672,3674],{"className":3673},[772,773,774,775],[638,3675,3677,3726,3729,3732,3735],{"className":3676},[681,775],[638,3678,3680,3692],{"className":3679},[734,775],[638,3681,3683,3686,3689],{"className":3682},[734,775],[638,3684,2663],{"className":3685},[775],[638,3687,2667],{"className":3688},[775],[638,3690,742],{"className":3691,"style":741},[775],[638,3693,3695],{"className":3694},[746],[638,3696,3698,3718],{"className":3697},[750,751],[638,3699,3701,3715],{"className":3700},[755],[638,3702,3704],{"className":3703,"style":2683},[759],[638,3705,3706,3709],{"style":2686},[638,3707],{"className":3708,"style":2690},[767],[638,3710,3712],{"className":3711},[772,2694,2695,775],[638,3713,895],{"className":3714},[681,682,775],[638,3716,783],{"className":3717},[782],[638,3719,3721],{"className":3720},[755],[638,3722,3724],{"className":3723,"style":2708},[759],[638,3725],{},[638,3727],{"className":3728,"style":2714},[796,775],[638,3730,859],{"className":3731},[681,682,775],[638,3733,1019],{"className":3734},[1084,775],[638,3736,2840],{"className":3737},[681,682,775],[638,3739,942],{"className":3740},[969],[2787,3742,3743],{},[638,3744,3746,3782],{"className":3745},[641],[638,3747,3749],{"className":3748},[645],[647,3750,3751],{"xmlns":649},[651,3752,3753,3779],{},[654,3754,3755,3757,3759,3761,3763,3765,3767,3769,3771,3773,3775,3777],{},[657,3756,990],{},[709,3758,937],{"stretchy":936},[657,3760,659],{},[709,3762,942],{"stretchy":936},[709,3764,999],{},[657,3766,2074],{"mathvariant":891},[709,3768,937],{"stretchy":936},[657,3770,933],{},[709,3772,937],{"stretchy":936},[657,3774,659],{},[709,3776,942],{"stretchy":936},[709,3778,942],{"stretchy":936},[661,3780,3781],{"encoding":663},"T(n) = \\Theta(f(n))",[638,3783,3785,3812],{"className":3784,"ariaHidden":668},[667],[638,3786,3788,3791,3794,3797,3800,3803,3806,3809],{"className":3787},[672],[638,3789],{"className":3790,"style":908},[676],[638,3792,990],{"className":3793,"style":1046},[681,682],[638,3795,937],{"className":3796},[962],[638,3798,659],{"className":3799},[681,682],[638,3801,942],{"className":3802},[969],[638,3804],{"className":3805,"style":1059},[796],[638,3807,999],{"className":3808},[1063],[638,3810],{"className":3811,"style":1059},[796],[638,3813,3815,3818,3821,3824,3827,3830,3833],{"className":3814},[672],[638,3816],{"className":3817,"style":908},[676],[638,3819,2074],{"className":3820},[681],[638,3822,937],{"className":3823},[962],[638,3825,933],{"className":3826,"style":958},[681,682],[638,3828,937],{"className":3829},[962],[638,3831,659],{"className":3832},[681,682],[638,3834,1989],{"className":3835},[969],[598,3837,3838,3839,3842],{},"En pratique, ",[601,3840,3841],{},"seul le cas 2 est à retenir"," : il couvre le tri fusion et la\nrecherche dichotomique, et c'est celui que vous rencontrerez au baccalauréat.\nLes cas 1 et 3 relèvent de la culture générale — utiles si vous croisez des\nrécurrences plus exotiques en études supérieures, hors-programme strict ici.",[593,3844,3846],{"id":3845},"tri-fusion-lexemple-canonique","Tri fusion — l'exemple canonique",[598,3848,2380,3849,604,3852,3855,3856,3859],{},[601,3850,3851],{},"tri fusion",[606,3853,3854],{},"merge sort",") est l'exemple type. On divise le tableau\nen deux moitiés, on trie chacune récursivement, puis on ",[601,3857,3858],{},"fusionne"," les\ndeux moitiés triées en une seule.",[3861,3862,3867],"pre",{"className":3863,"code":3864,"language":3865,"meta":3866,"style":3866},"language-python shiki shiki-themes github-light github-light github-dark","def fusionner(gauche, droite):\n    \"\"\"Fusionne deux listes déjà triées en une liste triée.\"\"\"\n    resultat = []\n    i, j = 0, 0\n    while i \u003C len(gauche) and j \u003C len(droite):\n        if gauche[i] \u003C= droite[j]:\n            resultat.append(gauche[i]); i += 1\n        else:\n            resultat.append(droite[j]); j += 1\n    resultat.extend(gauche[i:])\n    resultat.extend(droite[j:])\n    return resultat\n\ndef tri_fusion(tab):\n    n = len(tab)\n    if n \u003C= 1:                          # cas de base\n        return tab\n    milieu = n \u002F\u002F 2\n    g = tri_fusion(tab[:milieu])        # régner sur la gauche\n    d = tri_fusion(tab[milieu:])        # régner sur la droite\n    return fusionner(g, d)              # combiner\n\n# >>> tri_fusion([5, 2, 8, 1, 4, 7, 3, 6])\n# [1, 2, 3, 4, 5, 6, 7, 8]\n","python","",[3868,3869,3870,3886,3893,3904,3922,3953,3968,3980,3989,3999,4005,4011,4020,4027,4038,4051,4072,4081,4097,4111,4125,4136,4141,4147],"code",{"__ignoreMap":3866},[638,3871,3874,3878,3882],{"class":3872,"line":3873},"line",1,[638,3875,3877],{"class":3876},"s8jYJ","def",[638,3879,3881],{"class":3880},"snPdu"," fusionner",[638,3883,3885],{"class":3884},"sxrX7","(gauche, droite):\n",[638,3887,3889],{"class":3872,"line":3888},2,[638,3890,3892],{"class":3891},"sIIMD","    \"\"\"Fusionne deux listes déjà triées en une liste triée.\"\"\"\n",[638,3894,3896,3899,3901],{"class":3872,"line":3895},3,[638,3897,3898],{"class":3884},"    resultat ",[638,3900,999],{"class":3876},[638,3902,3903],{"class":3884}," []\n",[638,3905,3907,3910,3912,3916,3919],{"class":3872,"line":3906},4,[638,3908,3909],{"class":3884},"    i, j ",[638,3911,999],{"class":3876},[638,3913,3915],{"class":3914},"sBjJW"," 0",[638,3917,3918],{"class":3884},", ",[638,3920,3921],{"class":3914},"0\n",[638,3923,3925,3928,3931,3934,3937,3940,3943,3946,3948,3950],{"class":3872,"line":3924},5,[638,3926,3927],{"class":3876},"    while",[638,3929,3930],{"class":3884}," i ",[638,3932,3933],{"class":3876},"\u003C",[638,3935,3936],{"class":3914}," len",[638,3938,3939],{"class":3884},"(gauche) ",[638,3941,3942],{"class":3876},"and",[638,3944,3945],{"class":3884}," j ",[638,3947,3933],{"class":3876},[638,3949,3936],{"class":3914},[638,3951,3952],{"class":3884},"(droite):\n",[638,3954,3956,3959,3962,3965],{"class":3872,"line":3955},6,[638,3957,3958],{"class":3876},"        if",[638,3960,3961],{"class":3884}," gauche[i] ",[638,3963,3964],{"class":3876},"\u003C=",[638,3966,3967],{"class":3884}," droite[j]:\n",[638,3969,3971,3974,3977],{"class":3872,"line":3970},7,[638,3972,3973],{"class":3884},"            resultat.append(gauche[i]); i ",[638,3975,3976],{"class":3876},"+=",[638,3978,3979],{"class":3914}," 1\n",[638,3981,3983,3986],{"class":3872,"line":3982},8,[638,3984,3985],{"class":3876},"        else",[638,3987,3988],{"class":3884},":\n",[638,3990,3992,3995,3997],{"class":3872,"line":3991},9,[638,3993,3994],{"class":3884},"            resultat.append(droite[j]); j ",[638,3996,3976],{"class":3876},[638,3998,3979],{"class":3914},[638,4000,4002],{"class":3872,"line":4001},10,[638,4003,4004],{"class":3884},"    resultat.extend(gauche[i:])\n",[638,4006,4008],{"class":3872,"line":4007},11,[638,4009,4010],{"class":3884},"    resultat.extend(droite[j:])\n",[638,4012,4014,4017],{"class":3872,"line":4013},12,[638,4015,4016],{"class":3876},"    return",[638,4018,4019],{"class":3884}," resultat\n",[638,4021,4023],{"class":3872,"line":4022},13,[638,4024,4026],{"emptyLinePlaceholder":4025},true,"\n",[638,4028,4030,4032,4035],{"class":3872,"line":4029},14,[638,4031,3877],{"class":3876},[638,4033,4034],{"class":3880}," tri_fusion",[638,4036,4037],{"class":3884},"(tab):\n",[638,4039,4041,4044,4046,4048],{"class":3872,"line":4040},15,[638,4042,4043],{"class":3884},"    n ",[638,4045,999],{"class":3876},[638,4047,3936],{"class":3914},[638,4049,4050],{"class":3884},"(tab)\n",[638,4052,4054,4057,4060,4062,4065,4068],{"class":3872,"line":4053},16,[638,4055,4056],{"class":3876},"    if",[638,4058,4059],{"class":3884}," n ",[638,4061,3964],{"class":3876},[638,4063,4064],{"class":3914}," 1",[638,4066,4067],{"class":3884},":                          ",[638,4069,4071],{"class":4070},"sCsY4","# cas de base\n",[638,4073,4075,4078],{"class":3872,"line":4074},17,[638,4076,4077],{"class":3876},"        return",[638,4079,4080],{"class":3884}," tab\n",[638,4082,4084,4087,4089,4091,4094],{"class":3872,"line":4083},18,[638,4085,4086],{"class":3884},"    milieu ",[638,4088,999],{"class":3876},[638,4090,4059],{"class":3884},[638,4092,4093],{"class":3876},"\u002F\u002F",[638,4095,4096],{"class":3914}," 2\n",[638,4098,4100,4103,4105,4108],{"class":3872,"line":4099},19,[638,4101,4102],{"class":3884},"    g ",[638,4104,999],{"class":3876},[638,4106,4107],{"class":3884}," tri_fusion(tab[:milieu])        ",[638,4109,4110],{"class":4070},"# régner sur la gauche\n",[638,4112,4114,4117,4119,4122],{"class":3872,"line":4113},20,[638,4115,4116],{"class":3884},"    d ",[638,4118,999],{"class":3876},[638,4120,4121],{"class":3884}," tri_fusion(tab[milieu:])        ",[638,4123,4124],{"class":4070},"# régner sur la droite\n",[638,4126,4128,4130,4133],{"class":3872,"line":4127},21,[638,4129,4016],{"class":3876},[638,4131,4132],{"class":3884}," fusionner(g, d)              ",[638,4134,4135],{"class":4070},"# combiner\n",[638,4137,4139],{"class":3872,"line":4138},22,[638,4140,4026],{"emptyLinePlaceholder":4025},[638,4142,4144],{"class":3872,"line":4143},23,[638,4145,4146],{"class":4070},"# >>> tri_fusion([5, 2, 8, 1, 4, 7, 3, 6])\n",[638,4148,4150],{"class":3872,"line":4149},24,[638,4151,4152],{"class":4070},"# [1, 2, 3, 4, 5, 6, 7, 8]\n",[598,4154,4155,4158,4159,874,4210,4245,4246,4298,4299,4380,4381,4702,4703,4784],{},[601,4156,4157],{},"Analyse de coût"," : on découpe en ",[638,4160,4162,4180],{"className":4161},[641],[638,4163,4165],{"className":4164},[645],[647,4166,4167],{"xmlns":649},[651,4168,4169,4177],{},[654,4170,4171,4173,4175],{},[657,4172,859],{},[709,4174,999],{},[713,4176,715],{},[661,4178,4179],{"encoding":663},"a = 2",[638,4181,4183,4201],{"className":4182,"ariaHidden":668},[667],[638,4184,4186,4189,4192,4195,4198],{"className":4185},[672],[638,4187],{"className":4188,"style":677},[676],[638,4190,859],{"className":4191},[681,682],[638,4193],{"className":4194,"style":1059},[796],[638,4196,999],{"className":4197},[1063],[638,4199],{"className":4200,"style":1059},[796],[638,4202,4204,4207],{"className":4203},[672],[638,4205],{"className":4206,"style":1427},[676],[638,4208,715],{"className":4209},[681],[638,4211,4213,4230],{"className":4212},[641],[638,4214,4216],{"className":4215},[645],[647,4217,4218],{"xmlns":649},[651,4219,4220,4228],{},[654,4221,4222,4224,4226],{},[657,4223,659],{},[657,4225,892],{"mathvariant":891},[713,4227,715],{},[661,4229,1345],{"encoding":663},[638,4231,4233],{"className":4232,"ariaHidden":668},[667],[638,4234,4236,4239,4242],{"className":4235},[672],[638,4237],{"className":4238,"style":908},[676],[638,4240,659],{"className":4241},[681,682],[638,4243,1262],{"className":4244},[681],"\n(donc ",[638,4247,4249,4267],{"className":4248},[641],[638,4250,4252],{"className":4251},[645],[647,4253,4254],{"xmlns":649},[651,4255,4256,4264],{},[654,4257,4258,4260,4262],{},[657,4259,895],{},[709,4261,999],{},[713,4263,715],{},[661,4265,4266],{"encoding":663},"b = 2",[638,4268,4270,4289],{"className":4269,"ariaHidden":668},[667],[638,4271,4273,4277,4280,4283,4286],{"className":4272},[672],[638,4274],{"className":4275,"style":4276},[676],"height:0.6944em;",[638,4278,895],{"className":4279},[681,682],[638,4281],{"className":4282,"style":1059},[796],[638,4284,999],{"className":4285},[1063],[638,4287],{"className":4288,"style":1059},[796],[638,4290,4292,4295],{"className":4291},[672],[638,4293],{"className":4294,"style":1427},[676],[638,4296,715],{"className":4297},[681],"), la fusion linéaire coûte ",[638,4300,4302,4332],{"className":4301},[641],[638,4303,4305],{"className":4304},[645],[647,4306,4307],{"xmlns":649},[651,4308,4309,4329],{},[654,4310,4311,4313,4315,4317,4319,4321,4323,4325,4327],{},[657,4312,933],{},[709,4314,937],{"stretchy":936},[657,4316,659],{},[709,4318,942],{"stretchy":936},[709,4320,999],{},[657,4322,1198],{"mathvariant":1197},[709,4324,937],{"stretchy":936},[657,4326,659],{},[709,4328,942],{"stretchy":936},[661,4330,4331],{"encoding":663},"f(n) = \\mathcal{O}(n)",[638,4333,4335,4362],{"className":4334,"ariaHidden":668},[667],[638,4336,4338,4341,4344,4347,4350,4353,4356,4359],{"className":4337},[672],[638,4339],{"className":4340,"style":908},[676],[638,4342,933],{"className":4343,"style":958},[681,682],[638,4345,937],{"className":4346},[962],[638,4348,659],{"className":4349},[681,682],[638,4351,942],{"className":4352},[969],[638,4354],{"className":4355,"style":1059},[796],[638,4357,999],{"className":4358},[1063],[638,4360],{"className":4361,"style":1059},[796],[638,4363,4365,4368,4371,4374,4377],{"className":4364},[672],[638,4366],{"className":4367,"style":908},[676],[638,4369,1198],{"className":4370,"style":1285},[681,1284],[638,4372,937],{"className":4373},[962],[638,4375,659],{"className":4376},[681,682],[638,4378,942],{"className":4379},[969],". On a\n",[638,4382,4384,4446],{"className":4383},[641],[638,4385,4387],{"className":4386},[645],[647,4388,4389],{"xmlns":649},[651,4390,4391,4443],{},[654,4392,4393,4411,4413,4431,4433,4439,4441],{},[2593,4394,4395,4397],{},[657,4396,659],{},[654,4398,4399,4409],{},[701,4400,4401,4407],{},[654,4402,4403,4405],{},[657,4404,707],{},[709,4406,711],{},[657,4408,895],{},[657,4410,859],{},[709,4412,999],{},[2593,4414,4415,4417],{},[657,4416,659],{},[654,4418,4419,4429],{},[701,4420,4421,4427],{},[654,4422,4423,4425],{},[657,4424,707],{},[709,4426,711],{},[713,4428,715],{},[713,4430,715],{},[709,4432,999],{},[2593,4434,4435,4437],{},[657,4436,659],{},[713,4438,1415],{},[709,4440,999],{},[657,4442,659],{},[661,4444,4445],{"encoding":663},"n^{\\log_b a} = n^{\\log_2 2} = n^1 = n",[638,4447,4449,4548,4648,4693],{"className":4448,"ariaHidden":668},[667],[638,4450,4452,4455,4539,4542,4545],{"className":4451},[672],[638,4453],{"className":4454,"style":2623},[676],[638,4456,4458,4461],{"className":4457},[681],[638,4459,659],{"className":4460},[681,682],[638,4462,4464],{"className":4463},[746],[638,4465,4467],{"className":4466},[750],[638,4468,4470],{"className":4469},[755],[638,4471,4473],{"className":4472,"style":2623},[759],[638,4474,4475,4478],{"style":2644},[638,4476],{"className":4477,"style":768},[767],[638,4479,4481],{"className":4480},[772,773,774,775],[638,4482,4484,4533,4536],{"className":4483},[681,775],[638,4485,4487,4499],{"className":4486},[734,775],[638,4488,4490,4493,4496],{"className":4489},[734,775],[638,4491,2663],{"className":4492},[775],[638,4494,2667],{"className":4495},[775],[638,4497,742],{"className":4498,"style":741},[775],[638,4500,4502],{"className":4501},[746],[638,4503,4505,4525],{"className":4504},[750,751],[638,4506,4508,4522],{"className":4507},[755],[638,4509,4511],{"className":4510,"style":2683},[759],[638,4512,4513,4516],{"style":2686},[638,4514],{"className":4515,"style":2690},[767],[638,4517,4519],{"className":4518},[772,2694,2695,775],[638,4520,895],{"className":4521},[681,682,775],[638,4523,783],{"className":4524},[782],[638,4526,4528],{"className":4527},[755],[638,4529,4531],{"className":4530,"style":2708},[759],[638,4532],{},[638,4534],{"className":4535,"style":2714},[796,775],[638,4537,859],{"className":4538},[681,682,775],[638,4540],{"className":4541,"style":1059},[796],[638,4543,999],{"className":4544},[1063],[638,4546],{"className":4547,"style":1059},[796],[638,4549,4551,4554,4639,4642,4645],{"className":4550},[672],[638,4552],{"className":4553,"style":2623},[676],[638,4555,4557,4560],{"className":4556},[681],[638,4558,659],{"className":4559},[681,682],[638,4561,4563],{"className":4562},[746],[638,4564,4566],{"className":4565},[750],[638,4567,4569],{"className":4568},[755],[638,4570,4572],{"className":4571,"style":2623},[759],[638,4573,4574,4577],{"style":2644},[638,4575],{"className":4576,"style":768},[767],[638,4578,4580],{"className":4579},[772,773,774,775],[638,4581,4583,4633,4636],{"className":4582},[681,775],[638,4584,4586,4598],{"className":4585},[734,775],[638,4587,4589,4592,4595],{"className":4588},[734,775],[638,4590,2663],{"className":4591},[775],[638,4593,2667],{"className":4594},[775],[638,4596,742],{"className":4597,"style":741},[775],[638,4599,4601],{"className":4600},[746],[638,4602,4604,4625],{"className":4603},[750,751],[638,4605,4607,4622],{"className":4606},[755],[638,4608,4611],{"className":4609,"style":4610},[759],"height:0.1944em;",[638,4612,4613,4616],{"style":2686},[638,4614],{"className":4615,"style":2690},[767],[638,4617,4619],{"className":4618},[772,2694,2695,775],[638,4620,715],{"className":4621},[681,775],[638,4623,783],{"className":4624},[782],[638,4626,4628],{"className":4627},[755],[638,4629,4631],{"className":4630,"style":2708},[759],[638,4632],{},[638,4634],{"className":4635,"style":2714},[796,775],[638,4637,715],{"className":4638},[681,775],[638,4640],{"className":4641,"style":1059},[796],[638,4643,999],{"className":4644},[1063],[638,4646],{"className":4647,"style":1059},[796],[638,4649,4651,4655,4684,4687,4690],{"className":4650},[672],[638,4652],{"className":4653,"style":4654},[676],"height:0.8141em;",[638,4656,4658,4661],{"className":4657},[681],[638,4659,659],{"className":4660},[681,682],[638,4662,4664],{"className":4663},[746],[638,4665,4667],{"className":4666},[750],[638,4668,4670],{"className":4669},[755],[638,4671,4673],{"className":4672,"style":4654},[759],[638,4674,4675,4678],{"style":2644},[638,4676],{"className":4677,"style":768},[767],[638,4679,4681],{"className":4680},[772,773,774,775],[638,4682,1415],{"className":4683},[681,775],[638,4685],{"className":4686,"style":1059},[796],[638,4688,999],{"className":4689},[1063],[638,4691],{"className":4692,"style":1059},[796],[638,4694,4696,4699],{"className":4695},[672],[638,4697],{"className":4698,"style":677},[676],[638,4700,659],{"className":4701},[681,682],", donc ",[638,4704,4706,4736],{"className":4705},[641],[638,4707,4709],{"className":4708},[645],[647,4710,4711],{"xmlns":649},[651,4712,4713,4733],{},[654,4714,4715,4717,4719,4721,4723,4725,4727,4729,4731],{},[657,4716,933],{},[709,4718,937],{"stretchy":936},[657,4720,659],{},[709,4722,942],{"stretchy":936},[709,4724,999],{},[657,4726,2074],{"mathvariant":891},[709,4728,937],{"stretchy":936},[657,4730,659],{},[709,4732,942],{"stretchy":936},[661,4734,4735],{"encoding":663},"f(n) = \\Theta(n)",[638,4737,4739,4766],{"className":4738,"ariaHidden":668},[667],[638,4740,4742,4745,4748,4751,4754,4757,4760,4763],{"className":4741},[672],[638,4743],{"className":4744,"style":908},[676],[638,4746,933],{"className":4747,"style":958},[681,682],[638,4749,937],{"className":4750},[962],[638,4752,659],{"className":4753},[681,682],[638,4755,942],{"className":4756},[969],[638,4758],{"className":4759,"style":1059},[796],[638,4761,999],{"className":4762},[1063],[638,4764],{"className":4765,"style":1059},[796],[638,4767,4769,4772,4775,4778,4781],{"className":4768},[672],[638,4770],{"className":4771,"style":908},[676],[638,4773,2074],{"className":4774},[681],[638,4776,937],{"className":4777},[962],[638,4779,659],{"className":4780},[681,682],[638,4782,942],{"className":4783},[969]," — c'est\nle cas 2 du théorème maître :",[638,4786,4788],{"className":4787},[974],[638,4789,4791,4873],{"className":4790},[641],[638,4792,4794],{"className":4793},[645],[647,4795,4796],{"xmlns":649,"display":983},[651,4797,4798,4870],{},[654,4799,4800,4802,4804,4806,4808,4810,4812,4814,4816,4818,4820,4822,4824,4826,4828,4830,4832,4834,4836,4839,4842,4844,4846,4848,4850,4852,4854,4856,4858,4860,4862,4864,4866,4868],{},[657,4801,990],{},[709,4803,937],{"stretchy":936},[657,4805,659],{},[709,4807,942],{"stretchy":936},[709,4809,999],{},[713,4811,715],{},[709,4813,1004],{},[657,4815,990],{},[709,4817,937],{"stretchy":936},[657,4819,659],{},[657,4821,892],{"mathvariant":891},[713,4823,715],{},[709,4825,942],{"stretchy":936},[709,4827,1019],{},[657,4829,1198],{"mathvariant":1197},[709,4831,937],{"stretchy":936},[657,4833,659],{},[709,4835,942],{"stretchy":936},[1178,4837,4838],{},"  ",[709,4840,4841],{},"⟹",[1178,4843,4838],{},[657,4845,990],{},[709,4847,937],{"stretchy":936},[657,4849,659],{},[709,4851,942],{"stretchy":936},[709,4853,999],{},[657,4855,2074],{"mathvariant":891},[709,4857,937],{"stretchy":936},[657,4859,659],{},[657,4861,707],{},[709,4863,711],{},[657,4865,659],{},[709,4867,942],{"stretchy":936},[657,4869,1030],{"mathvariant":891},[661,4871,4872],{"encoding":663},"T(n) = 2 \\cdot T(n\u002F2) + \\mathcal{O}(n) \\;\\Longrightarrow\\; T(n) = \\Theta(n \\log n).",[638,4874,4876,4903,4921,4951,4984,5011],{"className":4875,"ariaHidden":668},[667],[638,4877,4879,4882,4885,4888,4891,4894,4897,4900],{"className":4878},[672],[638,4880],{"className":4881,"style":908},[676],[638,4883,990],{"className":4884,"style":1046},[681,682],[638,4886,937],{"className":4887},[962],[638,4889,659],{"className":4890},[681,682],[638,4892,942],{"className":4893},[969],[638,4895],{"className":4896,"style":1059},[796],[638,4898,999],{"className":4899},[1063],[638,4901],{"className":4902,"style":1059},[796],[638,4904,4906,4909,4912,4915,4918],{"className":4905},[672],[638,4907],{"className":4908,"style":1427},[676],[638,4910,715],{"className":4911},[681],[638,4913],{"className":4914,"style":1080},[796],[638,4916,1004],{"className":4917},[1084],[638,4919],{"className":4920,"style":1080},[796],[638,4922,4924,4927,4930,4933,4936,4939,4942,4945,4948],{"className":4923},[672],[638,4925],{"className":4926,"style":908},[676],[638,4928,990],{"className":4929,"style":1046},[681,682],[638,4931,937],{"className":4932},[962],[638,4934,659],{"className":4935},[681,682],[638,4937,1262],{"className":4938},[681],[638,4940,942],{"className":4941},[969],[638,4943],{"className":4944,"style":1080},[796],[638,4946,1019],{"className":4947},[1084],[638,4949],{"className":4950,"style":1080},[796],[638,4952,4954,4957,4960,4963,4966,4969,4972,4975,4978,4981],{"className":4953},[672],[638,4955],{"className":4956,"style":908},[676],[638,4958,1198],{"className":4959,"style":1285},[681,1284],[638,4961,937],{"className":4962},[962],[638,4964,659],{"className":4965},[681,682],[638,4967,942],{"className":4968},[969],[638,4970],{"className":4971,"style":1059},[796],[638,4973],{"className":4974,"style":1059},[796],[638,4976,4841],{"className":4977},[1063],[638,4979],{"className":4980,"style":1059},[796],[638,4982],{"className":4983,"style":1059},[796],[638,4985,4987,4990,4993,4996,4999,5002,5005,5008],{"className":4986},[672],[638,4988],{"className":4989,"style":908},[676],[638,4991,990],{"className":4992,"style":1046},[681,682],[638,4994,937],{"className":4995},[962],[638,4997,659],{"className":4998},[681,682],[638,5000,942],{"className":5001},[969],[638,5003],{"className":5004,"style":1059},[796],[638,5006,999],{"className":5007},[1063],[638,5009],{"className":5010,"style":1059},[796],[638,5012,5014,5017,5020,5023,5026,5029,5034,5037,5040,5043],{"className":5013},[672],[638,5015],{"className":5016,"style":908},[676],[638,5018,2074],{"className":5019},[681],[638,5021,937],{"className":5022},[962],[638,5024,659],{"className":5025},[681,682],[638,5027],{"className":5028,"style":797},[796],[638,5030,738,5032],{"className":5031},[734],[638,5033,742],{"style":741},[638,5035],{"className":5036,"style":797},[796],[638,5038,659],{"className":5039},[681,682],[638,5041,942],{"className":5042},[969],[638,5044,1030],{"className":5045},[681],[598,5047,5048,5049,5052],{},"Le tri fusion ",[601,5050,5051],{},"garantit"," ce coût en pire cas, ce qui le rend supérieur\nau tri rapide pour des données potentiellement adverses.",[5054,5055,5057,5063],"self-eval-qcm",{"correct":1415,"slug":5056},"diviser-pour-regner-c1",[5058,5059,5060],"template",{"v-slot:question":3866},[598,5061,5062],{},"Pour T(n) = 2 T(n\u002F2) + O(n), la complexité est…",[5058,5064,5065],{"v-slot:choices":3866},[1432,5066,5067,5070,5073],{},[816,5068,5069],{},"O(n)",[816,5071,5072],{},"O(n log n)",[816,5074,5075],{},"O(n²)",[593,5077,5079],{"id":5078},"un-mot-sur-le-coût-en-mémoire","Un mot sur le coût en mémoire",[598,5081,5082,5083,5126,5127,5130,5131,1030],{},"Le tri fusion exposé ci-dessus crée une nouvelle liste à chaque appel — son\ncoût mémoire est en ",[638,5084,5086,5105],{"className":5085},[641],[638,5087,5089],{"className":5088},[645],[647,5090,5091],{"xmlns":649},[651,5092,5093,5103],{},[654,5094,5095,5097,5099,5101],{},[657,5096,1198],{"mathvariant":1197},[709,5098,937],{"stretchy":936},[657,5100,659],{},[709,5102,942],{"stretchy":936},[661,5104,1463],{"encoding":663},[638,5106,5108],{"className":5107,"ariaHidden":668},[667],[638,5109,5111,5114,5117,5120,5123],{"className":5110},[672],[638,5112],{"className":5113,"style":908},[676],[638,5115,1198],{"className":5116,"style":1285},[681,1284],[638,5118,937],{"className":5119},[962],[638,5121,659],{"className":5122},[681,682],[638,5124,942],{"className":5125},[969],". Des implémentations ",[601,5128,5129],{},"en place"," (qui\nne créent pas de copies) existent mais sont plus délicates à écrire. C'est\nl'un des compromis classiques de l'algorithmique : ",[601,5132,5133],{},"gagner en mémoire au\nprix de la simplicité du code",[593,5135,5137],{"id":5136},"autre-exemple-rotation-dune-image-bitmap","Autre exemple : rotation d'une image bitmap",[598,5139,5140,5141,5144,5145,5220,5221,1030],{},"Une image bitmap est un tableau 2D de pixels. La ",[601,5142,5143],{},"rotation d'un quart de\ntour"," se résout naïvement en allouant une seconde image et en y recopiant\nles pixels permutés — coût mémoire ",[638,5146,5148,5172],{"className":5147},[641],[638,5149,5151],{"className":5150},[645],[647,5152,5153],{"xmlns":649},[651,5154,5155,5169],{},[654,5156,5157,5159,5161,5167],{},[657,5158,1198],{"mathvariant":1197},[709,5160,937],{"stretchy":936},[2593,5162,5163,5165],{},[657,5164,659],{},[713,5166,715],{},[709,5168,942],{"stretchy":936},[661,5170,5171],{"encoding":663},"\\mathcal{O}(n^2)",[638,5173,5175],{"className":5174,"ariaHidden":668},[667],[638,5176,5178,5182,5185,5188,5217],{"className":5177},[672],[638,5179],{"className":5180,"style":5181},[676],"height:1.0641em;vertical-align:-0.25em;",[638,5183,1198],{"className":5184,"style":1285},[681,1284],[638,5186,937],{"className":5187},[962],[638,5189,5191,5194],{"className":5190},[681],[638,5192,659],{"className":5193},[681,682],[638,5195,5197],{"className":5196},[746],[638,5198,5200],{"className":5199},[750],[638,5201,5203],{"className":5202},[755],[638,5204,5206],{"className":5205,"style":4654},[759],[638,5207,5208,5211],{"style":2644},[638,5209],{"className":5210,"style":768},[767],[638,5212,5214],{"className":5213},[772,773,774,775],[638,5215,715],{"className":5216},[681,775],[638,5218,942],{"className":5219},[969]," pour une image\n",[638,5222,5224,5242],{"className":5223},[641],[638,5225,5227],{"className":5226},[645],[647,5228,5229],{"xmlns":649},[651,5230,5231,5239],{},[654,5232,5233,5235,5237],{},[657,5234,659],{},[709,5236,1809],{},[657,5238,659],{},[661,5240,5241],{"encoding":663},"n \\times n",[638,5243,5245,5264],{"className":5244,"ariaHidden":668},[667],[638,5246,5248,5252,5255,5258,5261],{"className":5247},[672],[638,5249],{"className":5250,"style":5251},[676],"height:0.6667em;vertical-align:-0.0833em;",[638,5253,659],{"className":5254},[681,682],[638,5256],{"className":5257,"style":1080},[796],[638,5259,1809],{"className":5260},[1084],[638,5262],{"className":5263,"style":1080},[796],[638,5265,5267,5270],{"className":5266},[672],[638,5268],{"className":5269,"style":677},[676],[638,5271,659],{"className":5272},[681,682],[598,5274,5275,5276,5279,5280,5282,5283,5286],{},"L'approche ",[601,5277,5278],{},"diviser pour régner"," divise l'image en quatre quadrants,\nfait pivoter chaque quadrant récursivement, puis échange les quadrants\ndeux à deux ",[601,5281,5129],{},". Le coût mémoire devient ",[601,5284,5285],{},"constant"," (en\nignorant la pile d'appels), au prix d'un code plus subtil.",[5288,5289,5290],"warning",{},[598,5291,5292,5293,5343,5344,5400,5401,5456,5457,1030],{},"La récursion garde une pile d'appels — ce n'est jamais strictement de la\nmémoire « constante ». Pour une image ",[638,5294,5296,5313],{"className":5295},[641],[638,5297,5299],{"className":5298},[645],[647,5300,5301],{"xmlns":649},[651,5302,5303,5311],{},[654,5304,5305,5307,5309],{},[657,5306,659],{},[709,5308,1809],{},[657,5310,659],{},[661,5312,5241],{"encoding":663},[638,5314,5316,5334],{"className":5315,"ariaHidden":668},[667],[638,5317,5319,5322,5325,5328,5331],{"className":5318},[672],[638,5320],{"className":5321,"style":5251},[676],[638,5323,659],{"className":5324},[681,682],[638,5326],{"className":5327,"style":1080},[796],[638,5329,1809],{"className":5330},[1084],[638,5332],{"className":5333,"style":1080},[796],[638,5335,5337,5340],{"className":5336},[672],[638,5338],{"className":5339,"style":677},[676],[638,5341,659],{"className":5342},[681,682],", la profondeur récursive\nest ",[638,5345,5347,5371],{"className":5346},[641],[638,5348,5350],{"className":5349},[645],[647,5351,5352],{"xmlns":649},[651,5353,5354,5368],{},[654,5355,5356,5358,5360,5362,5364,5366],{},[657,5357,1198],{"mathvariant":1197},[709,5359,937],{"stretchy":936},[657,5361,707],{},[709,5363,711],{},[657,5365,659],{},[709,5367,942],{"stretchy":936},[661,5369,5370],{"encoding":663},"\\mathcal{O}(\\log n)",[638,5372,5374],{"className":5373,"ariaHidden":668},[667],[638,5375,5377,5380,5383,5386,5391,5394,5397],{"className":5376},[672],[638,5378],{"className":5379,"style":908},[676],[638,5381,1198],{"className":5382,"style":1285},[681,1284],[638,5384,937],{"className":5385},[962],[638,5387,738,5389],{"className":5388},[734],[638,5390,742],{"style":741},[638,5392],{"className":5393,"style":797},[796],[638,5395,659],{"className":5396},[681,682],[638,5398,942],{"className":5399},[969],", donc la mémoire totale est dominée par la\nrécursion, soit ",[638,5402,5404,5427],{"className":5403},[641],[638,5405,5407],{"className":5406},[645],[647,5408,5409],{"xmlns":649},[651,5410,5411,5425],{},[654,5412,5413,5415,5417,5419,5421,5423],{},[657,5414,1198],{"mathvariant":1197},[709,5416,937],{"stretchy":936},[657,5418,707],{},[709,5420,711],{},[657,5422,659],{},[709,5424,942],{"stretchy":936},[661,5426,5370],{"encoding":663},[638,5428,5430],{"className":5429,"ariaHidden":668},[667],[638,5431,5433,5436,5439,5442,5447,5450,5453],{"className":5432},[672],[638,5434],{"className":5435,"style":908},[676],[638,5437,1198],{"className":5438,"style":1285},[681,1284],[638,5440,937],{"className":5441},[962],[638,5443,738,5445],{"className":5444},[734],[638,5446,742],{"style":741},[638,5448],{"className":5449,"style":797},[796],[638,5451,659],{"className":5452},[681,682],[638,5454,942],{"className":5455},[969]," — très inférieur à ",[638,5458,5460,5483],{"className":5459},[641],[638,5461,5463],{"className":5462},[645],[647,5464,5465],{"xmlns":649},[651,5466,5467,5481],{},[654,5468,5469,5471,5473,5479],{},[657,5470,1198],{"mathvariant":1197},[709,5472,937],{"stretchy":936},[2593,5474,5475,5477],{},[657,5476,659],{},[713,5478,715],{},[709,5480,942],{"stretchy":936},[661,5482,5171],{"encoding":663},[638,5484,5486],{"className":5485,"ariaHidden":668},[667],[638,5487,5489,5492,5495,5498,5527],{"className":5488},[672],[638,5490],{"className":5491,"style":5181},[676],[638,5493,1198],{"className":5494,"style":1285},[681,1284],[638,5496,937],{"className":5497},[962],[638,5499,5501,5504],{"className":5500},[681],[638,5502,659],{"className":5503},[681,682],[638,5505,5507],{"className":5506},[746],[638,5508,5510],{"className":5509},[750],[638,5511,5513],{"className":5512},[755],[638,5514,5516],{"className":5515,"style":4654},[759],[638,5517,5518,5521],{"style":2644},[638,5519],{"className":5520,"style":768},[767],[638,5522,5524],{"className":5523},[772,773,774,775],[638,5525,715],{"className":5526},[681,775],[638,5528,942],{"className":5529},[969],[593,5531,5533],{"id":5532},"recette-pour-reconnaître-un-problème-diviser-pour-régner","Recette pour reconnaître un problème « diviser pour régner »",[598,5535,5536],{},"Trois indices :",[1432,5538,5539,5546,5549],{},[816,5540,5541,5542,5545],{},"Le problème se ",[601,5543,5544],{},"décompose naturellement"," en sous-problèmes du même type.",[816,5547,5548],{},"La résolution d'un cas de base (taille 0 ou 1) est triviale.",[816,5550,5551],{},"Recombiner les sous-solutions coûte beaucoup moins que le problème entier.",[5054,5553,5556,5561],{"correct":5554,"slug":5555},"0","diviser-pour-regner-c2",[5058,5557,5558],{"v-slot:question":3866},[598,5559,5560],{},"Le tri fusion garantit un coût en O(n log n) dans le pire cas.",[5058,5562,5563],{"v-slot:choices":3866},[1432,5564,5565,5568],{},[816,5566,5567],{},"vrai",[816,5569,5570],{},"faux",[593,5572,5574],{"id":5573},"pour-aller-plus-loin","Pour aller plus loin",[598,5576,5577,5578,5581,5582,5645,5646,5649,5650,5740,5741,5814,5815,5818,5819,831],{},"D'autres algorithmes célèbres reposent sur cette méthode : la ",[601,5579,5580],{},"transformée\nde Fourier rapide"," (FFT, ",[638,5583,5585,5610],{"className":5584},[641],[638,5586,5588],{"className":5587},[645],[647,5589,5590],{"xmlns":649},[651,5591,5592,5608],{},[654,5593,5594,5596,5598,5600,5602,5604,5606],{},[657,5595,1198],{"mathvariant":1197},[709,5597,937],{"stretchy":936},[657,5599,659],{},[657,5601,707],{},[709,5603,711],{},[657,5605,659],{},[709,5607,942],{"stretchy":936},[661,5609,1880],{"encoding":663},[638,5611,5613],{"className":5612,"ariaHidden":668},[667],[638,5614,5616,5619,5622,5625,5628,5631,5636,5639,5642],{"className":5615},[672],[638,5617],{"className":5618,"style":908},[676],[638,5620,1198],{"className":5621,"style":1285},[681,1284],[638,5623,937],{"className":5624},[962],[638,5626,659],{"className":5627},[681,682],[638,5629],{"className":5630,"style":797},[796],[638,5632,738,5634],{"className":5633},[734],[638,5635,742],{"style":741},[638,5637],{"className":5638,"style":797},[796],[638,5640,659],{"className":5641},[681,682],[638,5643,942],{"className":5644},[969]," pour traiter le signal),\nl'algorithme de ",[601,5647,5648],{},"Karatsuba"," pour la multiplication d'entiers en\n",[638,5651,5653,5678],{"className":5652},[641],[638,5654,5656],{"className":5655},[645],[647,5657,5658],{"xmlns":649},[651,5659,5660,5675],{},[654,5661,5662,5664,5666,5673],{},[657,5663,1198],{"mathvariant":1197},[709,5665,937],{"stretchy":936},[2593,5667,5668,5670],{},[657,5669,659],{},[713,5671,5672],{},"1,58",[709,5674,942],{"stretchy":936},[661,5676,5677],{"encoding":663},"\\mathcal{O}(n^{1{,}58})",[638,5679,5681],{"className":5680,"ariaHidden":668},[667],[638,5682,5684,5687,5690,5693,5737],{"className":5683},[672],[638,5685],{"className":5686,"style":5181},[676],[638,5688,1198],{"className":5689,"style":1285},[681,1284],[638,5691,937],{"className":5692},[962],[638,5694,5696,5699],{"className":5695},[681],[638,5697,659],{"className":5698},[681,682],[638,5700,5702],{"className":5701},[746],[638,5703,5705],{"className":5704},[750],[638,5706,5708],{"className":5707},[755],[638,5709,5711],{"className":5710,"style":4654},[759],[638,5712,5713,5716],{"style":2644},[638,5714],{"className":5715,"style":768},[767],[638,5717,5719],{"className":5718},[772,773,774,775],[638,5720,5722,5725,5733],{"className":5721},[681,775],[638,5723,1415],{"className":5724},[681,775],[638,5726,5728],{"className":5727},[681,775],[638,5729,5732],{"className":5730},[5731,775],"mpunct",",",[638,5734,5736],{"className":5735},[681,775],"58",[638,5738,942],{"className":5739},[969]," au lieu de ",[638,5742,5744,5767],{"className":5743},[641],[638,5745,5747],{"className":5746},[645],[647,5748,5749],{"xmlns":649},[651,5750,5751,5765],{},[654,5752,5753,5755,5757,5763],{},[657,5754,1198],{"mathvariant":1197},[709,5756,937],{"stretchy":936},[2593,5758,5759,5761],{},[657,5760,659],{},[713,5762,715],{},[709,5764,942],{"stretchy":936},[661,5766,5171],{"encoding":663},[638,5768,5770],{"className":5769,"ariaHidden":668},[667],[638,5771,5773,5776,5779,5782,5811],{"className":5772},[672],[638,5774],{"className":5775,"style":5181},[676],[638,5777,1198],{"className":5778,"style":1285},[681,1284],[638,5780,937],{"className":5781},[962],[638,5783,5785,5788],{"className":5784},[681],[638,5786,659],{"className":5787},[681,682],[638,5789,5791],{"className":5790},[746],[638,5792,5794],{"className":5793},[750],[638,5795,5797],{"className":5796},[755],[638,5798,5800],{"className":5799,"style":4654},[759],[638,5801,5802,5805],{"style":2644},[638,5803],{"className":5804,"style":768},[767],[638,5806,5808],{"className":5807},[772,773,774,775],[638,5809,715],{"className":5810},[681,775],[638,5812,942],{"className":5813},[969],", ou encore la\n",[601,5816,5817],{},"recherche dichotomique"," (cas particulier où ",[638,5820,5822,5840],{"className":5821},[641],[638,5823,5825],{"className":5824},[645],[647,5826,5827],{"xmlns":649},[651,5828,5829,5837],{},[654,5830,5831,5833,5835],{},[657,5832,859],{},[709,5834,999],{},[713,5836,1415],{},[661,5838,5839],{"encoding":663},"a = 1",[638,5841,5843,5861],{"className":5842,"ariaHidden":668},[667],[638,5844,5846,5849,5852,5855,5858],{"className":5845},[672],[638,5847],{"className":5848,"style":677},[676],[638,5850,859],{"className":5851},[681,682],[638,5853],{"className":5854,"style":1059},[796],[638,5856,999],{"className":5857},[1063],[638,5859],{"className":5860,"style":1059},[796],[638,5862,5864,5867],{"className":5863},[672],[638,5865],{"className":5866,"style":1427},[676],[638,5868,1415],{"className":5869},[681],[5871,5872,5873],"style",{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html 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fusion",[638,6531,6533,6606],{"className":6532},[641],[638,6534,6536],{"className":6535},[645],[647,6537,6538],{"xmlns":649},[651,6539,6540,6603],{},[654,6541,6542,6544,6546,6548,6550,6552,6554,6556,6558,6560,6562,6564,6566,6568,6570,6572,6574,6576,6579,6581,6583,6585,6587,6589,6591,6593,6595,6597,6599,6601],{},[657,6543,990],{},[709,6545,937],{"stretchy":936},[657,6547,659],{},[709,6549,942],{"stretchy":936},[709,6551,999],{},[713,6553,715],{},[657,6555,990],{},[709,6557,937],{"stretchy":936},[657,6559,659],{},[657,6561,892],{"mathvariant":891},[713,6563,715],{},[709,6565,942],{"stretchy":936},[709,6567,1019],{},[657,6569,1198],{"mathvariant":1197},[709,6571,937],{"stretchy":936},[657,6573,659],{},[709,6575,942],{"stretchy":936},[709,6577,6578],{},"⇒",[657,6580,990],{},[709,6582,937],{"stretchy":936},[657,6584,659],{},[709,6586,942],{"stretchy":936},[709,6588,999],{},[657,6590,2074],{"mathvariant":891},[709,6592,937],{"stretchy":936},[657,6594,659],{},[657,6596,707],{},[709,6598,711],{},[657,6600,659],{},[709,6602,942],{"stretchy":936},[661,6604,6605],{"encoding":663},"T(n) = 2 T(n\u002F2) + \\mathcal{O}(n) \\Rightarrow T(n) = \\Theta(n \\log n)",[638,6607,6609,6636,6669,6696,6723],{"className":6608,"ariaHidden":668},[667],[638,6610,6612,6615,6618,6621,6624,6627,6630,6633],{"className":6611},[672],[638,6613],{"className":6614,"style":908},[676],[638,6616,990],{"className":6617,"style":1046},[681,682],[638,6619,937],{"className":6620},[962],[638,6622,659],{"className":6623},[681,682],[638,6625,942],{"className":6626},[969],[638,6628],{"className":6629,"style":1059},[796],[638,6631,999],{"className":6632},[1063],[638,6634],{"className":6635,"style":1059},[796],[638,6637,6639,6642,6645,6648,6651,6654,6657,6660,6663,6666],{"className":6638},[672],[638,6640],{"className":6641,"style":908},[676],[638,6643,715],{"className":6644},[681],[638,6646,990],{"className":6647,"style":1046},[681,682],[638,6649,937],{"className":6650},[962],[638,6652,659],{"className":6653},[681,682],[638,6655,1262],{"className":6656},[681],[638,6658,942],{"className":6659},[969],[638,6661],{"className":6662,"style":1080},[796],[638,6664,1019],{"className":6665},[1084],[638,6667],{"className":6668,"style":1080},[796],[638,6670,6672,6675,6678,6681,6684,6687,6690,6693],{"className":6671},[672],[638,6673],{"className":6674,"style":908},[676],[638,6676,1198],{"className":6677,"style":1285},[681,1284],[638,6679,937],{"className":6680},[962],[638,6682,659],{"className":6683},[681,682],[638,6685,942],{"className":6686},[969],[638,6688],{"className":6689,"style":1059},[796],[638,6691,6578],{"className":6692},[1063],[638,6694],{"className":6695,"style":1059},[796],[638,6697,6699,6702,6705,6708,6711,6714,6717,6720],{"className":6698},[672],[638,6700],{"className":6701,"style":908},[676],[638,6703,990],{"className":6704,"style":1046},[681,682],[638,6706,937],{"className":6707},[962],[638,6709,659],{"className":6710},[681,682],[638,6712,942],{"className":6713},[969],[638,6715],{"className":6716,"style":1059},[796],[638,6718,999],{"className":6719},[1063],[638,6721],{"className":6722,"style":1059},[796],[638,6724,6726,6729,6732,6735,6738,6741,6746,6749,6752],{"className":6725},[672],[638,6727],{"className":6728,"style":908},[676],[638,6730,2074],{"className":6731},[681],[638,6733,937],{"className":6734},[962],[638,6736,659],{"className":6737},[681,682],[638,6739],{"className":6740,"style":797},[796],[638,6742,738,6744],{"className":6743},[734],[638,6745,742],{"style":741},[638,6747],{"className":6748,"style":797},[796],[638,6750,659],{"className":6751},[681,682],[638,6753,942],{"className":6754},[969],[816,6756,6757,1710,6760,6845,6846,1030],{},[601,6758,6759],{},"Profondeur récursive",[638,6761,6763,6787],{"className":6762},[641],[638,6764,6766],{"className":6765},[645],[647,6767,6768],{"xmlns":649},[651,6769,6770,6784],{},[654,6771,6772,6782],{},[701,6773,6774,6780],{},[654,6775,6776,6778],{},[657,6777,707],{},[709,6779,711],{},[657,6781,895],{},[657,6783,659],{},[661,6785,6786],{"encoding":663},"\\log_b n",[638,6788,6790],{"className":6789,"ariaHidden":668},[667],[638,6791,6793,6796,6839,6842],{"className":6792},[672],[638,6794],{"className":6795,"style":730},[676],[638,6797,6799,6804],{"className":6798},[734],[638,6800,738,6802],{"className":6801},[734],[638,6803,742],{"style":741},[638,6805,6807],{"className":6806},[746],[638,6808,6810,6831],{"className":6809},[750,751],[638,6811,6813,6828],{"className":6812},[755],[638,6814,6817],{"className":6815,"style":6816},[759],"height:0.242em;",[638,6818,6819,6822],{"style":763},[638,6820],{"className":6821,"style":768},[767],[638,6823,6825],{"className":6824},[772,773,774,775],[638,6826,895],{"className":6827},[681,682,775],[638,6829,783],{"className":6830},[782],[638,6832,6834],{"className":6833},[755],[638,6835,6837],{"className":6836,"style":790},[759],[638,6838],{},[638,6840],{"className":6841,"style":797},[796],[638,6843,659],{"className":6844},[681,682]," niveaux pour une découpe en parts de\ntaille ",[638,6847,6849,6866],{"className":6848},[641],[638,6850,6852],{"className":6851},[645],[647,6853,6854],{"xmlns":649},[651,6855,6856,6864],{},[654,6857,6858,6860,6862],{},[657,6859,659],{},[657,6861,892],{"mathvariant":891},[657,6863,895],{},[661,6865,898],{"encoding":663},[638,6867,6869],{"className":6868,"ariaHidden":668},[667],[638,6870,6872,6875,6878,6881],{"className":6871},[672],[638,6873],{"className":6874,"style":908},[676],[638,6876,659],{"className":6877},[681,682],[638,6879,892],{"className":6880},[681],[638,6882,895],{"className":6883},[681,682],[593,6885,6887],{"id":6886},"le-squelette-dun-algorithme","Le squelette d'un algorithme",[2720,6889,6890,6900],{},[2723,6891,6892],{},[2726,6893,6894,6897],{},[2729,6895,6896],{},"Étape",[2729,6898,6899],{},"Action",[2782,6901,6902,6963,6970,6977],{},[2726,6903,6904,6907],{},[2787,6905,6906],{},"Cas de base",[2787,6908,6909,6910],{},"renvoyer la réponse pour ",[638,6911,6913,6932],{"className":6912},[641],[638,6914,6916],{"className":6915},[645],[647,6917,6918],{"xmlns":649},[651,6919,6920,6929],{},[654,6921,6922,6924,6927],{},[657,6923,659],{},[709,6925,6926],{},"≤",[713,6928,1415],{},[661,6930,6931],{"encoding":663},"n \\le 1",[638,6933,6935,6954],{"className":6934,"ariaHidden":668},[667],[638,6936,6938,6942,6945,6948,6951],{"className":6937},[672],[638,6939],{"className":6940,"style":6941},[676],"height:0.7719em;vertical-align:-0.136em;",[638,6943,659],{"className":6944},[681,682],[638,6946],{"className":6947,"style":1059},[796],[638,6949,6926],{"className":6950},[1063],[638,6952],{"className":6953,"style":1059},[796],[638,6955,6957,6960],{"className":6956},[672],[638,6958],{"className":6959,"style":1427},[676],[638,6961,1415],{"className":6962},[681],[2726,6964,6965,6967],{},[2787,6966,820],{},[2787,6968,6969],{},"partitionner l'entrée",[2726,6971,6972,6974],{},[2787,6973,826],{},[2787,6975,6976],{},"appels récursifs",[2726,6978,6979,6981],{},[2787,6980,836],{},[2787,6982,6983],{},"fusionner les sous-solutions",[593,6985,6987],{"id":6986},"coûts-à-connaître","Coûts à connaître",[2720,6989,6990,7003],{},[2723,6991,6992],{},[2726,6993,6994,6997,7000],{},[2729,6995,6996],{},"Algorithme",[2729,6998,6999],{},"Récurrence",[2729,7001,7002],{},"Coût",[2782,7004,7005,7169,7320],{},[2726,7006,7007,7009,7104],{},[2787,7008,6529],{},[2787,7010,7011],{},[638,7012,7014,7050],{"className":7013},[641],[638,7015,7017],{"className":7016},[645],[647,7018,7019],{"xmlns":649},[651,7020,7021,7047],{},[654,7022,7023,7025,7027,7029,7031,7033,7035,7037,7039,7041,7043,7045],{},[713,7024,715],{},[657,7026,990],{},[709,7028,937],{"stretchy":936},[657,7030,659],{},[657,7032,892],{"mathvariant":891},[713,7034,715],{},[709,7036,942],{"stretchy":936},[709,7038,1019],{},[657,7040,1198],{"mathvariant":1197},[709,7042,937],{"stretchy":936},[657,7044,659],{},[709,7046,942],{"stretchy":936},[661,7048,7049],{"encoding":663},"2 T(n\u002F2) + \\mathcal{O}(n)",[638,7051,7053,7086],{"className":7052,"ariaHidden":668},[667],[638,7054,7056,7059,7062,7065,7068,7071,7074,7077,7080,7083],{"className":7055},[672],[638,7057],{"className":7058,"style":908},[676],[638,7060,715],{"className":7061},[681],[638,7063,990],{"className":7064,"style":1046},[681,682],[638,7066,937],{"className":7067},[962],[638,7069,659],{"className":7070},[681,682],[638,7072,1262],{"className":7073},[681],[638,7075,942],{"className":7076},[969],[638,7078],{"className":7079,"style":1080},[796],[638,7081,1019],{"className":7082},[1084],[638,7084],{"className":7085,"style":1080},[796],[638,7087,7089,7092,7095,7098,7101],{"className":7088},[672],[638,7090],{"className":7091,"style":908},[676],[638,7093,1198],{"className":7094,"style":1285},[681,1284],[638,7096,937],{"className":7097},[962],[638,7099,659],{"className":7100},[681,682],[638,7102,942],{"className":7103},[969],[2787,7105,7106],{},[638,7107,7109,7134],{"className":7108},[641],[638,7110,7112],{"className":7111},[645],[647,7113,7114],{"xmlns":649},[651,7115,7116,7132],{},[654,7117,7118,7120,7122,7124,7126,7128,7130],{},[657,7119,1198],{"mathvariant":1197},[709,7121,937],{"stretchy":936},[657,7123,659],{},[657,7125,707],{},[709,7127,711],{},[657,7129,659],{},[709,7131,942],{"stretchy":936},[661,7133,1880],{"encoding":663},[638,7135,7137],{"className":7136,"ariaHidden":668},[667],[638,7138,7140,7143,7146,7149,7152,7155,7160,7163,7166],{"className":7139},[672],[638,7141],{"className":7142,"style":908},[676],[638,7144,1198],{"className":7145,"style":1285},[681,1284],[638,7147,937],{"className":7148},[962],[638,7150,659],{"className":7151},[681,682],[638,7153],{"className":7154,"style":797},[796],[638,7156,738,7158],{"className":7157},[734],[638,7159,742],{"style":741},[638,7161],{"className":7162,"style":797},[796],[638,7164,659],{"className":7165},[681,682],[638,7167,942],{"className":7168},[969],[2726,7170,7171,7173,7263],{},[2787,7172,429],{},[2787,7174,7175],{},[638,7176,7178,7212],{"className":7177},[641],[638,7179,7181],{"className":7180},[645],[647,7182,7183],{"xmlns":649},[651,7184,7185,7209],{},[654,7186,7187,7189,7191,7193,7195,7197,7199,7201,7203,7205,7207],{},[657,7188,990],{},[709,7190,937],{"stretchy":936},[657,7192,659],{},[657,7194,892],{"mathvariant":891},[713,7196,715],{},[709,7198,942],{"stretchy":936},[709,7200,1019],{},[657,7202,1198],{"mathvariant":1197},[709,7204,937],{"stretchy":936},[713,7206,1415],{},[709,7208,942],{"stretchy":936},[661,7210,7211],{"encoding":663},"T(n\u002F2) + \\mathcal{O}(1)",[638,7213,7215,7245],{"className":7214,"ariaHidden":668},[667],[638,7216,7218,7221,7224,7227,7230,7233,7236,7239,7242],{"className":7217},[672],[638,7219],{"className":7220,"style":908},[676],[638,7222,990],{"className":7223,"style":1046},[681,682],[638,7225,937],{"className":7226},[962],[638,7228,659],{"className":7229},[681,682],[638,7231,1262],{"className":7232},[681],[638,7234,942],{"className":7235},[969],[638,7237],{"className":7238,"style":1080},[796],[638,7240,1019],{"className":7241},[1084],[638,7243],{"className":7244,"style":1080},[796],[638,7246,7248,7251,7254,7257,7260],{"className":7247},[672],[638,7249],{"className":7250,"style":908},[676],[638,7252,1198],{"className":7253,"style":1285},[681,1284],[638,7255,937],{"className":7256},[962],[638,7258,1415],{"className":7259},[681],[638,7261,942],{"className":7262},[969],[2787,7264,7265],{},[638,7266,7268,7291],{"className":7267},[641],[638,7269,7271],{"className":7270},[645],[647,7272,7273],{"xmlns":649},[651,7274,7275,7289],{},[654,7276,7277,7279,7281,7283,7285,7287],{},[657,7278,1198],{"mathvariant":1197},[709,7280,937],{"stretchy":936},[657,7282,707],{},[709,7284,711],{},[657,7286,659],{},[709,7288,942],{"stretchy":936},[661,7290,5370],{"encoding":663},[638,7292,7294],{"className":7293,"ariaHidden":668},[667],[638,7295,7297,7300,7303,7306,7311,7314,7317],{"className":7296},[672],[638,7298],{"className":7299,"style":908},[676],[638,7301,1198],{"className":7302,"style":1285},[681,1284],[638,7304,937],{"className":7305},[962],[638,7307,738,7309],{"className":7308},[734],[638,7310,742],{"style":741},[638,7312],{"className":7313,"style":797},[796],[638,7315,659],{"className":7316},[681,682],[638,7318,942],{"className":7319},[969],[2726,7321,7322,7324,7419],{},[2787,7323,5648],{},[2787,7325,7326],{},[638,7327,7329,7365],{"className":7328},[641],[638,7330,7332],{"className":7331},[645],[647,7333,7334],{"xmlns":649},[651,7335,7336,7362],{},[654,7337,7338,7340,7342,7344,7346,7348,7350,7352,7354,7356,7358,7360],{},[713,7339,3550],{},[657,7341,990],{},[709,7343,937],{"stretchy":936},[657,7345,659],{},[657,7347,892],{"mathvariant":891},[713,7349,715],{},[709,7351,942],{"stretchy":936},[709,7353,1019],{},[657,7355,1198],{"mathvariant":1197},[709,7357,937],{"stretchy":936},[657,7359,659],{},[709,7361,942],{"stretchy":936},[661,7363,7364],{"encoding":663},"3 T(n\u002F2) + \\mathcal{O}(n)",[638,7366,7368,7401],{"className":7367,"ariaHidden":668},[667],[638,7369,7371,7374,7377,7380,7383,7386,7389,7392,7395,7398],{"className":7370},[672],[638,7372],{"className":7373,"style":908},[676],[638,7375,3550],{"className":7376},[681],[638,7378,990],{"className":7379,"style":1046},[681,682],[638,7381,937],{"className":7382},[962],[638,7384,659],{"className":7385},[681,682],[638,7387,1262],{"className":7388},[681],[638,7390,942],{"className":7391},[969],[638,7393],{"className":7394,"style":1080},[796],[638,7396,1019],{"className":7397},[1084],[638,7399],{"className":7400,"style":1080},[796],[638,7402,7404,7407,7410,7413,7416],{"className":7403},[672],[638,7405],{"className":7406,"style":908},[676],[638,7408,1198],{"className":7409,"style":1285},[681,1284],[638,7411,937],{"className":7412},[962],[638,7414,659],{"className":7415},[681,682],[638,7417,942],{"className":7418},[969],[2787,7420,7421],{},[638,7422,7424,7460],{"className":7423},[641],[638,7425,7427],{"className":7426},[645],[647,7428,7429],{"xmlns":649},[651,7430,7431,7457],{},[654,7432,7433,7435,7437,7455],{},[657,7434,1198],{"mathvariant":1197},[709,7436,937],{"stretchy":936},[2593,7438,7439,7441],{},[657,7440,659],{},[654,7442,7443,7453],{},[701,7444,7445,7451],{},[654,7446,7447,7449],{},[657,7448,707],{},[709,7450,711],{},[713,7452,715],{},[713,7454,3550],{},[709,7456,942],{"stretchy":936},[661,7458,7459],{"encoding":663},"\\mathcal{O}(n^{\\log_2 3})",[638,7461,7463],{"className":7462,"ariaHidden":668},[667],[638,7464,7466,7469,7472,7475,7559],{"className":7465},[672],[638,7467],{"className":7468,"style":2882},[676],[638,7470,1198],{"className":7471,"style":1285},[681,1284],[638,7473,937],{"className":7474},[962],[638,7476,7478,7481],{"className":7477},[681],[638,7479,659],{"className":7480},[681,682],[638,7482,7484],{"className":7483},[746],[638,7485,7487],{"className":7486},[750],[638,7488,7490],{"className":7489},[755],[638,7491,7493],{"className":7492,"style":2623},[759],[638,7494,7495,7498],{"style":2644},[638,7496],{"className":7497,"style":768},[767],[638,7499,7501],{"className":7500},[772,773,774,775],[638,7502,7504,7553,7556],{"className":7503},[681,775],[638,7505,7507,7519],{"className":7506},[734,775],[638,7508,7510,7513,7516],{"className":7509},[734,775],[638,7511,2663],{"className":7512},[775],[638,7514,2667],{"className":7515},[775],[638,7517,742],{"className":7518,"style":741},[775],[638,7520,7522],{"className":7521},[746],[638,7523,7525,7545],{"className":7524},[750,751],[638,7526,7528,7542],{"className":7527},[755],[638,7529,7531],{"className":7530,"style":4610},[759],[638,7532,7533,7536],{"style":2686},[638,7534],{"className":7535,"style":2690},[767],[638,7537,7539],{"className":7538},[772,2694,2695,775],[638,7540,715],{"className":7541},[681,775],[638,7543,783],{"className":7544},[782],[638,7546,7548],{"className":7547},[755],[638,7549,7551],{"className":7550,"style":2708},[759],[638,7552],{},[638,7554],{"className":7555,"style":2714},[796,775],[638,7557,3550],{"className":7558},[681,775],[638,7560,942],{"className":7561},[969],[593,7563,7565],{"id":7564},"à-savoir-reconnaître","À savoir reconnaître",[2720,7567,7568,7578],{},[2723,7569,7570],{},[2726,7571,7572,7575],{},[2729,7573,7574],{},"Indice",[2729,7576,7577],{},"Méthode",[2782,7579,7580,7587,7692],{},[2726,7581,7582,7585],{},[2787,7583,7584],{},"Le problème se décompose en deux moitiés",[2787,7586,5278],{},[2726,7588,7589,7627],{},[2787,7590,7591,7592],{},"La combinaison est linéaire et il y a 2 appels sur ",[638,7593,7595,7612],{"className":7594},[641],[638,7596,7598],{"className":7597},[645],[647,7599,7600],{"xmlns":649},[651,7601,7602,7610],{},[654,7603,7604,7606,7608],{},[657,7605,659],{},[657,7607,892],{"mathvariant":891},[713,7609,715],{},[661,7611,1345],{"encoding":663},[638,7613,7615],{"className":7614,"ariaHidden":668},[667],[638,7616,7618,7621,7624],{"className":7617},[672],[638,7619],{"className":7620,"style":908},[676],[638,7622,659],{"className":7623},[681,682],[638,7625,1262],{"className":7626},[681],[2787,7628,7629],{},[638,7630,7632,7657],{"className":7631},[641],[638,7633,7635],{"className":7634},[645],[647,7636,7637],{"xmlns":649},[651,7638,7639,7655],{},[654,7640,7641,7643,7645,7647,7649,7651,7653],{},[657,7642,1198],{"mathvariant":1197},[709,7644,937],{"stretchy":936},[657,7646,659],{},[657,7648,707],{},[709,7650,711],{},[657,7652,659],{},[709,7654,942],{"stretchy":936},[661,7656,1880],{"encoding":663},[638,7658,7660],{"className":7659,"ariaHidden":668},[667],[638,7661,7663,7666,7669,7672,7675,7678,7683,7686,7689],{"className":7662},[672],[638,7664],{"className":7665,"style":908},[676],[638,7667,1198],{"className":7668,"style":1285},[681,1284],[638,7670,937],{"className":7671},[962],[638,7673,659],{"className":7674},[681,682],[638,7676],{"className":7677,"style":797},[796],[638,7679,738,7681],{"className":7680},[734],[638,7682,742],{"style":741},[638,7684],{"className":7685,"style":797},[796],[638,7687,659],{"className":7688},[681,682],[638,7690,942],{"className":7691},[969],[2726,7693,7694,7733],{},[2787,7695,7696,7697,7732],{},"Un seul appel récursif sur ",[638,7698,7700,7717],{"className":7699},[641],[638,7701,7703],{"className":7702},[645],[647,7704,7705],{"xmlns":649},[651,7706,7707,7715],{},[654,7708,7709,7711,7713],{},[657,7710,659],{},[657,7712,892],{"mathvariant":891},[713,7714,715],{},[661,7716,1345],{"encoding":663},[638,7718,7720],{"className":7719,"ariaHidden":668},[667],[638,7721,7723,7726,7729],{"className":7722},[672],[638,7724],{"className":7725,"style":908},[676],[638,7727,659],{"className":7728},[681,682],[638,7730,1262],{"className":7731},[681],", combinaison constante",[2787,7734,7735],{},[638,7736,7738,7761],{"className":7737},[641],[638,7739,7741],{"className":7740},[645],[647,7742,7743],{"xmlns":649},[651,7744,7745,7759],{},[654,7746,7747,7749,7751,7753,7755,7757],{},[657,7748,1198],{"mathvariant":1197},[709,7750,937],{"stretchy":936},[657,7752,707],{},[709,7754,711],{},[657,7756,659],{},[709,7758,942],{"stretchy":936},[661,7760,5370],{"encoding":663},[638,7762,7764],{"className":7763,"ariaHidden":668},[667],[638,7765,7767,7770,7773,7776,7781,7784,7787],{"className":7766},[672],[638,7768],{"className":7769,"style":908},[676],[638,7771,1198],{"className":7772,"style":1285},[681,1284],[638,7774,937],{"className":7775},[962],[638,7777,738,7779],{"className":7778},[734],[638,7780,742],{"style":741},[638,7782],{"className":7783,"style":797},[796],[638,7785,659],{"className":7786},[681,682],[638,7788,942],{"className":7789},[969],[593,7791,7793],{"id":7792},"pièges-courants","Pièges courants",[1432,7795,7796,7799,7831],{},[816,7797,7798],{},"Oublier le cas de base → récursion infinie.",[816,7800,7801,7802,7830],{},"Croire que diviser pour régner est toujours plus efficace que l'itératif —\nc'est vrai sur le coût asymptotique, faux sur les petits ",[638,7803,7805,7818],{"className":7804},[641],[638,7806,7808],{"className":7807},[645],[647,7809,7810],{"xmlns":649},[651,7811,7812,7816],{},[654,7813,7814],{},[657,7815,659],{},[661,7817,659],{"encoding":663},[638,7819,7821],{"className":7820,"ariaHidden":668},[667],[638,7822,7824,7827],{"className":7823},[672],[638,7825],{"className":7826,"style":677},[676],[638,7828,659],{"className":7829},[681,682]," (constantes\ncachées et coût des appels).",[816,7832,7833,7834,7837,7838,1030],{},"Confondre coût en ",[601,7835,7836],{},"temps"," et coût en ",[601,7839,7840],{},"mémoire",[593,7842,7844],{"id":7843},"syntaxe-python-à-retenir","Syntaxe Python (à retenir)",[3861,7846,7848],{"className":3863,"code":7847,"language":3865,"meta":3866,"style":3866},"def diviser_regner(donnees):\n    if cas_de_base(donnees):\n        return solution_directe(donnees)\n    sous_problemes = diviser(donnees)\n    sous_solutions = [diviser_regner(s) for s in sous_problemes]\n    return combiner(sous_solutions)\n",[3868,7849,7850,7860,7867,7874,7884,7906],{"__ignoreMap":3866},[638,7851,7852,7854,7857],{"class":3872,"line":3873},[638,7853,3877],{"class":3876},[638,7855,7856],{"class":3880}," diviser_regner",[638,7858,7859],{"class":3884},"(donnees):\n",[638,7861,7862,7864],{"class":3872,"line":3888},[638,7863,4056],{"class":3876},[638,7865,7866],{"class":3884}," cas_de_base(donnees):\n",[638,7868,7869,7871],{"class":3872,"line":3895},[638,7870,4077],{"class":3876},[638,7872,7873],{"class":3884}," solution_directe(donnees)\n",[638,7875,7876,7879,7881],{"class":3872,"line":3906},[638,7877,7878],{"class":3884},"    sous_problemes ",[638,7880,999],{"class":3876},[638,7882,7883],{"class":3884}," diviser(donnees)\n",[638,7885,7886,7889,7891,7894,7897,7900,7903],{"class":3872,"line":3924},[638,7887,7888],{"class":3884},"    sous_solutions ",[638,7890,999],{"class":3876},[638,7892,7893],{"class":3884}," [diviser_regner(s) ",[638,7895,7896],{"class":3876},"for",[638,7898,7899],{"class":3884}," s ",[638,7901,7902],{"class":3876},"in",[638,7904,7905],{"class":3884}," sous_problemes]\n",[638,7907,7908,7910],{"class":3872,"line":3955},[638,7909,4016],{"class":3876},[638,7911,7912],{"class":3884}," combiner(sous_solutions)\n",[5871,7914,7915],{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":3866,"searchDepth":3888,"depth":3888,"links":7917},[7918,7919,7920,7921,7922,7923,7924],{"id":5906,"depth":3888,"text":5907},{"id":5986,"depth":3888,"text":5987},{"id":6886,"depth":3888,"text":6887},{"id":6986,"depth":3888,"text":6987},{"id":7564,"depth":3888,"text":7565},{"id":7792,"depth":3888,"text":7793},{"id":7843,"depth":3888,"text":7844},"Mémo de révision sur la méthode diviser pour régner et le tri fusion.",{},"\u002Fnsi-tale\u002Fte-algorithmique\u002Fdiviser-pour-regner.memo",[],{"title":5900,"description":7925},"2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner.memo","2SXc-khvPcixkTmEtTp4pmRvhPop5envytrLyr44uwA",{"id":7933,"title":7934,"body":7935,"description":10171,"extension":5887,"meta":10172,"navigation":5893,"path":10173,"seo":10174,"stem":10175,"__hash__":10176},"coursSlides\u002F2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner.slides.md","Diviser pour régner — diapositives",{"type":590,"value":7936,"toc":10169},[7937,7941,7944,7947,7951,7968,7970,7974,8126,8236,8238,8242,8391,8457,8459,8463,8539,8541,8545,8552,8554,8558,8564,8787,8792,8794,8798,8978,9181,9187,9189,9193,9576,9680,9685,9687,9690,9805,9807,9811,10149,10151,10155,10166],[7938,7939,574],"h1",{"id":7940},"diviser-pour-régner",[598,7942,7943],{},"TE03 — tri fusion, théorème maître.",[7945,7946],"hr",{},[7938,7948,7950],{"id":7949},"le-schéma","Le schéma",[813,7952,7953,7958,7963],{},[816,7954,7955,7957],{},[601,7956,820],{}," : casser en sous-problèmes plus petits.",[816,7959,7960,7962],{},[601,7961,826],{}," : résoudre par récursion.",[816,7964,7965,7967],{},[601,7966,836],{}," : fusionner les sous-solutions.",[7945,7969],{},[7938,7971,7973],{"id":7972},"la-récurrence-générale","La récurrence générale",[638,7975,7977],{"className":7976},[974],[638,7978,7980,8027],{"className":7979},[641],[638,7981,7983],{"className":7982},[645],[647,7984,7985],{"xmlns":649,"display":983},[651,7986,7987,8025],{},[654,7988,7989,7991,7993,7995,7997,7999,8001,8003,8005,8007,8009,8011,8013,8015,8017,8019,8021,8023],{},[657,7990,990],{},[709,7992,937],{"stretchy":936},[657,7994,659],{},[709,7996,942],{"stretchy":936},[709,7998,999],{},[657,8000,859],{},[709,8002,1004],{},[657,8004,990],{},[709,8006,937],{"stretchy":936},[657,8008,659],{},[657,8010,892],{"mathvariant":891},[657,8012,895],{},[709,8014,942],{"stretchy":936},[709,8016,1019],{},[657,8018,933],{},[709,8020,937],{"stretchy":936},[657,8022,659],{},[709,8024,942],{"stretchy":936},[661,8026,2435],{"encoding":663},[638,8028,8030,8057,8075,8108],{"className":8029,"ariaHidden":668},[667],[638,8031,8033,8036,8039,8042,8045,8048,8051,8054],{"className":8032},[672],[638,8034],{"className":8035,"style":908},[676],[638,8037,990],{"className":8038,"style":1046},[681,682],[638,8040,937],{"className":8041},[962],[638,8043,659],{"className":8044},[681,682],[638,8046,942],{"className":8047},[969],[638,8049],{"className":8050,"style":1059},[796],[638,8052,999],{"className":8053},[1063],[638,8055],{"className":8056,"style":1059},[796],[638,8058,8060,8063,8066,8069,8072],{"className":8059},[672],[638,8061],{"className":8062,"style":1073},[676],[638,8064,859],{"className":8065},[681,682],[638,8067],{"className":8068,"style":1080},[796],[638,8070,1004],{"className":8071},[1084],[638,8073],{"className":8074,"style":1080},[796],[638,8076,8078,8081,8084,8087,8090,8093,8096,8099,8102,8105],{"className":8077},[672],[638,8079],{"className":8080,"style":908},[676],[638,8082,990],{"className":8083,"style":1046},[681,682],[638,8085,937],{"className":8086},[962],[638,8088,659],{"className":8089},[681,682],[638,8091,892],{"className":8092},[681],[638,8094,895],{"className":8095},[681,682],[638,8097,942],{"className":8098},[969],[638,8100],{"className":8101,"style":1080},[796],[638,8103,1019],{"className":8104},[1084],[638,8106],{"className":8107,"style":1080},[796],[638,8109,8111,8114,8117,8120,8123],{"className":8110},[672],[638,8112],{"className":8113,"style":908},[676],[638,8115,933],{"className":8116,"style":958},[681,682],[638,8118,937],{"className":8119},[962],[638,8121,659],{"className":8122},[681,682],[638,8124,942],{"className":8125},[969],[1432,8127,8128,8159,8190],{},[816,8129,8130,8158],{},[638,8131,8133,8146],{"className":8132},[641],[638,8134,8136],{"className":8135},[645],[647,8137,8138],{"xmlns":649},[651,8139,8140,8144],{},[654,8141,8142],{},[657,8143,859],{},[661,8145,859],{"encoding":663},[638,8147,8149],{"className":8148,"ariaHidden":668},[667],[638,8150,8152,8155],{"className":8151},[672],[638,8153],{"className":8154,"style":677},[676],[638,8156,859],{"className":8157},[681,682]," : nombre de sous-problèmes.",[816,8160,8161,8189],{},[638,8162,8164,8177],{"className":8163},[641],[638,8165,8167],{"className":8166},[645],[647,8168,8169],{"xmlns":649},[651,8170,8171,8175],{},[654,8172,8173],{},[657,8174,895],{},[661,8176,895],{"encoding":663},[638,8178,8180],{"className":8179,"ariaHidden":668},[667],[638,8181,8183,8186],{"className":8182},[672],[638,8184],{"className":8185,"style":4276},[676],[638,8187,895],{"className":8188},[681,682]," : facteur de division de la taille.",[816,8191,8192,8235],{},[638,8193,8195,8214],{"className":8194},[641],[638,8196,8198],{"className":8197},[645],[647,8199,8200],{"xmlns":649},[651,8201,8202,8212],{},[654,8203,8204,8206,8208,8210],{},[657,8205,933],{},[709,8207,937],{"stretchy":936},[657,8209,659],{},[709,8211,942],{"stretchy":936},[661,8213,945],{"encoding":663},[638,8215,8217],{"className":8216,"ariaHidden":668},[667],[638,8218,8220,8223,8226,8229,8232],{"className":8219},[672],[638,8221],{"className":8222,"style":908},[676],[638,8224,933],{"className":8225,"style":958},[681,682],[638,8227,937],{"className":8228},[962],[638,8230,659],{"className":8231},[681,682],[638,8233,942],{"className":8234},[969]," : coût de diviser + combiner.",[7945,8237],{},[7938,8239,8241],{"id":8240},"notations-asymptotiques","Notations asymptotiques",[1432,8243,8244,8293,8341],{},[816,8245,8246,1710,8290,8292],{},[638,8247,8249,8269],{"className":8248},[641],[638,8250,8252],{"className":8251},[645],[647,8253,8254],{"xmlns":649},[651,8255,8256,8266],{},[654,8257,8258,8260,8262,8264],{},[657,8259,1198],{"mathvariant":1197},[709,8261,937],{"stretchy":936},[657,8263,933],{},[709,8265,942],{"stretchy":936},[661,8267,8268],{"encoding":663},"\\mathcal{O}(f)",[638,8270,8272],{"className":8271,"ariaHidden":668},[667],[638,8273,8275,8278,8281,8284,8287],{"className":8274},[672],[638,8276],{"className":8277,"style":908},[676],[638,8279,1198],{"className":8280,"style":1285},[681,1284],[638,8282,937],{"className":8283},[962],[638,8285,933],{"className":8286,"style":958},[681,682],[638,8288,942],{"className":8289},[969],[601,8291,1993],{}," (au plus).",[816,8294,8295,1710,8339,2057],{},[638,8296,8298,8318],{"className":8297},[641],[638,8299,8301],{"className":8300},[645],[647,8302,8303],{"xmlns":649},[651,8304,8305,8315],{},[654,8306,8307,8309,8311,8313],{},[657,8308,2011],{"mathvariant":891},[709,8310,937],{"stretchy":936},[657,8312,933],{},[709,8314,942],{"stretchy":936},[661,8316,8317],{"encoding":663},"\\Omega(f)",[638,8319,8321],{"className":8320,"ariaHidden":668},[667],[638,8322,8324,8327,8330,8333,8336],{"className":8323},[672],[638,8325],{"className":8326,"style":908},[676],[638,8328,2011],{"className":8329},[681],[638,8331,937],{"className":8332},[962],[638,8334,933],{"className":8335,"style":958},[681,682],[638,8337,942],{"className":8338},[969],[601,8340,2056],{},[816,8342,8343,1710,8387,8390],{},[638,8344,8346,8366],{"className":8345},[641],[638,8347,8349],{"className":8348},[645],[647,8350,8351],{"xmlns":649},[651,8352,8353,8363],{},[654,8354,8355,8357,8359,8361],{},[657,8356,2074],{"mathvariant":891},[709,8358,937],{"stretchy":936},[657,8360,933],{},[709,8362,942],{"stretchy":936},[661,8364,8365],{"encoding":663},"\\Theta(f)",[638,8367,8369],{"className":8368,"ariaHidden":668},[667],[638,8370,8372,8375,8378,8381,8384],{"className":8371},[672],[638,8373],{"className":8374,"style":908},[676],[638,8376,2074],{"className":8377},[681],[638,8379,937],{"className":8380},[962],[638,8382,933],{"className":8383,"style":958},[681,682],[638,8385,942],{"className":8386},[969],[601,8388,8389],{},"encadrement"," strict.",[598,8392,8393,8456],{},[638,8394,8396,8421],{"className":8395},[641],[638,8397,8399],{"className":8398},[645],[647,8400,8401],{"xmlns":649},[651,8402,8403,8419],{},[654,8404,8405,8407,8409,8411,8413,8415,8417],{},[657,8406,2074],{"mathvariant":891},[709,8408,937],{"stretchy":936},[657,8410,659],{},[657,8412,707],{},[709,8414,711],{},[657,8416,659],{},[709,8418,942],{"stretchy":936},[661,8420,2337],{"encoding":663},[638,8422,8424],{"className":8423,"ariaHidden":668},[667],[638,8425,8427,8430,8433,8436,8439,8442,8447,8450,8453],{"className":8426},[672],[638,8428],{"className":8429,"style":908},[676],[638,8431,2074],{"className":8432},[681],[638,8434,937],{"className":8435},[962],[638,8437,659],{"className":8438},[681,682],[638,8440],{"className":8441,"style":797},[796],[638,8443,738,8445],{"className":8444},[734],[638,8446,742],{"style":741},[638,8448],{"className":8449,"style":797},[796],[638,8451,659],{"className":8452},[681,682],[638,8454,942],{"className":8455},[969]," = « ni plus, ni moins, en ordre de grandeur ».",[7945,8458],{},[7938,8460,8462],{"id":8461},"tri-fusion-lalgo","Tri fusion : l'algo",[3861,8464,8466],{"className":3863,"code":8465,"language":3865,"meta":3866,"style":3866},"def tri_fusion(t):\n    if len(t) \u003C= 1: return t\n    m = len(t) \u002F\u002F 2\n    g = tri_fusion(t[:m])\n    d = tri_fusion(t[m:])\n    return fusionner(g, d)\n",[3868,8467,8468,8477,8499,8514,8523,8532],{"__ignoreMap":3866},[638,8469,8470,8472,8474],{"class":3872,"line":3873},[638,8471,3877],{"class":3876},[638,8473,4034],{"class":3880},[638,8475,8476],{"class":3884},"(t):\n",[638,8478,8479,8481,8483,8486,8488,8490,8493,8496],{"class":3872,"line":3888},[638,8480,4056],{"class":3876},[638,8482,3936],{"class":3914},[638,8484,8485],{"class":3884},"(t) ",[638,8487,3964],{"class":3876},[638,8489,4064],{"class":3914},[638,8491,8492],{"class":3884},": ",[638,8494,8495],{"class":3876},"return",[638,8497,8498],{"class":3884}," 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intuition compte plus que la table abstraite.",[7945,8793],{},[7938,8795,8797],{"id":8796},"théorème-maître-cas-2","Théorème maître — cas 2",[598,8799,8800,8801,2718],{},"Si 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ssName":9126},[746],[638,9128,9130,9150],{"className":9129},[750,751],[638,9131,9133,9147],{"className":9132},[755],[638,9134,9136],{"className":9135,"style":2683},[759],[638,9137,9138,9141],{"style":2686},[638,9139],{"className":9140,"style":2690},[767],[638,9142,9144],{"className":9143},[772,2694,2695,775],[638,9145,895],{"className":9146},[681,682,775],[638,9148,783],{"className":9149},[782],[638,9151,9153],{"className":9152},[755],[638,9154,9156],{"className":9155,"style":2708},[759],[638,9157],{},[638,9159],{"className":9160,"style":2714},[796,775],[638,9162,859],{"className":9163},[681,682,775],[638,9165],{"className":9166,"style":797},[796],[638,9168,738,9170],{"className":9169},[734],[638,9171,742],{"style":741},[638,9173],{"className":9174,"style":797},[796],[638,9176,659],{"className":9177},[681,682],[638,9179,942],{"className":9180},[969],[598,9182,9183,9186],{},[601,9184,9185],{},"Seul cas à retenir"," — couvre tri fusion et dichotomie. Cas 1 et 3 : culture générale.",[7945,9188],{},[7938,9190,9192],{"id":9191},"analyse-du-tri-fusion","Analyse du tri fusion",[598,9194,9195,3918,9245,3918,9295,9375,9575],{},[638,9196,9198,9215],{"className":9197},[641],[638,9199,9201],{"className":9200},[645],[647,9202,9203],{"xmlns":649},[651,9204,9205,9213],{},[654,9206,9207,9209,9211],{},[657,9208,859],{},[709,9210,999],{},[713,9212,715],{},[661,9214,4179],{"encoding":663},[638,9216,9218,9236],{"className":9217,"ariaHidden":668},[667],[638,9219,9221,9224,9227,9230,9233],{"className":9220},[672],[638,9222],{"className":9223,"style":677},[676],[638,9225,859],{"className":9226},[681,682],[638,9228],{"className":9229,"style":1059},[796],[638,9231,999],{"className":9232},[1063],[638,9234],{"className":9235,"style":1059},[796],[638,9237,9239,9242],{"className":9238},[672],[638,9240],{"className":9241,"style":1427},[676],[638,9243,715],{"className":9244},[681],[638,9246,9248,9265],{"className":9247},[641],[638,9249,9251],{"className":9250},[645],[647,9252,9253],{"xmlns":649},[651,9254,9255,9263],{},[654,9256,9257,9259,9261],{},[657,9258,895],{},[709,9260,999],{},[713,9262,715],{},[661,9264,4266],{"encoding":663},[638,9266,9268,9286],{"className":9267,"ariaHidden":668},[667],[638,9269,9271,9274,9277,9280,9283],{"className":9270},[672],[638,9272],{"className":9273,"style":4276},[676],[638,9275,895],{"className":9276},[681,682],[638,9278],{"className":9279,"style":1059},[796],[638,9281,999],{"className":9282},[1063],[638,9284],{"className":9285,"style":1059},[796],[638,9287,9289,9292],{"className":9288},[672],[638,9290],{"className":9291,"style":1427},[676],[638,9293,715],{"className":9294},[681],[638,9296,9298,9327],{"className":9297},[641],[638,9299,9301],{"className":9300},[645],[647,9302,9303],{"xmlns":649},[651,9304,9305,9325],{},[654,9306,9307,9309,9311,9313,9315,9317,9319,9321,9323],{},[657,9308,933],{},[709,9310,937],{"stretchy":936},[657,9312,659],{},[709,9314,942],{"stretchy":936},[709,9316,999],{},[657,9318,1198],{"mathvariant":1197},[709,9320,937],{"stretchy":936},[657,9322,659],{},[709,9324,942],{"stretchy":936},[661,9326,4331],{"encoding":663},[638,9328,9330,9357],{"className":9329,"ariaHidden":668},[667],[638,9331,9333,9336,9339,9342,9345,9348,9351,9354],{"className":9332},[672],[638,9334],{"className":9335,"style":908},[676],[638,9337,933],{"className":9338,"style":958},[681,682],[638,9340,937],{"className":9341},[962],[638,9343,659],{"className":9344},[681,682],[638,9346,942],{"className":9347},[969],[638,9349],{"className":9350,"style":1059},[796],[638,9352,999],{"className":9353},[1063],[638,9355],{"className":9356,"style":1059},[796],[638,9358,9360,9363,9366,9369,9372],{"className":9359},[672],[638,9361],{"className":9362,"style":908},[676],[638,9364,1198],{"className":9365,"style":1285},[681,1284],[638,9367,937],{"className":9368},[962],[638,9370,659],{"className":9371},[681,682],[638,9373,942],{"className":9374},[969],[638,9376,9378,9420],{"className":9377},[641],[638,9379,9381],{"className":9380},[645],[647,9382,9383],{"xmlns":649},[651,9384,9385,9417],{},[654,9386,9387,9405,9407,9413,9415],{},[2593,9388,9389,9391],{},[657,9390,659],{},[654,9392,9393,9403],{},[701,9394,9395,9401],{},[654,9396,9397,9399],{},[657,9398,707],{},[709,9400,711],{},[657,9402,895],{},[657,9404,859],{},[709,9406,999],{},[2593,9408,9409,9411],{},[657,9410,659],{},[713,9412,1415],{},[709,9414,999],{},[657,9416,659],{},[661,9418,9419],{"encoding":663},"n^{\\log_b a} = n^1 = n",[638,9421,9423,9522,9566],{"className":9422,"ariaHidden":668},[667],[638,9424,9426,9429,9513,9516,9519],{"className":9425},[672],[638,9427],{"className":9428,"style":2623},[676],[638,9430,9432,9435],{"className":9431},[681],[638,9433,659],{"className":9434},[681,682],[638,9436,9438],{"className":9437},[746],[638,9439,9441],{"className":9440},[750],[638,9442,9444],{"className":9443},[755],[638,9445,9447],{"className":9446,"style":2623},[759],[638,9448,9449,9452],{"style":2644},[638,9450],{"className":9451,"style":768},[767],[638,9453,9455],{"className":9454},[772,773,774,775],[638,9456,9458,9507,9510],{"className":9457},[681,775],[638,9459,9461,9473],{"className":9460},[734,775],[638,9462,9464,9467,9470],{"className":9463},[734,775],[638,9465,2663],{"className":9466},[775],[638,9468,2667],{"className":9469},[775],[638,9471,742],{"className":9472,"style":741},[775],[638,9474,9476],{"className":9475},[746],[638,9477,9479,9499],{"className":9478},[750,751],[638,9480,9482,9496],{"className":9481},[755],[638,9483,9485],{"className":9484,"style":2683},[759],[638,9486,9487,9490],{"style":2686},[638,9488],{"className":9489,"style":2690},[767],[638,9491,9493],{"className":9492},[772,2694,2695,775],[638,9494,895],{"className":9495},[681,682,775],[638,9497,783],{"className":9498},[782],[638,9500,9502],{"className":9501},[755],[638,9503,9505],{"className":9504,"style":2708},[759],[638,9506],{},[638,9508],{"className":9509,"style":2714},[796,775],[638,9511,859],{"className":9512},[681,682,775],[638,9514],{"className":9515,"style":1059},[796],[638,9517,999],{"className":9518},[1063],[638,9520],{"className":9521,"style":1059},[796],[638,9523,9525,9528,9557,9560,9563],{"className":9524},[672],[638,9526],{"className":9527,"style":4654},[676],[638,9529,9531,9534],{"className":9530},[681],[638,9532,659],{"className":9533},[681,682],[638,9535,9537],{"className":9536},[746],[638,9538,9540],{"className":9539},[750],[638,9541,9543],{"className":9542},[755],[638,9544,9546],{"className":9545,"style":4654},[759],[638,9547,9548,9551],{"style":2644},[638,9549],{"className":9550,"style":768},[767],[638,9552,9554],{"className":9553},[772,773,774,775],[638,9555,1415],{"className":9556},[681,775],[638,9558],{"className":9559,"style":1059},[796],[638,9561,999],{"className":9562},[1063],[638,9564],{"className":9565,"style":1059},[796],[638,9567,9569,9572],{"className":9568},[672],[638,9570],{"className":9571,"style":677},[676],[638,9573,659],{"className":9574},[681,682]," → cas 2.",[638,9577,9579],{"className":9578},[974],[638,9580,9582,9618],{"className":9581},[641],[638,9583,9585],{"className":9584},[645],[647,9586,9587],{"xmlns":649,"display":983},[651,9588,9589,9615],{},[654,9590,9591,9593,9595,9597,9599,9601,9603,9605,9607,9609,9611,9613],{},[657,9592,990],{},[709,9594,937],{"stretchy":936},[657,9596,659],{},[709,9598,942],{"stretchy":936},[709,9600,999],{},[657,9602,2074],{"mathvariant":891},[709,9604,937],{"stretchy":936},[657,9606,659],{},[657,9608,707],{},[709,9610,711],{},[657,9612,659],{},[709,9614,942],{"stretchy":936},[661,9616,9617],{"encoding":663},"T(n) = \\Theta(n \\log n)",[638,9619,9621,9648],{"className":9620,"ariaHidden":668},[667],[638,9622,9624,9627,9630,9633,9636,9639,9642,9645],{"className":9623},[672],[638,9625],{"className":9626,"style":908},[676],[638,9628,990],{"className":9629,"style":1046},[681,682],[638,9631,937],{"className":9632},[962],[638,9634,659],{"className":9635},[681,682],[638,9637,942],{"className":9638},[969],[638,9640],{"className":9641,"style":1059},[796],[638,9643,999],{"className":9644},[1063],[638,9646],{"className":9647,"style":1059},[796],[638,9649,9651,9654,9657,9660,9663,9666,9671,9674,9677],{"className":9650},[672],[638,9652],{"className":9653,"style":908},[676],[638,9655,2074],{"className":9656},[681],[638,9658,937],{"className":9659},[962],[638,9661,659],{"className":9662},[681,682],[638,9664],{"className":9665,"style":797},[796],[638,9667,738,9669],{"className":9668},[734],[638,9670,742],{"style":741},[638,9672],{"className":9673,"style":797},[796],[638,9675,659],{"className":9676},[681,682],[638,9678,942],{"className":9679},[969],[598,9681,9682],{},[601,9683,9684],{},"Garanti en pire cas.",[7945,9686],{},[7938,9688,9689],{"id":7840},"Mémoire",[1432,9691,9692,9739,9798],{},[816,9693,9694,9695,9738],{},"Tri fusion naïf : ",[638,9696,9698,9717],{"className":9697},[641],[638,9699,9701],{"className":9700},[645],[647,9702,9703],{"xmlns":649},[651,9704,9705,9715],{},[654,9706,9707,9709,9711,9713],{},[657,9708,1198],{"mathvariant":1197},[709,9710,937],{"stretchy":936},[657,9712,659],{},[709,9714,942],{"stretchy":936},[661,9716,1463],{"encoding":663},[638,9718,9720],{"className":9719,"ariaHidden":668},[667],[638,9721,9723,9726,9729,9732,9735],{"className":9722},[672],[638,9724],{"className":9725,"style":908},[676],[638,9727,1198],{"className":9728,"style":1285},[681,1284],[638,9730,937],{"className":9731},[962],[638,9733,659],{"className":9734},[681,682],[638,9736,942],{"className":9737},[969]," (copies).",[816,9740,9741,9742,9797],{},"Rotation bitmap en place : ",[638,9743,9745,9768],{"className":9744},[641],[638,9746,9748],{"className":9747},[645],[647,9749,9750],{"xmlns":649},[651,9751,9752,9766],{},[654,9753,9754,9756,9758,9760,9762,9764],{},[657,9755,1198],{"mathvariant":1197},[709,9757,937],{"stretchy":936},[657,9759,707],{},[709,9761,711],{},[657,9763,659],{},[709,9765,942],{"stretchy":936},[661,9767,5370],{"encoding":663},[638,9769,9771],{"className":9770,"ariaHidden":668},[667],[638,9772,9774,9777,9780,9783,9788,9791,9794],{"className":9773},[672],[638,9775],{"className":9776,"style":908},[676],[638,9778,1198],{"className":9779,"style":1285},[681,1284],[638,9781,937],{"className":9782},[962],[638,9784,738,9786],{"className":9785},[734],[638,9787,742],{"style":741},[638,9789],{"className":9790,"style":797},[796],[638,9792,659],{"className":9793},[681,682],[638,9795,942],{"className":9796},[969]," (juste la pile).",[816,9799,9800,9801,9804],{},"Le BO insiste : on peut ",[601,9802,9803],{},"gagner en mémoire"," au prix de code subtil.",[7945,9806],{},[7938,9808,9810],{"id":9809},"autres-exemples","Autres exemples",[1432,9812,9813,9991,10081],{},[816,9814,9815,1710,9817,1030],{},[601,9816,429],{},[638,9818,9820,9878],{"className":9819},[641],[638,9821,9823],{"className":9822},[645],[647,9824,9825],{"xmlns":649},[651,9826,9827,9875],{},[654,9828,9829,9831,9833,9835,9837,9839,9841,9843,9845,9847,9849,9851,9853,9855,9857,9859,9861,9863,9865,9867,9869,9871,9873],{},[657,9830,990],{},[709,9832,937],{"stretchy":936},[657,9834,659],{},[709,9836,942],{"stretchy":936},[709,9838,999],{},[657,9840,990],{},[709,9842,937],{"stretchy":936},[657,9844,659],{},[657,9846,892],{"mathvariant":891},[713,9848,715],{},[709,9850,942],{"stretchy":936},[709,9852,1019],{},[657,9854,1198],{"mathvariant":1197},[709,9856,937],{"stretchy":936},[713,9858,1415],{},[709,9860,942],{"stretchy":936},[709,9862,999],{},[657,9864,1198],{"mathvariant":1197},[709,9866,937],{"stretchy":936},[657,9868,707],{},[709,9870,711],{},[657,9872,659],{},[709,9874,942],{"stretchy":936},[661,9876,9877],{"encoding":663},"T(n) = T(n\u002F2) + \\mathcal{O}(1) = \\mathcal{O}(\\log 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