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sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F0.index",[71,72,76,80,84,88],{"title":67,"path":68,"stem":69},{"title":73,"path":74,"stem":75},"Panorama des réseaux sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fpanorama-rs","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F1.panorama-rs",{"title":77,"path":78,"stem":79},"Identité numérique et authentification","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fidentite-numerique","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F2.identite-numerique",{"title":81,"path":82,"stem":83},"Le modèle économique des réseaux sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fmodele-economique","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F3.modele-economique",{"title":85,"path":86,"stem":87},"Petit monde et graphes","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fpetit-monde-graphes","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F4.petit-monde-graphes",{"title":89,"path":90,"stem":91},"La 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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":549,"bo":587,"body":589,"description":3675,"eval":3676,"extension":3677,"labo":3676,"meta":3678,"navigation":724,"path":550,"quizzes":3679,"readingTime":687,"seo":3686,"stem":551,"tp":3676,"__hash__":3687},"cours\u002F2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite.md",[588],"TD05",{"type":590,"value":591,"toc":3665},"minimark",[592,597,610,621,636,640,643,667,673,837,862,950,954,973,981,1244,1265,1287,1291,1353,1579,1971,1975,2410,2493,2500,2701,3078,3096,3219,3273,3407,3410,3509,3517,3599,3610,3623,3644,3648,3659,3661],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,605,606,609],"p",{},"Une fonction ",[602,603,604],"strong",{},"récursive"," est une fonction qui s'appelle elle-même. Cette\nnotion d'auto-référence, étrangère au calcul itératif, est l'un des outils les\nplus puissants du programmeur — et l'un des plus subtils à analyser. Elle est\nle cœur de l'item ",[607,608],"bo-ref",{"code":588},".",[598,611,612,613,616,617,620],{},"Beaucoup de problèmes ont une ",[602,614,615],{},"structure récursive naturelle"," : la taille\nd'un arbre, le parcours d'un graphe, le calcul d'une factorielle, la\ndécomposition d'un problème en sous-problèmes plus petits. Écrire la solution\nrécursive est souvent plus court et plus lisible que sa version itérative —\nmais l'",[602,618,619],{},"analyse de coût"," demande un outillage spécifique.",[598,622,623,624,627,628,631,632,635],{},"L'idée est partout dans la culture : les ",[602,625,626],{},"matriochkas"," russes (chaque poupée\ncontient une poupée plus petite, jusqu'à une dernière insécable), l'",[602,629,630],{},"effet\nDroste"," (la vieille boîte de cacao Van Houten où une infirmière tient une\nboîte… qui contient l'infirmière qui tient la boîte), ou encore la\n",[602,633,634],{},"fractale de Sierpinski"," (chaque triangle est composé de trois triangles\nidentiques à plus petite échelle).",[593,637,639],{"id":638},"anatomie-dune-fonction-récursive","Anatomie d'une fonction récursive",[598,641,642],{},"Toute fonction récursive bien formée a deux ingrédients :",[644,645,646,657],"ol",{},[647,648,649,652,653,656],"li",{},[602,650,651],{},"Un ou plusieurs cas de base"," — une situation où la fonction retourne\ndirectement, ",[602,654,655],{},"sans s'appeler",". C'est la condition de terminaison.",[647,658,659,662,663,666],{},[602,660,661],{},"Un cas récursif"," — la fonction se rappelle sur une entrée ",[602,664,665],{},"strictement\nplus petite"," (au sens d'un ordre bien fondé), et combine le résultat.",[598,668,669,670,609],{},"Exemple canonique : la ",[602,671,672],{},"factorielle",[674,675,680],"pre",{"className":676,"code":677,"language":678,"meta":679,"style":679},"language-python shiki shiki-themes github-light github-light github-dark","def factorielle(n: int) -> int:\n    \"\"\"Calcule n! par récurrence.\n\n    >>> factorielle(0)\n    1\n    >>> factorielle(5)\n    120\n    \"\"\"\n    assert n >= 0, \"n doit être positif ou nul\"\n    if n == 0:              # cas de base\n        return 1\n    return n * factorielle(n - 1)   # cas récursif\n","python","",[681,682,683,712,719,726,735,741,749,755,761,782,802,811],"code",{"__ignoreMap":679},[684,685,688,692,696,700,704,707,709],"span",{"class":686,"line":687},"line",1,[684,689,691],{"class":690},"s8jYJ","def",[684,693,695],{"class":694},"snPdu"," factorielle",[684,697,699],{"class":698},"sxrX7","(n: ",[684,701,703],{"class":702},"sBjJW","int",[684,705,706],{"class":698},") -> ",[684,708,703],{"class":702},[684,710,711],{"class":698},":\n",[684,713,715],{"class":686,"line":714},2,[684,716,718],{"class":717},"sIIMD","    \"\"\"Calcule n! par récurrence.\n",[684,720,722],{"class":686,"line":721},3,[684,723,725],{"emptyLinePlaceholder":724},true,"\n",[684,727,729,732],{"class":686,"line":728},4,[684,730,731],{"class":690},"    >>> ",[684,733,734],{"class":717},"factorielle(0)\n",[684,736,738],{"class":686,"line":737},5,[684,739,740],{"class":717},"    1\n",[684,742,744,746],{"class":686,"line":743},6,[684,745,731],{"class":690},[684,747,748],{"class":717},"factorielle(5)\n",[684,750,752],{"class":686,"line":751},7,[684,753,754],{"class":717},"    120\n",[684,756,758],{"class":686,"line":757},8,[684,759,760],{"class":717},"    \"\"\"\n",[684,762,764,767,770,773,776,779],{"class":686,"line":763},9,[684,765,766],{"class":690},"    assert",[684,768,769],{"class":698}," n ",[684,771,772],{"class":690},">=",[684,774,775],{"class":702}," 0",[684,777,778],{"class":698},", ",[684,780,781],{"class":717},"\"n doit être positif ou nul\"\n",[684,783,785,788,790,793,795,798],{"class":686,"line":784},10,[684,786,787],{"class":690},"    if",[684,789,769],{"class":698},[684,791,792],{"class":690},"==",[684,794,775],{"class":702},[684,796,797],{"class":698},":              ",[684,799,801],{"class":800},"sCsY4","# cas de base\n",[684,803,805,808],{"class":686,"line":804},11,[684,806,807],{"class":690},"        return",[684,809,810],{"class":702}," 1\n",[684,812,814,817,819,822,825,828,831,834],{"class":686,"line":813},12,[684,815,816],{"class":690},"    return",[684,818,769],{"class":698},[684,820,821],{"class":690},"*",[684,823,824],{"class":698}," factorielle(n ",[684,826,827],{"class":690},"-",[684,829,830],{"class":702}," 1",[684,832,833],{"class":698},")   ",[684,835,836],{"class":800},"# cas récursif\n",[838,839,840],"note",{},[598,841,842,843,846,847,850,851,854,855,858,859,609],{},"La ",[602,844,845],{},"terminaison"," est garantie ici car chaque appel diminue strictement ",[681,848,849],{},"n",",\net ",[681,852,853],{},"n == 0"," est le cas de base. Sans cas de base, ou si l'appel récursif ne\nréduit pas l'entrée, on obtient une ",[602,856,857],{},"récursion infinie"," — Python finit par\nlever ",[681,860,861],{},"RecursionError",[863,864,865],"python-snippet",{},[674,866,868],{"className":676,"code":867,"language":678,"meta":679,"style":679},"def factorielle(n):\n    if n == 0:\n        return 1\n    return n * factorielle(n - 1)\n\nprint(factorielle(5))       # 120\nprint(factorielle(10))      # 3628800\n",[681,869,870,879,891,897,914,918,935],{"__ignoreMap":679},[684,871,872,874,876],{"class":686,"line":687},[684,873,691],{"class":690},[684,875,695],{"class":694},[684,877,878],{"class":698},"(n):\n",[684,880,881,883,885,887,889],{"class":686,"line":714},[684,882,787],{"class":690},[684,884,769],{"class":698},[684,886,792],{"class":690},[684,888,775],{"class":702},[684,890,711],{"class":698},[684,892,893,895],{"class":686,"line":721},[684,894,807],{"class":690},[684,896,810],{"class":702},[684,898,899,901,903,905,907,909,911],{"class":686,"line":728},[684,900,816],{"class":690},[684,902,769],{"class":698},[684,904,821],{"class":690},[684,906,824],{"class":698},[684,908,827],{"class":690},[684,910,830],{"class":702},[684,912,913],{"class":698},")\n",[684,915,916],{"class":686,"line":737},[684,917,725],{"emptyLinePlaceholder":724},[684,919,920,923,926,929,932],{"class":686,"line":743},[684,921,922],{"class":702},"print",[684,924,925],{"class":698},"(factorielle(",[684,927,928],{"class":702},"5",[684,930,931],{"class":698},"))       ",[684,933,934],{"class":800},"# 120\n",[684,936,937,939,941,944,947],{"class":686,"line":751},[684,938,922],{"class":702},[684,940,925],{"class":698},[684,942,943],{"class":702},"10",[684,945,946],{"class":698},"))      ",[684,948,949],{"class":800},"# 3628800\n",[593,951,953],{"id":952},"la-pile-dappels-ce-qui-se-passe-vraiment","La pile d'appels — ce qui se passe vraiment",[598,955,956,957,960,961,964,965,968,969,972],{},"Chaque appel de fonction empile un ",[602,958,959],{},"cadre d'activation"," sur la ",[602,962,963],{},"pile\nd'appels"," (stack en anglais). On l'imagine comme une ",[602,966,967],{},"pile d'assiettes"," dans\nun placard : on ne pose et on ne retire qu'au sommet, et l'assiette tout en\nbas — le premier appel — n'est libérée qu'en dernier. Pour ",[681,970,971],{},"factorielle(3)",",\nla pile se déploie ainsi :",[674,974,979],{"className":975,"code":977,"language":978},[976],"language-text","factorielle(3)        ← retour : 3 * factorielle(2)\n  factorielle(2)      ← retour : 2 * factorielle(1)\n    factorielle(1)    ← retour : 1 * factorielle(0)\n      factorielle(0)  ← retour : 1\n","text",[681,980,977],{"__ignoreMap":679},[598,982,983,984,987,988,609],{},"Puis la pile se ",[602,985,986],{},"dépile"," dans l'ordre inverse :\n",[684,989,992,1058],{"className":990},[991],"katex",[684,993,996],{"className":994},[995],"katex-mathml",[997,998,1000],"math",{"xmlns":999},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1001,1002,1003,1053],"semantics",{},[1004,1005,1006,1010,1014,1016,1019,1021,1024,1026,1028,1031,1033,1035,1037,1039,1041,1044,1046,1048,1050],"mrow",{},[1007,1008,1009],"mn",{},"1",[1011,1012,1013],"mo",{},"→",[1007,1015,1009],{},[1011,1017,1018],{},"×",[1007,1020,1009],{},[1011,1022,1023],{},"=",[1007,1025,1009],{},[1011,1027,1013],{},[1007,1029,1030],{},"2",[1011,1032,1018],{},[1007,1034,1009],{},[1011,1036,1023],{},[1007,1038,1030],{},[1011,1040,1013],{},[1007,1042,1043],{},"3",[1011,1045,1018],{},[1007,1047,1030],{},[1011,1049,1023],{},[1007,1051,1052],{},"6",[1054,1055,1057],"annotation",{"encoding":1056},"application\u002Fx-tex","1 \\to 1 \\times 1 = 1 \\to 2 \\times 1 = 2 \\to 3 \\times 2 = 6",[684,1059,1063,1088,1109,1127,1145,1163,1181,1199,1217,1235],{"className":1060,"ariaHidden":1062},[1061],"katex-html","true",[684,1064,1067,1072,1076,1081,1085],{"className":1065},[1066],"base",[684,1068],{"className":1069,"style":1071},[1070],"strut","height:0.6444em;",[684,1073,1009],{"className":1074},[1075],"mord",[684,1077],{"className":1078,"style":1080},[1079],"mspace","margin-right:0.2778em;",[684,1082,1013],{"className":1083},[1084],"mrel",[684,1086],{"className":1087,"style":1080},[1079],[684,1089,1091,1095,1098,1102,1106],{"className":1090},[1066],[684,1092],{"className":1093,"style":1094},[1070],"height:0.7278em;vertical-align:-0.0833em;",[684,1096,1009],{"className":1097},[1075],[684,1099],{"className":1100,"style":1101},[1079],"margin-right:0.2222em;",[684,1103,1018],{"className":1104},[1105],"mbin",[684,1107],{"className":1108,"style":1101},[1079],[684,1110,1112,1115,1118,1121,1124],{"className":1111},[1066],[684,1113],{"className":1114,"style":1071},[1070],[684,1116,1009],{"className":1117},[1075],[684,1119],{"className":1120,"style":1080},[1079],[684,1122,1023],{"className":1123},[1084],[684,1125],{"className":1126,"style":1080},[1079],[684,1128,1130,1133,1136,1139,1142],{"className":1129},[1066],[684,1131],{"className":1132,"style":1071},[1070],[684,1134,1009],{"className":1135},[1075],[684,1137],{"className":1138,"style":1080},[1079],[684,1140,1013],{"className":1141},[1084],[684,1143],{"className":1144,"style":1080},[1079],[684,1146,1148,1151,1154,1157,1160],{"className":1147},[1066],[684,1149],{"className":1150,"style":1094},[1070],[684,1152,1030],{"className":1153},[1075],[684,1155],{"className":1156,"style":1101},[1079],[684,1158,1018],{"className":1159},[1105],[684,1161],{"className":1162,"style":1101},[1079],[684,1164,1166,1169,1172,1175,1178],{"className":1165},[1066],[684,1167],{"className":1168,"style":1071},[1070],[684,1170,1009],{"className":1171},[1075],[684,1173],{"className":1174,"style":1080},[1079],[684,1176,1023],{"className":1177},[1084],[684,1179],{"className":1180,"style":1080},[1079],[684,1182,1184,1187,1190,1193,1196],{"className":1183},[1066],[684,1185],{"className":1186,"style":1071},[1070],[684,1188,1030],{"className":1189},[1075],[684,1191],{"className":1192,"style":1080},[1079],[684,1194,1013],{"className":1195},[1084],[684,1197],{"className":1198,"style":1080},[1079],[684,1200,1202,1205,1208,1211,1214],{"className":1201},[1066],[684,1203],{"className":1204,"style":1094},[1070],[684,1206,1043],{"className":1207},[1075],[684,1209],{"className":1210,"style":1101},[1079],[684,1212,1018],{"className":1213},[1105],[684,1215],{"className":1216,"style":1101},[1079],[684,1218,1220,1223,1226,1229,1232],{"className":1219},[1066],[684,1221],{"className":1222,"style":1071},[1070],[684,1224,1030],{"className":1225},[1075],[684,1227],{"className":1228,"style":1080},[1079],[684,1230,1023],{"className":1231},[1084],[684,1233],{"className":1234,"style":1080},[1079],[684,1236,1238,1241],{"className":1237},[1066],[684,1239],{"className":1240,"style":1071},[1070],[684,1242,1052],{"className":1243},[1075],[1245,1246,1247],"warning",{},[598,1248,1249,1250,1253,1254,1257,1258,1260,1261,1264],{},"Python limite la profondeur de pile à ",[602,1251,1252],{},"1000 appels"," par défaut\n(",[681,1255,1256],{},"sys.getrecursionlimit()","). Au-delà, ",[681,1259,861],{},". C'est une limite\n",[602,1262,1263],{},"pratique"," de l'implémentation, pas une limite théorique de la récursion.",[1266,1267,1269,1275],"self-eval-qcm",{"correct":1009,"slug":1268},"recursivite-c1",[1270,1271,1272],"template",{"v-slot:question":679},[598,1273,1274],{},"Combien d'appels récursifs imbriqués pour factorielle(4) ?",[1270,1276,1277],{"v-slot:choices":679},[1278,1279,1280,1282,1285],"ul",{},[647,1281,1043],{},[647,1283,1284],{},"4",[647,1286,928],{},[593,1288,1290],{"id":1289},"complexité-la-factorielle-est-linéaire","Complexité — la factorielle est linéaire",[598,1292,1293,1294,1348,1349,1352],{},"Notons ",[684,1295,1297,1322],{"className":1296},[991],[684,1298,1300],{"className":1299},[995],[997,1301,1302],{"xmlns":999},[1001,1303,1304,1319],{},[1004,1305,1306,1310,1314,1316],{},[1307,1308,1309],"mi",{},"T",[1011,1311,1313],{"stretchy":1312},"false","(",[1307,1315,849],{},[1011,1317,1318],{"stretchy":1312},")",[1054,1320,1321],{"encoding":1056},"T(n)",[684,1323,1325],{"className":1324,"ariaHidden":1062},[1061],[684,1326,1328,1332,1337,1341,1344],{"className":1327},[1066],[684,1329],{"className":1330,"style":1331},[1070],"height:1em;vertical-align:-0.25em;",[684,1333,1309],{"className":1334,"style":1336},[1075,1335],"mathnormal","margin-right:0.1389em;",[684,1338,1313],{"className":1339},[1340],"mopen",[684,1342,849],{"className":1343},[1075,1335],[684,1345,1318],{"className":1346},[1347],"mclose"," le nombre d'opérations pour calculer ",[681,1350,1351],{},"factorielle(n)",". La\ndéfinition récursive donne directement :",[598,1354,1355],{},[684,1356,1358,1433],{"className":1357},[991],[684,1359,1361],{"className":1360},[995],[997,1362,1363],{"xmlns":999},[1001,1364,1365,1430],{},[1004,1366,1367,1369,1371,1373,1375,1377,1379,1381,1383,1386,1388,1390,1393,1396,1398,1400,1402,1405,1409,1411,1413,1415,1418,1420,1422,1424,1426,1428],{},[1307,1368,1309],{},[1011,1370,1313],{"stretchy":1312},[1307,1372,849],{},[1011,1374,1318],{"stretchy":1312},[1011,1376,1023],{},[1307,1378,1309],{},[1011,1380,1313],{"stretchy":1312},[1307,1382,849],{},[1011,1384,1385],{},"−",[1007,1387,1009],{},[1011,1389,1318],{"stretchy":1312},[1011,1391,1392],{},"+",[1307,1394,1395],{},"O",[1011,1397,1313],{"stretchy":1312},[1007,1399,1009],{},[1011,1401,1318],{"stretchy":1312},[1079,1403],{"width":1404},"1em",[1406,1407,1408],"mtext",{},"avec",[1079,1410],{"width":1404},[1307,1412,1309],{},[1011,1414,1313],{"stretchy":1312},[1007,1416,1417],{},"0",[1011,1419,1318],{"stretchy":1312},[1011,1421,1023],{},[1307,1423,1395],{},[1011,1425,1313],{"stretchy":1312},[1007,1427,1009],{},[1011,1429,1318],{"stretchy":1312},[1054,1431,1432],{"encoding":1056},"T(n) = T(n-1) + O(1) \\quad \\text{avec} \\quad T(0) = O(1)",[684,1434,1436,1463,1487,1508,1561],{"className":1435,"ariaHidden":1062},[1061],[684,1437,1439,1442,1445,1448,1451,1454,1457,1460],{"className":1438},[1066],[684,1440],{"className":1441,"style":1331},[1070],[684,1443,1309],{"className":1444,"style":1336},[1075,1335],[684,1446,1313],{"className":1447},[1340],[684,1449,849],{"className":1450},[1075,1335],[684,1452,1318],{"className":1453},[1347],[684,1455],{"className":1456,"style":1080},[1079],[684,1458,1023],{"className":1459},[1084],[684,1461],{"className":1462,"style":1080},[1079],[684,1464,1466,1469,1472,1475,1478,1481,1484],{"className":1465},[1066],[684,1467],{"className":1468,"style":1331},[1070],[684,1470,1309],{"className":1471,"style":1336},[1075,1335],[684,1473,1313],{"className":1474},[1340],[684,1476,849],{"className":1477},[1075,1335],[684,1479],{"className":1480,"style":1101},[1079],[684,1482,1385],{"className":1483},[1105],[684,1485],{"className":1486,"style":1101},[1079],[684,1488,1490,1493,1496,1499,1502,1505],{"className":1489},[1066],[684,1491],{"className":1492,"style":1331},[1070],[684,1494,1009],{"className":1495},[1075],[684,1497,1318],{"className":1498},[1347],[684,1500],{"className":1501,"style":1101},[1079],[684,1503,1392],{"className":1504},[1105],[684,1506],{"className":1507,"style":1101},[1079],[684,1509,1511,1514,1518,1521,1524,1527,1531,1537,1540,1543,1546,1549,1552,1555,1558],{"className":1510},[1066],[684,1512],{"className":1513,"style":1331},[1070],[684,1515,1395],{"className":1516,"style":1517},[1075,1335],"margin-right:0.0278em;",[684,1519,1313],{"className":1520},[1340],[684,1522,1009],{"className":1523},[1075],[684,1525,1318],{"className":1526},[1347],[684,1528],{"className":1529,"style":1530},[1079],"margin-right:1em;",[684,1532,1534],{"className":1533},[1075,978],[684,1535,1408],{"className":1536},[1075],[684,1538],{"className":1539,"style":1530},[1079],[684,1541,1309],{"className":1542,"style":1336},[1075,1335],[684,1544,1313],{"className":1545},[1340],[684,1547,1417],{"className":1548},[1075],[684,1550,1318],{"className":1551},[1347],[684,1553],{"className":1554,"style":1080},[1079],[684,1556,1023],{"className":1557},[1084],[684,1559],{"className":1560,"style":1080},[1079],[684,1562,1564,1567,1570,1573,1576],{"className":1563},[1066],[684,1565],{"className":1566,"style":1331},[1070],[684,1568,1395],{"className":1569,"style":1517},[1075,1335],[684,1571,1313],{"className":1572},[1340],[684,1574,1009],{"className":1575},[1075],[684,1577,1318],{"className":1578},[1347],[598,1580,1581,1582,1884,1885,1966,1967,1970],{},"En déroulant : ",[684,1583,1585,1665],{"className":1584},[991],[684,1586,1588],{"className":1587},[995],[997,1589,1590],{"xmlns":999},[1001,1591,1592,1662],{},[1004,1593,1594,1596,1598,1600,1602,1604,1606,1608,1610,1612,1614,1616,1618,1621,1623,1625,1627,1629,1631,1633,1635,1637,1639,1641,1643,1646,1648,1650,1652,1654,1656,1658,1660],{},[1307,1595,1309],{},[1011,1597,1313],{"stretchy":1312},[1307,1599,849],{},[1011,1601,1318],{"stretchy":1312},[1011,1603,1023],{},[1307,1605,1309],{},[1011,1607,1313],{"stretchy":1312},[1307,1609,849],{},[1011,1611,1385],{},[1007,1613,1009],{},[1011,1615,1318],{"stretchy":1312},[1011,1617,1392],{},[1307,1619,1620],{},"c",[1011,1622,1023],{},[1307,1624,1309],{},[1011,1626,1313],{"stretchy":1312},[1307,1628,849],{},[1011,1630,1385],{},[1007,1632,1030],{},[1011,1634,1318],{"stretchy":1312},[1011,1636,1392],{},[1007,1638,1030],{},[1307,1640,1620],{},[1011,1642,1023],{},[1011,1644,1645],{},"⋯",[1011,1647,1023],{},[1307,1649,1309],{},[1011,1651,1313],{"stretchy":1312},[1007,1653,1417],{},[1011,1655,1318],{"stretchy":1312},[1011,1657,1392],{},[1307,1659,849],{},[1307,1661,1620],{},[1054,1663,1664],{"encoding":1056},"T(n) = T(n-1) + c = T(n-2) + 2c = \\dots = T(0) + nc",[684,1666,1668,1695,1719,1740,1759,1783,1804,1825,1845,1872],{"className":1667,"ariaHidden":1062},[1061],[684,1669,1671,1674,1677,1680,1683,1686,1689,1692],{"className":1670},[1066],[684,1672],{"className":1673,"style":1331},[1070],[684,1675,1309],{"className":1676,"style":1336},[1075,1335],[684,1678,1313],{"className":1679},[1340],[684,1681,849],{"className":1682},[1075,1335],[684,1684,1318],{"className":1685},[1347],[684,1687],{"className":1688,"style":1080},[1079],[684,1690,1023],{"className":1691},[1084],[684,1693],{"className":1694,"style":1080},[1079],[684,1696,1698,1701,1704,1707,1710,1713,1716],{"className":1697},[1066],[684,1699],{"className":1700,"style":1331},[1070],[684,1702,1309],{"className":1703,"style":1336},[1075,1335],[684,1705,1313],{"className":1706},[1340],[684,1708,849],{"className":1709},[1075,1335],[684,1711],{"className":1712,"style":1101},[1079],[684,1714,1385],{"className":1715},[1105],[684,1717],{"className":1718,"style":1101},[1079],[684,1720,1722,1725,1728,1731,1734,1737],{"className":1721},[1066],[684,1723],{"className":1724,"style":1331},[1070],[684,1726,1009],{"className":1727},[1075],[684,1729,1318],{"className":1730},[1347],[684,1732],{"className":1733,"style":1101},[1079],[684,1735,1392],{"className":1736},[1105],[684,1738],{"className":1739,"style":1101},[1079],[684,1741,1743,1747,1750,1753,1756],{"className":1742},[1066],[684,1744],{"className":1745,"style":1746},[1070],"height:0.4306em;",[684,1748,1620],{"className":1749},[1075,1335],[684,1751],{"className":1752,"style":1080},[1079],[684,1754,1023],{"className":1755},[1084],[684,1757],{"className":1758,"style":1080},[1079],[684,1760,1762,1765,1768,1771,1774,1777,1780],{"className":1761},[1066],[684,1763],{"className":1764,"style":1331},[1070],[684,1766,1309],{"className":1767,"style":1336},[1075,1335],[684,1769,1313],{"className":1770},[1340],[684,1772,849],{"className":1773},[1075,1335],[684,1775],{"className":1776,"style":1101},[1079],[684,1778,1385],{"className":1779},[1105],[684,1781],{"className":1782,"style":1101},[1079],[684,1784,1786,1789,1792,1795,1798,1801],{"className":1785},[1066],[684,1787],{"className":1788,"style":1331},[1070],[684,1790,1030],{"className":1791},[1075],[684,1793,1318],{"className":1794},[1347],[684,1796],{"className":1797,"style":1101},[1079],[684,1799,1392],{"className":1800},[1105],[684,1802],{"className":1803,"style":1101},[1079],[684,1805,1807,1810,1813,1816,1819,1822],{"className":1806},[1066],[684,1808],{"className":1809,"style":1071},[1070],[684,1811,1030],{"className":1812},[1075],[684,1814,1620],{"className":1815},[1075,1335],[684,1817],{"className":1818,"style":1080},[1079],[684,1820,1023],{"className":1821},[1084],[684,1823],{"className":1824,"style":1080},[1079],[684,1826,1828,1832,1836,1839,1842],{"className":1827},[1066],[684,1829],{"className":1830,"style":1831},[1070],"height:0.3669em;",[684,1833,1645],{"className":1834},[1835],"minner",[684,1837],{"className":1838,"style":1080},[1079],[684,1840,1023],{"className":1841},[1084],[684,1843],{"className":1844,"style":1080},[1079],[684,1846,1848,1851,1854,1857,1860,1863,1866,1869],{"className":1847},[1066],[684,1849],{"className":1850,"style":1331},[1070],[684,1852,1309],{"className":1853,"style":1336},[1075,1335],[684,1855,1313],{"className":1856},[1340],[684,1858,1417],{"className":1859},[1075],[684,1861,1318],{"className":1862},[1347],[684,1864],{"className":1865,"style":1101},[1079],[684,1867,1392],{"className":1868},[1105],[684,1870],{"className":1871,"style":1101},[1079],[684,1873,1875,1878,1881],{"className":1874},[1066],[684,1876],{"className":1877,"style":1746},[1070],[684,1879,849],{"className":1880},[1075,1335],[684,1882,1620],{"className":1883},[1075,1335],", soit\n",[684,1886,1888,1918],{"className":1887},[991],[684,1889,1891],{"className":1890},[995],[997,1892,1893],{"xmlns":999},[1001,1894,1895,1915],{},[1004,1896,1897,1899,1901,1903,1905,1907,1909,1911,1913],{},[1307,1898,1309],{},[1011,1900,1313],{"stretchy":1312},[1307,1902,849],{},[1011,1904,1318],{"stretchy":1312},[1011,1906,1023],{},[1307,1908,1395],{},[1011,1910,1313],{"stretchy":1312},[1307,1912,849],{},[1011,1914,1318],{"stretchy":1312},[1054,1916,1917],{"encoding":1056},"T(n) = O(n)",[684,1919,1921,1948],{"className":1920,"ariaHidden":1062},[1061],[684,1922,1924,1927,1930,1933,1936,1939,1942,1945],{"className":1923},[1066],[684,1925],{"className":1926,"style":1331},[1070],[684,1928,1309],{"className":1929,"style":1336},[1075,1335],[684,1931,1313],{"className":1932},[1340],[684,1934,849],{"className":1935},[1075,1335],[684,1937,1318],{"className":1938},[1347],[684,1940],{"className":1941,"style":1080},[1079],[684,1943,1023],{"className":1944},[1084],[684,1946],{"className":1947,"style":1080},[1079],[684,1949,1951,1954,1957,1960,1963],{"className":1950},[1066],[684,1952],{"className":1953,"style":1331},[1070],[684,1955,1395],{"className":1956,"style":1517},[1075,1335],[684,1958,1313],{"className":1959},[1340],[684,1961,849],{"className":1962},[1075,1335],[684,1964,1318],{"className":1965},[1347],". La factorielle récursive est ",[602,1968,1969],{},"linéaire"," — exactement comme\nsa version itérative.",[593,1972,1974],{"id":1973},"fibonacci-naïf-lexplosion-exponentielle","Fibonacci naïf — l'explosion exponentielle",[598,1976,1977,1978,778,2089,778,2181,609],{},"Considérons Fibonacci : ",[684,1979,1981,2005],{"className":1980},[991],[684,1982,1984],{"className":1983},[995],[997,1985,1986],{"xmlns":999},[1001,1987,1988,2002],{},[1004,1989,1990,1998,2000],{},[1991,1992,1993,1996],"msub",{},[1307,1994,1995],{},"F",[1007,1997,1417],{},[1011,1999,1023],{},[1007,2001,1417],{},[1054,2003,2004],{"encoding":1056},"F_0 = 0",[684,2006,2008,2080],{"className":2007,"ariaHidden":1062},[1061],[684,2009,2011,2015,2071,2074,2077],{"className":2010},[1066],[684,2012],{"className":2013,"style":2014},[1070],"height:0.8333em;vertical-align:-0.15em;",[684,2016,2018,2021],{"className":2017},[1075],[684,2019,1995],{"className":2020,"style":1336},[1075,1335],[684,2022,2025],{"className":2023},[2024],"msupsub",[684,2026,2030,2062],{"className":2027},[2028,2029],"vlist-t","vlist-t2",[684,2031,2034,2057],{"className":2032},[2033],"vlist-r",[684,2035,2039],{"className":2036,"style":2038},[2037],"vlist","height:0.3011em;",[684,2040,2042,2047],{"style":2041},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[684,2043],{"className":2044,"style":2046},[2045],"pstrut","height:2.7em;",[684,2048,2054],{"className":2049},[2050,2051,2052,2053],"sizing","reset-size6","size3","mtight",[684,2055,1417],{"className":2056},[1075,2053],[684,2058,2061],{"className":2059},[2060],"vlist-s","​",[684,2063,2065],{"className":2064},[2033],[684,2066,2069],{"className":2067,"style":2068},[2037],"height:0.15em;",[684,2070],{},[684,2072],{"className":2073,"style":1080},[1079],[684,2075,1023],{"className":2076},[1084],[684,2078],{"className":2079,"style":1080},[1079],[684,2081,2083,2086],{"className":2082},[1066],[684,2084],{"className":2085,"style":1071},[1070],[684,2087,1417],{"className":2088},[1075],[684,2090,2092,2114],{"className":2091},[991],[684,2093,2095],{"className":2094},[995],[997,2096,2097],{"xmlns":999},[1001,2098,2099,2111],{},[1004,2100,2101,2107,2109],{},[1991,2102,2103,2105],{},[1307,2104,1995],{},[1007,2106,1009],{},[1011,2108,1023],{},[1007,2110,1009],{},[1054,2112,2113],{"encoding":1056},"F_1 = 1",[684,2115,2117,2172],{"className":2116,"ariaHidden":1062},[1061],[684,2118,2120,2123,2163,2166,2169],{"className":2119},[1066],[684,2121],{"className":2122,"style":2014},[1070],[684,2124,2126,2129],{"className":2125},[1075],[684,2127,1995],{"className":2128,"style":1336},[1075,1335],[684,2130,2132],{"className":2131},[2024],[684,2133,2135,2155],{"className":2134},[2028,2029],[684,2136,2138,2152],{"className":2137},[2033],[684,2139,2141],{"className":2140,"style":2038},[2037],[684,2142,2143,2146],{"style":2041},[684,2144],{"className":2145,"style":2046},[2045],[684,2147,2149],{"className":2148},[2050,2051,2052,2053],[684,2150,1009],{"className":2151},[1075,2053],[684,2153,2061],{"className":2154},[2060],[684,2156,2158],{"className":2157},[2033],[684,2159,2161],{"className":2160,"style":2068},[2037],[684,2162],{},[684,2164],{"className":2165,"style":1080},[1079],[684,2167,1023],{"className":2168},[1084],[684,2170],{"className":2171,"style":1080},[1079],[684,2173,2175,2178],{"className":2174},[1066],[684,2176],{"className":2177,"style":1071},[1070],[684,2179,1009],{"className":2180},[1075],[684,2182,2184,2230],{"className":2183},[991],[684,2185,2187],{"className":2186},[995],[997,2188,2189],{"xmlns":999},[1001,2190,2191,2227],{},[1004,2192,2193,2199,2201,2213,2215],{},[1991,2194,2195,2197],{},[1307,2196,1995],{},[1307,2198,849],{},[1011,2200,1023],{},[1991,2202,2203,2205],{},[1307,2204,1995],{},[1004,2206,2207,2209,2211],{},[1307,2208,849],{},[1011,2210,1385],{},[1007,2212,1009],{},[1011,2214,1392],{},[1991,2216,2217,2219],{},[1307,2218,1995],{},[1004,2220,2221,2223,2225],{},[1307,2222,849],{},[1011,2224,1385],{},[1007,2226,1030],{},[1054,2228,2229],{"encoding":1056},"F_n = F_{n-1} + F_{n-2}",[684,2231,2233,2289,2355],{"className":2232,"ariaHidden":1062},[1061],[684,2234,2236,2239,2280,2283,2286],{"className":2235},[1066],[684,2237],{"className":2238,"style":2014},[1070],[684,2240,2242,2245],{"className":2241},[1075],[684,2243,1995],{"className":2244,"style":1336},[1075,1335],[684,2246,2248],{"className":2247},[2024],[684,2249,2251,2272],{"className":2250},[2028,2029],[684,2252,2254,2269],{"className":2253},[2033],[684,2255,2258],{"className":2256,"style":2257},[2037],"height:0.1514em;",[684,2259,2260,2263],{"style":2041},[684,2261],{"className":2262,"style":2046},[2045],[684,2264,2266],{"className":2265},[2050,2051,2052,2053],[684,2267,849],{"className":2268},[1075,1335,2053],[684,2270,2061],{"className":2271},[2060],[684,2273,2275],{"className":2274},[2033],[684,2276,2278],{"className":2277,"style":2068},[2037],[684,2279],{},[684,2281],{"className":2282,"style":1080},[1079],[684,2284,1023],{"className":2285},[1084],[684,2287],{"className":2288,"style":1080},[1079],[684,2290,2292,2296,2346,2349,2352],{"className":2291},[1066],[684,2293],{"className":2294,"style":2295},[1070],"height:0.8917em;vertical-align:-0.2083em;",[684,2297,2299,2302],{"className":2298},[1075],[684,2300,1995],{"className":2301,"style":1336},[1075,1335],[684,2303,2305],{"className":2304},[2024],[684,2306,2308,2337],{"className":2307},[2028,2029],[684,2309,2311,2334],{"className":2310},[2033],[684,2312,2314],{"className":2313,"style":2038},[2037],[684,2315,2316,2319],{"style":2041},[684,2317],{"className":2318,"style":2046},[2045],[684,2320,2322],{"className":2321},[2050,2051,2052,2053],[684,2323,2325,2328,2331],{"className":2324},[1075,2053],[684,2326,849],{"className":2327},[1075,1335,2053],[684,2329,1385],{"className":2330},[1105,2053],[684,2332,1009],{"className":2333},[1075,2053],[684,2335,2061],{"className":2336},[2060],[684,2338,2340],{"className":2339},[2033],[684,2341,2344],{"className":2342,"style":2343},[2037],"height:0.2083em;",[684,2345],{},[684,2347],{"className":2348,"style":1101},[1079],[684,2350,1392],{"className":2351},[1105],[684,2353],{"className":2354,"style":1101},[1079],[684,2356,2358,2361],{"className":2357},[1066],[684,2359],{"className":2360,"style":2295},[1070],[684,2362,2364,2367],{"className":2363},[1075],[684,2365,1995],{"className":2366,"style":1336},[1075,1335],[684,2368,2370],{"className":2369},[2024],[684,2371,2373,2402],{"className":2372},[2028,2029],[684,2374,2376,2399],{"className":2375},[2033],[684,2377,2379],{"className":2378,"style":2038},[2037],[684,2380,2381,2384],{"style":2041},[684,2382],{"className":2383,"style":2046},[2045],[684,2385,2387],{"className":2386},[2050,2051,2052,2053],[684,2388,2390,2393,2396],{"className":2389},[1075,2053],[684,2391,849],{"className":2392},[1075,1335,2053],[684,2394,1385],{"className":2395},[1105,2053],[684,2397,1030],{"className":2398},[1075,2053],[684,2400,2061],{"className":2401},[2060],[684,2403,2405],{"className":2404},[2033],[684,2406,2408],{"className":2407,"style":2343},[2037],[684,2409],{},[674,2411,2413],{"className":676,"code":2412,"language":678,"meta":679,"style":679},"def fib_naif(n: int) -> int:\n    \"\"\"Fibonacci, version récursive directe.\"\"\"\n    assert n >= 0\n    if n \u003C 2:\n        return n\n    return fib_naif(n - 1) + fib_naif(n - 2)\n",[681,2414,2415,2432,2437,2448,2462,2469],{"__ignoreMap":679},[684,2416,2417,2419,2422,2424,2426,2428,2430],{"class":686,"line":687},[684,2418,691],{"class":690},[684,2420,2421],{"class":694}," fib_naif",[684,2423,699],{"class":698},[684,2425,703],{"class":702},[684,2427,706],{"class":698},[684,2429,703],{"class":702},[684,2431,711],{"class":698},[684,2433,2434],{"class":686,"line":714},[684,2435,2436],{"class":717},"    \"\"\"Fibonacci, version récursive directe.\"\"\"\n",[684,2438,2439,2441,2443,2445],{"class":686,"line":721},[684,2440,766],{"class":690},[684,2442,769],{"class":698},[684,2444,772],{"class":690},[684,2446,2447],{"class":702}," 0\n",[684,2449,2450,2452,2454,2457,2460],{"class":686,"line":728},[684,2451,787],{"class":690},[684,2453,769],{"class":698},[684,2455,2456],{"class":690},"\u003C",[684,2458,2459],{"class":702}," 2",[684,2461,711],{"class":698},[684,2463,2464,2466],{"class":686,"line":737},[684,2465,807],{"class":690},[684,2467,2468],{"class":698}," n\n",[684,2470,2471,2473,2476,2478,2480,2483,2485,2487,2489,2491],{"class":686,"line":743},[684,2472,816],{"class":690},[684,2474,2475],{"class":698}," fib_naif(n ",[684,2477,827],{"class":690},[684,2479,830],{"class":702},[684,2481,2482],{"class":698},") ",[684,2484,1392],{"class":690},[684,2486,2475],{"class":698},[684,2488,827],{"class":690},[684,2490,2459],{"class":702},[684,2492,913],{"class":698},[598,2494,2495,2496,2499],{},"Chaque appel en déclenche ",[602,2497,2498],{},"deux",". La relation de récurrence devient :",[598,2501,2502],{},[684,2503,2505,2563],{"className":2504},[991],[684,2506,2508],{"className":2507},[995],[997,2509,2510],{"xmlns":999},[1001,2511,2512,2560],{},[1004,2513,2514,2516,2518,2520,2522,2524,2526,2528,2530,2532,2534,2536,2538,2540,2542,2544,2546,2548,2550,2552,2554,2556,2558],{},[1307,2515,1309],{},[1011,2517,1313],{"stretchy":1312},[1307,2519,849],{},[1011,2521,1318],{"stretchy":1312},[1011,2523,1023],{},[1307,2525,1309],{},[1011,2527,1313],{"stretchy":1312},[1307,2529,849],{},[1011,2531,1385],{},[1007,2533,1009],{},[1011,2535,1318],{"stretchy":1312},[1011,2537,1392],{},[1307,2539,1309],{},[1011,2541,1313],{"stretchy":1312},[1307,2543,849],{},[1011,2545,1385],{},[1007,2547,1030],{},[1011,2549,1318],{"stretchy":1312},[1011,2551,1392],{},[1307,2553,1395],{},[1011,2555,1313],{"stretchy":1312},[1007,2557,1009],{},[1011,2559,1318],{"stretchy":1312},[1054,2561,2562],{"encoding":1056},"T(n) = T(n-1) + T(n-2) + O(1)",[684,2564,2566,2593,2617,2638,2662,2683],{"className":2565,"ariaHidden":1062},[1061],[684,2567,2569,2572,2575,2578,2581,2584,2587,2590],{"className":2568},[1066],[684,2570],{"className":2571,"style":1331},[1070],[684,2573,1309],{"className":2574,"style":1336},[1075,1335],[684,2576,1313],{"className":2577},[1340],[684,2579,849],{"className":2580},[1075,1335],[684,2582,1318],{"className":2583},[1347],[684,2585],{"className":2586,"style":1080},[1079],[684,2588,1023],{"className":2589},[1084],[684,2591],{"className":2592,"style":1080},[1079],[684,2594,2596,2599,2602,2605,2608,2611,2614],{"className":2595},[1066],[684,2597],{"className":2598,"style":1331},[1070],[684,2600,1309],{"className":2601,"style":1336},[1075,1335],[684,2603,1313],{"className":2604},[1340],[684,2606,849],{"className":2607},[1075,1335],[684,2609],{"className":2610,"style":1101},[1079],[684,2612,1385],{"className":2613},[1105],[684,2615],{"className":2616,"style":1101},[1079],[684,2618,2620,2623,2626,2629,2632,2635],{"className":2619},[1066],[684,2621],{"className":2622,"style":1331},[1070],[684,2624,1009],{"className":2625},[1075],[684,2627,1318],{"className":2628},[1347],[684,2630],{"className":2631,"style":1101},[1079],[684,2633,1392],{"className":2634},[1105],[684,2636],{"className":2637,"style":1101},[1079],[684,2639,2641,2644,2647,2650,2653,2656,2659],{"className":2640},[1066],[684,2642],{"className":2643,"style":1331},[1070],[684,2645,1309],{"className":2646,"style":1336},[1075,1335],[684,2648,1313],{"className":2649},[1340],[684,2651,849],{"className":2652},[1075,1335],[684,2654],{"className":2655,"style":1101},[1079],[684,2657,1385],{"className":2658},[1105],[684,2660],{"className":2661,"style":1101},[1079],[684,2663,2665,2668,2671,2674,2677,2680],{"className":2664},[1066],[684,2666],{"className":2667,"style":1331},[1070],[684,2669,1030],{"className":2670},[1075],[684,2672,1318],{"className":2673},[1347],[684,2675],{"className":2676,"style":1101},[1079],[684,2678,1392],{"className":2679},[1105],[684,2681],{"className":2682,"style":1101},[1079],[684,2684,2686,2689,2692,2695,2698],{"className":2685},[1066],[684,2687],{"className":2688,"style":1331},[1070],[684,2690,1395],{"className":2691,"style":1517},[1075,1335],[684,2693,1313],{"className":2694},[1340],[684,2696,1009],{"className":2697},[1075],[684,2699,1318],{"className":2700},[1347],[598,2702,2703,2704,2819,2820,3065,3066,3069,3070,3073,3074,3077],{},"On montre que ",[684,2705,2707,2743],{"className":2706},[991],[684,2708,2710],{"className":2709},[995],[997,2711,2712],{"xmlns":999},[1001,2713,2714,2740],{},[1004,2715,2716,2718,2720,2722,2724,2726,2728,2730,2738],{},[1307,2717,1309],{},[1011,2719,1313],{"stretchy":1312},[1307,2721,849],{},[1011,2723,1318],{"stretchy":1312},[1011,2725,1023],{},[1307,2727,1395],{},[1011,2729,1313],{"stretchy":1312},[2731,2732,2733,2736],"msup",{},[1307,2734,2735],{},"φ",[1307,2737,849],{},[1011,2739,1318],{"stretchy":1312},[1054,2741,2742],{"encoding":1056},"T(n) = O(\\varphi^n)",[684,2744,2746,2773],{"className":2745,"ariaHidden":1062},[1061],[684,2747,2749,2752,2755,2758,2761,2764,2767,2770],{"className":2748},[1066],[684,2750],{"className":2751,"style":1331},[1070],[684,2753,1309],{"className":2754,"style":1336},[1075,1335],[684,2756,1313],{"className":2757},[1340],[684,2759,849],{"className":2760},[1075,1335],[684,2762,1318],{"className":2763},[1347],[684,2765],{"className":2766,"style":1080},[1079],[684,2768,1023],{"className":2769},[1084],[684,2771],{"className":2772,"style":1080},[1079],[684,2774,2776,2779,2782,2785,2816],{"className":2775},[1066],[684,2777],{"className":2778,"style":1331},[1070],[684,2780,1395],{"className":2781,"style":1517},[1075,1335],[684,2783,1313],{"className":2784},[1340],[684,2786,2788,2791],{"className":2787},[1075],[684,2789,2735],{"className":2790},[1075,1335],[684,2792,2794],{"className":2793},[2024],[684,2795,2797],{"className":2796},[2028],[684,2798,2800],{"className":2799},[2033],[684,2801,2804],{"className":2802,"style":2803},[2037],"height:0.6644em;",[684,2805,2807,2810],{"style":2806},"top:-3.063em;margin-right:0.05em;",[684,2808],{"className":2809,"style":2046},[2045],[684,2811,2813],{"className":2812},[2050,2051,2052,2053],[684,2814,849],{"className":2815},[1075,1335,2053],[684,2817,1318],{"className":2818},[1347]," avec ",[684,2821,2823,2861],{"className":2822},[991],[684,2824,2826],{"className":2825},[995],[997,2827,2828],{"xmlns":999},[1001,2829,2830,2858],{},[1004,2831,2832,2834,2836,2852,2855],{},[1307,2833,2735],{},[1011,2835,1023],{},[2837,2838,2839,2850],"mfrac",{},[1004,2840,2841,2843,2845],{},[1007,2842,1009],{},[1011,2844,1392],{},[2846,2847,2848],"msqrt",{},[1007,2849,928],{},[1007,2851,1030],{},[1011,2853,2854],{},"≈",[1007,2856,2857],{},"1,618",[1054,2859,2860],{"encoding":1056},"\\varphi = \\frac{1 + \\sqrt{5}}{2}\n\\approx 1{,}618",[684,2862,2864,2883,3043],{"className":2863,"ariaHidden":1062},[1061],[684,2865,2867,2871,2874,2877,2880],{"className":2866},[1066],[684,2868],{"className":2869,"style":2870},[1070],"height:0.625em;vertical-align:-0.1944em;",[684,2872,2735],{"className":2873},[1075,1335],[684,2875],{"className":2876,"style":1080},[1079],[684,2878,1023],{"className":2879},[1084],[684,2881],{"className":2882,"style":1080},[1079],[684,2884,2886,2890,3034,3037,3040],{"className":2885},[1066],[684,2887],{"className":2888,"style":2889},[1070],"height:1.383em;vertical-align:-0.345em;",[684,2891,2893,2897,3031],{"className":2892},[1075],[684,2894],{"className":2895},[1340,2896],"nulldelimiter",[684,2898,2900],{"className":2899},[2837],[684,2901,2903,3022],{"className":2902},[2028,2029],[684,2904,2906,3019],{"className":2905},[2033],[684,2907,2910,2926,2937],{"className":2908,"style":2909},[2037],"height:1.038em;",[684,2911,2913,2917],{"style":2912},"top:-2.655em;",[684,2914],{"className":2915,"style":2916},[2045],"height:3em;",[684,2918,2920],{"className":2919},[2050,2051,2052,2053],[684,2921,2923],{"className":2922},[1075,2053],[684,2924,1030],{"className":2925},[1075,2053],[684,2927,2929,2932],{"style":2928},"top:-3.23em;",[684,2930],{"className":2931,"style":2916},[2045],[684,2933],{"className":2934,"style":2936},[2935],"frac-line","border-bottom-width:0.04em;",[684,2938,2940,2943],{"style":2939},"top:-3.399em;",[684,2941],{"className":2942,"style":2916},[2045],[684,2944,2946],{"className":2945},[2050,2051,2052,2053],[684,2947,2949,2952,2955],{"className":2948},[1075,2053],[684,2950,1009],{"className":2951},[1075,2053],[684,2953,1392],{"className":2954},[1105,2053],[684,2956,2959],{"className":2957},[1075,2958,2053],"sqrt",[684,2960,2962,3010],{"className":2961},[2028,2029],[684,2963,2965,3007],{"className":2964},[2033],[684,2966,2969,2984],{"className":2967,"style":2968},[2037],"height:0.9128em;",[684,2970,2974,2977],{"className":2971,"style":2973},[2972],"svg-align","top:-3em;",[684,2975],{"className":2976,"style":2916},[2045],[684,2978,2981],{"className":2979,"style":2980},[1075,2053],"padding-left:0.833em;",[684,2982,928],{"className":2983},[1075,2053],[684,2985,2987,2990],{"style":2986},"top:-2.8728em;",[684,2988],{"className":2989,"style":2916},[2045],[684,2991,2995],{"className":2992,"style":2994},[2993,2053],"hide-tail","min-width:0.853em;height:1.08em;",[2996,2997,3003],"svg",{"xmlns":2998,"width":2999,"height":3000,"viewBox":3001,"preserveAspectRatio":3002},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[3004,3005],"path",{"d":3006},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[684,3008,2061],{"className":3009},[2060],[684,3011,3013],{"className":3012},[2033],[684,3014,3017],{"className":3015,"style":3016},[2037],"height:0.1272em;",[684,3018],{},[684,3020,2061],{"className":3021},[2060],[684,3023,3025],{"className":3024},[2033],[684,3026,3029],{"className":3027,"style":3028},[2037],"height:0.345em;",[684,3030],{},[684,3032],{"className":3033},[1347,2896],[684,3035],{"className":3036,"style":1080},[1079],[684,3038,2854],{"className":3039},[1084],[684,3041],{"className":3042,"style":1080},[1079],[684,3044,3046,3050,3053,3061],{"className":3045},[1066],[684,3047],{"className":3048,"style":3049},[1070],"height:0.8389em;vertical-align:-0.1944em;",[684,3051,1009],{"className":3052},[1075],[684,3054,3056],{"className":3055},[1075],[684,3057,3060],{"className":3058},[3059],"mpunct",",",[684,3062,3064],{"className":3063},[1075],"618"," — autrement dit, ",[602,3067,3068],{},"exponentielle",". En pratique, ",[681,3071,3072],{},"fib_naif(40)","\nprend déjà plusieurs secondes ; ",[681,3075,3076],{},"fib_naif(60)"," est inatteignable.",[598,3079,3080,3081,3083,3084,3087,3088,3091,3092,3095],{},"La raison : ",[681,3082,3072],{}," recalcule ",[681,3085,3086],{},"fib_naif(38)"," deux fois,\n",[681,3089,3090],{},"fib_naif(37)"," trois fois, ",[681,3093,3094],{},"fib_naif(36)"," cinq fois… exponentiellement.",[593,3097,3099,3100,3174,3175],{"id":3098},"mémoïsation-de-o2no2no2n-à-ononon","Mémoïsation — de ",[684,3101,3103,3127],{"className":3102},[991],[684,3104,3106],{"className":3105},[995],[997,3107,3108],{"xmlns":999},[1001,3109,3110,3124],{},[1004,3111,3112,3114,3116,3122],{},[1307,3113,1395],{},[1011,3115,1313],{"stretchy":1312},[2731,3117,3118,3120],{},[1007,3119,1030],{},[1307,3121,849],{},[1011,3123,1318],{"stretchy":1312},[1054,3125,3126],{"encoding":1056},"O(2^n)",[684,3128,3130],{"className":3129,"ariaHidden":1062},[1061],[684,3131,3133,3136,3139,3142,3171],{"className":3132},[1066],[684,3134],{"className":3135,"style":1331},[1070],[684,3137,1395],{"className":3138,"style":1517},[1075,1335],[684,3140,1313],{"className":3141},[1340],[684,3143,3145,3148],{"className":3144},[1075],[684,3146,1030],{"className":3147},[1075],[684,3149,3151],{"className":3150},[2024],[684,3152,3154],{"className":3153},[2028],[684,3155,3157],{"className":3156},[2033],[684,3158,3160],{"className":3159,"style":2803},[2037],[684,3161,3162,3165],{"style":2806},[684,3163],{"className":3164,"style":2046},[2045],[684,3166,3168],{"className":3167},[2050,2051,2052,2053],[684,3169,849],{"className":3170},[1075,1335,2053],[684,3172,1318],{"className":3173},[1347]," à ",[684,3176,3178,3198],{"className":3177},[991],[684,3179,3181],{"className":3180},[995],[997,3182,3183],{"xmlns":999},[1001,3184,3185,3195],{},[1004,3186,3187,3189,3191,3193],{},[1307,3188,1395],{},[1011,3190,1313],{"stretchy":1312},[1307,3192,849],{},[1011,3194,1318],{"stretchy":1312},[1054,3196,3197],{"encoding":1056},"O(n)",[684,3199,3201],{"className":3200,"ariaHidden":1062},[1061],[684,3202,3204,3207,3210,3213,3216],{"className":3203},[1066],[684,3205],{"className":3206,"style":1331},[1070],[684,3208,1395],{"className":3209,"style":1517},[1075,1335],[684,3211,1313],{"className":3212},[1340],[684,3214,849],{"className":3215},[1075,1335],[684,3217,1318],{"className":3218},[1347],[598,3220,842,3221,3224,3225,3228,3229,609],{},[602,3222,3223],{},"mémoïsation"," mémorise chaque résultat déjà calculé. Une valeur n'est\ncalculée qu'",[602,3226,3227],{},"une fois"," ; les appels suivants sont ",[684,3230,3232,3252],{"className":3231},[991],[684,3233,3235],{"className":3234},[995],[997,3236,3237],{"xmlns":999},[1001,3238,3239,3249],{},[1004,3240,3241,3243,3245,3247],{},[1307,3242,1395],{},[1011,3244,1313],{"stretchy":1312},[1007,3246,1009],{},[1011,3248,1318],{"stretchy":1312},[1054,3250,3251],{"encoding":1056},"O(1)",[684,3253,3255],{"className":3254,"ariaHidden":1062},[1061],[684,3256,3258,3261,3264,3267,3270],{"className":3257},[1066],[684,3259],{"className":3260,"style":1331},[1070],[684,3262,1395],{"className":3263,"style":1517},[1075,1335],[684,3265,1313],{"className":3266},[1340],[684,3268,1009],{"className":3269},[1075],[684,3271,1318],{"className":3272},[1347],[674,3274,3276],{"className":676,"code":3275,"language":678,"meta":679,"style":679},"def fib_memo(n: int, cache: dict = None) -> int:\n    \"\"\"Fibonacci mémoïsé via un dictionnaire.\"\"\"\n    if cache is None:\n        cache = {}\n    if n in cache:\n        return cache[n]\n    if n \u003C 2:\n        return n\n    cache[n] = fib_memo(n - 1, cache) + fib_memo(n - 2, cache)\n    return cache[n]\n",[681,3277,3278,3307,3312,3326,3336,3348,3355,3367,3373,3401],{"__ignoreMap":679},[684,3279,3280,3282,3285,3287,3289,3292,3295,3298,3301,3303,3305],{"class":686,"line":687},[684,3281,691],{"class":690},[684,3283,3284],{"class":694}," fib_memo",[684,3286,699],{"class":698},[684,3288,703],{"class":702},[684,3290,3291],{"class":698},", cache: ",[684,3293,3294],{"class":702},"dict",[684,3296,3297],{"class":690}," =",[684,3299,3300],{"class":702}," None",[684,3302,706],{"class":698},[684,3304,703],{"class":702},[684,3306,711],{"class":698},[684,3308,3309],{"class":686,"line":714},[684,3310,3311],{"class":717},"    \"\"\"Fibonacci mémoïsé via un dictionnaire.\"\"\"\n",[684,3313,3314,3316,3319,3322,3324],{"class":686,"line":721},[684,3315,787],{"class":690},[684,3317,3318],{"class":698}," cache ",[684,3320,3321],{"class":690},"is",[684,3323,3300],{"class":702},[684,3325,711],{"class":698},[684,3327,3328,3331,3333],{"class":686,"line":728},[684,3329,3330],{"class":698},"        cache ",[684,3332,1023],{"class":690},[684,3334,3335],{"class":698}," {}\n",[684,3337,3338,3340,3342,3345],{"class":686,"line":737},[684,3339,787],{"class":690},[684,3341,769],{"class":698},[684,3343,3344],{"class":690},"in",[684,3346,3347],{"class":698}," cache:\n",[684,3349,3350,3352],{"class":686,"line":743},[684,3351,807],{"class":690},[684,3353,3354],{"class":698}," cache[n]\n",[684,3356,3357,3359,3361,3363,3365],{"class":686,"line":751},[684,3358,787],{"class":690},[684,3360,769],{"class":698},[684,3362,2456],{"class":690},[684,3364,2459],{"class":702},[684,3366,711],{"class":698},[684,3368,3369,3371],{"class":686,"line":757},[684,3370,807],{"class":690},[684,3372,2468],{"class":698},[684,3374,3375,3378,3380,3383,3385,3387,3390,3392,3394,3396,3398],{"class":686,"line":763},[684,3376,3377],{"class":698},"    cache[n] ",[684,3379,1023],{"class":690},[684,3381,3382],{"class":698}," fib_memo(n ",[684,3384,827],{"class":690},[684,3386,830],{"class":702},[684,3388,3389],{"class":698},", cache) ",[684,3391,1392],{"class":690},[684,3393,3382],{"class":698},[684,3395,827],{"class":690},[684,3397,2459],{"class":702},[684,3399,3400],{"class":698},", cache)\n",[684,3402,3403,3405],{"class":686,"line":784},[684,3404,816],{"class":690},[684,3406,3354],{"class":698},[598,3408,3409],{},"Plus idiomatique en Python : utiliser le décorateur de la bibliothèque\nstandard.",[674,3411,3413],{"className":676,"code":3412,"language":678,"meta":679,"style":679},"from functools import lru_cache\n\n@lru_cache(maxsize=None)\ndef fib(n: int) -> int:\n    if n \u003C 2:\n        return n\n    return fib(n - 1) + fib(n - 2)\n",[681,3414,3415,3429,3433,3451,3468,3480,3486],{"__ignoreMap":679},[684,3416,3417,3420,3423,3426],{"class":686,"line":687},[684,3418,3419],{"class":690},"from",[684,3421,3422],{"class":698}," functools ",[684,3424,3425],{"class":690},"import",[684,3427,3428],{"class":698}," lru_cache\n",[684,3430,3431],{"class":686,"line":714},[684,3432,725],{"emptyLinePlaceholder":724},[684,3434,3435,3438,3440,3444,3446,3449],{"class":686,"line":721},[684,3436,3437],{"class":694},"@lru_cache",[684,3439,1313],{"class":698},[684,3441,3443],{"class":3442},"sP4rz","maxsize",[684,3445,1023],{"class":690},[684,3447,3448],{"class":702},"None",[684,3450,913],{"class":698},[684,3452,3453,3455,3458,3460,3462,3464,3466],{"class":686,"line":728},[684,3454,691],{"class":690},[684,3456,3457],{"class":694}," fib",[684,3459,699],{"class":698},[684,3461,703],{"class":702},[684,3463,706],{"class":698},[684,3465,703],{"class":702},[684,3467,711],{"class":698},[684,3469,3470,3472,3474,3476,3478],{"class":686,"line":737},[684,3471,787],{"class":690},[684,3473,769],{"class":698},[684,3475,2456],{"class":690},[684,3477,2459],{"class":702},[684,3479,711],{"class":698},[684,3481,3482,3484],{"class":686,"line":743},[684,3483,807],{"class":690},[684,3485,2468],{"class":698},[684,3487,3488,3490,3493,3495,3497,3499,3501,3503,3505,3507],{"class":686,"line":751},[684,3489,816],{"class":690},[684,3491,3492],{"class":698}," fib(n ",[684,3494,827],{"class":690},[684,3496,830],{"class":702},[684,3498,2482],{"class":698},[684,3500,1392],{"class":690},[684,3502,3492],{"class":698},[684,3504,827],{"class":690},[684,3506,2459],{"class":702},[684,3508,913],{"class":698},[598,3510,3511,3512,3174,3514,3516],{},"Nouveau coût : chaque valeur de ",[681,3513,1417],{},[681,3515,849],{}," est calculée une fois.",[598,3518,3519],{},[684,3520,3522,3551],{"className":3521},[991],[684,3523,3525],{"className":3524},[995],[997,3526,3527],{"xmlns":999},[1001,3528,3529,3549],{},[1004,3530,3531,3533,3535,3537,3539,3541,3543,3545,3547],{},[1307,3532,1309],{},[1011,3534,1313],{"stretchy":1312},[1307,3536,849],{},[1011,3538,1318],{"stretchy":1312},[1011,3540,1023],{},[1307,3542,1395],{},[1011,3544,1313],{"stretchy":1312},[1307,3546,849],{},[1011,3548,1318],{"stretchy":1312},[1054,3550,1917],{"encoding":1056},[684,3552,3554,3581],{"className":3553,"ariaHidden":1062},[1061],[684,3555,3557,3560,3563,3566,3569,3572,3575,3578],{"className":3556},[1066],[684,3558],{"className":3559,"style":1331},[1070],[684,3561,1309],{"className":3562,"style":1336},[1075,1335],[684,3564,1313],{"className":3565},[1340],[684,3567,849],{"className":3568},[1075,1335],[684,3570,1318],{"className":3571},[1347],[684,3573],{"className":3574,"style":1080},[1079],[684,3576,1023],{"className":3577},[1084],[684,3579],{"className":3580,"style":1080},[1079],[684,3582,3584,3587,3590,3593,3596],{"className":3583},[1066],[684,3585],{"className":3586,"style":1331},[1070],[684,3588,1395],{"className":3589,"style":1517},[1075,1335],[684,3591,1313],{"className":3592},[1340],[684,3594,849],{"className":3595},[1075,1335],[684,3597,1318],{"className":3598},[1347],[598,3600,3601,3602,3605,3606,3609],{},"Concrètement, ",[681,3603,3604],{},"fib(100)"," est instantané — alors que ",[681,3607,3608],{},"fib_naif(100)"," ne\nterminerait jamais.",[3611,3612,3613],"tip",{},[598,3614,3615,3618,3619,3622],{},[602,3616,3617],{},"Règle générale"," : dès qu'une récursion recalcule plusieurs fois la même\nsous-instance, mémoïser. C'est l'idée centrale de la ",[602,3620,3621],{},"programmation\ndynamique",", formalisée au chapitre TE.",[1266,3624,3626,3631],{"correct":1417,"slug":3625},"recursivite-c2",[1270,3627,3628],{"v-slot:question":679},[598,3629,3630],{},"Pourquoi fib_naif(n) est-il en O(φ^n) ?",[1270,3632,3633],{"v-slot:choices":679},[1278,3634,3635,3638,3641],{},[647,3636,3637],{},"Chaque appel en déclenche deux qui ne se réduisent que de 1 ou 2",[647,3639,3640],{},"Python est lent en récursion",[647,3642,3643],{},"Le cache est désactivé",[593,3645,3647],{"id":3646},"pour-aller-plus-loin","Pour aller plus loin",[598,3649,3650,3651,3654,3655,3658],{},"La récursivité éclaire les structures arborescentes (chapitre TA) — la\nhauteur d'un arbre se calcule récursivement par\n",[681,3652,3653],{},"1 + max(hauteur(gauche), hauteur(droite))",". Certains langages\n(Haskell, OCaml) optimisent la ",[602,3656,3657],{},"récursion terminale"," en boucle — Python\nne le fait pas.",[607,3660],{"code":588},[3662,3663,3664],"style",{},"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 pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}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);}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":679,"searchDepth":714,"depth":714,"links":3666},[3667,3668,3669,3670,3671,3672,3674],{"id":595,"depth":714,"text":596},{"id":638,"depth":714,"text":639},{"id":952,"depth":714,"text":953},{"id":1289,"depth":714,"text":1290},{"id":1973,"depth":714,"text":1974},{"id":3098,"depth":714,"text":3673},"Mémoïsation — de O(2n)O(2^n)O(2n) à O(n)O(n)O(n)",{"id":3646,"depth":714,"text":3647},"Écrire et analyser un programme récursif — pile d'appels, terminaison, factorielle linéaire et Fibonacci naïf vs mémoïsé.",null,"md",{},[3680,3684],{"slug":1268,"kind":3681,"question":1274,"choices":3682,"correct":1009,"multi":3683,"explanation":679},"qcm",[1043,1284,928],false,{"slug":3625,"kind":3681,"question":3630,"choices":3685,"correct":1417,"multi":3683,"explanation":679},[3637,3640,3643],{"title":549,"description":3675},"svW80dTXse1MaQHCk0YQek4hQt0s4pS3aKWkqxmkOy0",{"id":3689,"title":3690,"bo":3691,"body":3692,"description":4508,"eval":3676,"extension":3677,"labo":3676,"meta":4509,"navigation":3683,"path":4510,"quizzes":4511,"readingTime":687,"seo":4512,"stem":4513,"tp":3676,"__hash__":4514},"coursMemo\u002F2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite.memo.md","Récursivité — l'essentiel",[588],{"type":590,"value":3693,"toc":4501},[3694,3698,3703,3707,4348,4352,4398,4402,4413,4417,4498],[593,3695,3697],{"id":3696},"en-une-phrase","En une phrase",[598,3699,3700],{},[602,3701,3702],{},"Une fonction récursive s'appelle sur une entrée strictement plus petite jusqu'à un cas de base ; bien analysée par une relation de récurrence, mémoïsée si nécessaire.",[593,3704,3706],{"id":3705},"à-mémoriser","À mémoriser",[1278,3708,3709,3715,3721,3942,4251,4337],{},[647,3710,3711,3714],{},[602,3712,3713],{},"Deux ingrédients"," : cas de base (terminaison) + cas récursif (réduction).",[647,3716,3717,3720],{},[602,3718,3719],{},"Pile d'appels"," : chaque appel empile un cadre ; profondeur max ~1000 en Python.",[647,3722,3723,3726,3727,609],{},[602,3724,3725],{},"Factorielle"," : ",[684,3728,3730,3795],{"className":3729},[991],[684,3731,3733],{"className":3732},[995],[997,3734,3735],{"xmlns":999},[1001,3736,3737,3792],{},[1004,3738,3739,3741,3743,3745,3747,3749,3751,3753,3755,3757,3759,3761,3763,3765,3767,3769,3771,3774,3776,3778,3780,3782,3784,3786,3788,3790],{},[1307,3740,1309],{},[1011,3742,1313],{"stretchy":1312},[1307,3744,849],{},[1011,3746,1318],{"stretchy":1312},[1011,3748,1023],{},[1307,3750,1309],{},[1011,3752,1313],{"stretchy":1312},[1307,3754,849],{},[1011,3756,1385],{},[1007,3758,1009],{},[1011,3760,1318],{"stretchy":1312},[1011,3762,1392],{},[1307,3764,1395],{},[1011,3766,1313],{"stretchy":1312},[1007,3768,1009],{},[1011,3770,1318],{"stretchy":1312},[1011,3772,3773],{},"⇒",[1307,3775,1309],{},[1011,3777,1313],{"stretchy":1312},[1307,3779,849],{},[1011,3781,1318],{"stretchy":1312},[1011,3783,1023],{},[1307,3785,1395],{},[1011,3787,1313],{"stretchy":1312},[1307,3789,849],{},[1011,3791,1318],{"stretchy":1312},[1054,3793,3794],{"encoding":1056},"T(n) = T(n-1) + O(1) \\Rightarrow T(n) = O(n)",[684,3796,3798,3825,3849,3870,3897,3924],{"className":3797,"ariaHidden":1062},[1061],[684,3799,3801,3804,3807,3810,3813,3816,3819,3822],{"className":3800},[1066],[684,3802],{"className":3803,"style":1331},[1070],[684,3805,1309],{"className":3806,"style":1336},[1075,1335],[684,3808,1313],{"className":3809},[1340],[684,3811,849],{"className":3812},[1075,1335],[684,3814,1318],{"className":3815},[1347],[684,3817],{"className":3818,"style":1080},[1079],[684,3820,1023],{"className":3821},[1084],[684,3823],{"className":3824,"style":1080},[1079],[684,3826,3828,3831,3834,3837,3840,3843,3846],{"className":3827},[1066],[684,3829],{"className":3830,"style":1331},[1070],[684,3832,1309],{"className":3833,"style":1336},[1075,1335],[684,3835,1313],{"className":3836},[1340],[684,3838,849],{"className":3839},[1075,1335],[684,3841],{"className":3842,"style":1101},[1079],[684,3844,1385],{"className":3845},[1105],[684,3847],{"className":3848,"style":1101},[1079],[684,3850,3852,3855,3858,3861,3864,3867],{"className":3851},[1066],[684,3853],{"className":3854,"style":1331},[1070],[684,3856,1009],{"className":3857},[1075],[684,3859,1318],{"className":3860},[1347],[684,3862],{"className":3863,"style":1101},[1079],[684,3865,1392],{"className":3866},[1105],[684,3868],{"className":3869,"style":1101},[1079],[684,3871,3873,3876,3879,3882,3885,3888,3891,3894],{"className":3872},[1066],[684,3874],{"className":3875,"style":1331},[1070],[684,3877,1395],{"className":3878,"style":1517},[1075,1335],[684,3880,1313],{"className":3881},[1340],[684,3883,1009],{"className":3884},[1075],[684,3886,1318],{"className":3887},[1347],[684,3889],{"className":3890,"style":1080},[1079],[684,3892,3773],{"className":3893},[1084],[684,3895],{"className":3896,"style":1080},[1079],[684,3898,3900,3903,3906,3909,3912,3915,3918,3921],{"className":3899},[1066],[684,3901],{"className":3902,"style":1331},[1070],[684,3904,1309],{"className":3905,"style":1336},[1075,1335],[684,3907,1313],{"className":3908},[1340],[684,3910,849],{"className":3911},[1075,1335],[684,3913,1318],{"className":3914},[1347],[684,3916],{"className":3917,"style":1080},[1079],[684,3919,1023],{"className":3920},[1084],[684,3922],{"className":3923,"style":1080},[1079],[684,3925,3927,3930,3933,3936,3939],{"className":3926},[1066],[684,3928],{"className":3929,"style":1331},[1070],[684,3931,1395],{"className":3932,"style":1517},[1075,1335],[684,3934,1313],{"className":3935},[1340],[684,3937,849],{"className":3938},[1075,1335],[684,3940,1318],{"className":3941},[1347],[647,3943,3944,3726,3947,4250],{},[602,3945,3946],{},"Fibonacci naïf",[684,3948,3950,4032],{"className":3949},[991],[684,3951,3953],{"className":3952},[995],[997,3954,3955],{"xmlns":999},[1001,3956,3957,4029],{},[1004,3958,3959,3961,3963,3965,3967,3969,3971,3973,3975,3977,3979,3981,3983,3985,3987,3989,3991,3993,3995,3997,3999,4001,4003,4005,4007,4009,4011,4013,4015,4017,4019,4021,4027],{},[1307,3960,1309],{},[1011,3962,1313],{"stretchy":1312},[1307,3964,849],{},[1011,3966,1318],{"stretchy":1312},[1011,3968,1023],{},[1307,3970,1309],{},[1011,3972,1313],{"stretchy":1312},[1307,3974,849],{},[1011,3976,1385],{},[1007,3978,1009],{},[1011,3980,1318],{"stretchy":1312},[1011,3982,1392],{},[1307,3984,1309],{},[1011,3986,1313],{"stretchy":1312},[1307,3988,849],{},[1011,3990,1385],{},[1007,3992,1030],{},[1011,3994,1318],{"stretchy":1312},[1011,3996,1392],{},[1307,3998,1395],{},[1011,4000,1313],{"stretchy":1312},[1007,4002,1009],{},[1011,4004,1318],{"stretchy":1312},[1011,4006,3773],{},[1307,4008,1309],{},[1011,4010,1313],{"stretchy":1312},[1307,4012,849],{},[1011,4014,1318],{"stretchy":1312},[1011,4016,1023],{},[1307,4018,1395],{},[1011,4020,1313],{"stretchy":1312},[2731,4022,4023,4025],{},[1307,4024,2735],{},[1307,4026,849],{},[1011,4028,1318],{"stretchy":1312},[1054,4030,4031],{"encoding":1056},"T(n) = T(n-1) + T(n-2) + O(1) \\Rightarrow T(n) = O(\\varphi^n)",[684,4033,4035,4062,4086,4107,4131,4152,4179,4206],{"className":4034,"ariaHidden":1062},[1061],[684,4036,4038,4041,4044,4047,4050,4053,4056,4059],{"className":4037},[1066],[684,4039],{"className":4040,"style":1331},[1070],[684,4042,1309],{"className":4043,"style":1336},[1075,1335],[684,4045,1313],{"className":4046},[1340],[684,4048,849],{"className":4049},[1075,1335],[684,4051,1318],{"className":4052},[1347],[684,4054],{"className":4055,"style":1080},[1079],[684,4057,1023],{"className":4058},[1084],[684,4060],{"className":4061,"style":1080},[1079],[684,4063,4065,4068,4071,4074,4077,4080,4083],{"className":4064},[1066],[684,4066],{"className":4067,"style":1331},[1070],[684,4069,1309],{"className":4070,"style":1336},[1075,1335],[684,4072,1313],{"className":4073},[1340],[684,4075,849],{"className":4076},[1075,1335],[684,4078],{"className":4079,"style":1101},[1079],[684,4081,1385],{"className":4082},[1105],[684,4084],{"className":4085,"style":1101},[1079],[684,4087,4089,4092,4095,4098,4101,4104],{"className":4088},[1066],[684,4090],{"className":4091,"style":1331},[1070],[684,4093,1009],{"className":4094},[1075],[684,4096,1318],{"className":4097},[1347],[684,4099],{"className":4100,"style":1101},[1079],[684,4102,1392],{"className":4103},[1105],[684,4105],{"className":4106,"style":1101},[1079],[684,4108,4110,4113,4116,4119,4122,4125,4128],{"className":4109},[1066],[684,4111],{"className":4112,"style":1331},[1070],[684,4114,1309],{"className":4115,"style":1336},[1075,1335],[684,4117,1313],{"className":4118},[1340],[684,4120,849],{"className":4121},[1075,1335],[684,4123],{"className":4124,"style":1101},[1079],[684,4126,1385],{"className":4127},[1105],[684,4129],{"className":4130,"style":1101},[1079],[684,4132,4134,4137,4140,4143,4146,4149],{"className":4133},[1066],[684,4135],{"className":4136,"style":1331},[1070],[684,4138,1030],{"className":4139},[1075],[684,4141,1318],{"className":4142},[1347],[684,4144],{"className":4145,"style":1101},[1079],[684,4147,1392],{"className":4148},[1105],[684,4150],{"className":4151,"style":1101},[1079],[684,4153,4155,4158,4161,4164,4167,4170,4173,4176],{"className":4154},[1066],[684,4156],{"className":4157,"style":1331},[1070],[684,4159,1395],{"className":4160,"style":1517},[1075,1335],[684,4162,1313],{"className":4163},[1340],[684,4165,1009],{"className":4166},[1075],[684,4168,1318],{"className":4169},[1347],[684,4171],{"className":4172,"style":1080},[1079],[684,4174,3773],{"className":4175},[1084],[684,4177],{"className":4178,"style":1080},[1079],[684,4180,4182,4185,4188,4191,4194,4197,4200,4203],{"className":4181},[1066],[684,4183],{"className":4184,"style":1331},[1070],[684,4186,1309],{"className":4187,"style":1336},[1075,1335],[684,4189,1313],{"className":4190},[1340],[684,4192,849],{"className":4193},[1075,1335],[684,4195,1318],{"className":4196},[1347],[684,4198],{"className":4199,"style":1080},[1079],[684,4201,1023],{"className":4202},[1084],[684,4204],{"className":4205,"style":1080},[1079],[684,4207,4209,4212,4215,4218,4247],{"className":4208},[1066],[684,4210],{"className":4211,"style":1331},[1070],[684,4213,1395],{"className":4214,"style":1517},[1075,1335],[684,4216,1313],{"className":4217},[1340],[684,4219,4221,4224],{"className":4220},[1075],[684,4222,2735],{"className":4223},[1075,1335],[684,4225,4227],{"className":4226},[2024],[684,4228,4230],{"className":4229},[2028],[684,4231,4233],{"className":4232},[2033],[684,4234,4236],{"className":4235,"style":2803},[2037],[684,4237,4238,4241],{"style":2806},[684,4239],{"className":4240,"style":2046},[2045],[684,4242,4244],{"className":4243},[2050,2051,2052,2053],[684,4245,849],{"className":4246},[1075,1335,2053],[684,4248,1318],{"className":4249},[1347]," — exponentiel.",[647,4252,4253,4256,4257,609],{},[602,4254,4255],{},"Fibonacci mémoïsé"," : chaque valeur calculée une seule fois, ",[684,4258,4260,4289],{"className":4259},[991],[684,4261,4263],{"className":4262},[995],[997,4264,4265],{"xmlns":999},[1001,4266,4267,4287],{},[1004,4268,4269,4271,4273,4275,4277,4279,4281,4283,4285],{},[1307,4270,1309],{},[1011,4272,1313],{"stretchy":1312},[1307,4274,849],{},[1011,4276,1318],{"stretchy":1312},[1011,4278,1023],{},[1307,4280,1395],{},[1011,4282,1313],{"stretchy":1312},[1307,4284,849],{},[1011,4286,1318],{"stretchy":1312},[1054,4288,1917],{"encoding":1056},[684,4290,4292,4319],{"className":4291,"ariaHidden":1062},[1061],[684,4293,4295,4298,4301,4304,4307,4310,4313,4316],{"className":4294},[1066],[684,4296],{"className":4297,"style":1331},[1070],[684,4299,1309],{"className":4300,"style":1336},[1075,1335],[684,4302,1313],{"className":4303},[1340],[684,4305,849],{"className":4306},[1075,1335],[684,4308,1318],{"className":4309},[1347],[684,4311],{"className":4312,"style":1080},[1079],[684,4314,1023],{"className":4315},[1084],[684,4317],{"className":4318,"style":1080},[1079],[684,4320,4322,4325,4328,4331,4334],{"className":4321},[1066],[684,4323],{"className":4324,"style":1331},[1070],[684,4326,1395],{"className":4327,"style":1517},[1075,1335],[684,4329,1313],{"className":4330},[1340],[684,4332,849],{"className":4333},[1075,1335],[684,4335,1318],{"className":4336},[1347],[647,4338,4339,4343,4344,4347],{},[602,4340,4341],{},[681,4342,3437],{}," (module ",[681,4345,4346],{},"functools",") : mémoïsation déclarative.",[593,4349,4351],{"id":4350},"à-savoir-reconnaître","À savoir reconnaître",[4353,4354,4355,4368],"table",{},[4356,4357,4358],"thead",{},[4359,4360,4361,4365],"tr",{},[4362,4363,4364],"th",{},"Situation",[4362,4366,4367],{},"Action",[4369,4370,4371,4382,4390],"tbody",{},[4359,4372,4373,4377],{},[4374,4375,4376],"td",{},"Récursion sans cas de base",[4374,4378,4379,4381],{},[681,4380,861],{}," garantie.",[4359,4383,4384,4387],{},[4374,4385,4386],{},"Sous-problèmes recalculés (Fibonacci, chemins)",[4374,4388,4389],{},"Mémoïser.",[4359,4391,4392,4395],{},[4374,4393,4394],{},"Profondeur d'appel proche de 1000",[4374,4396,4397],{},"Repenser en itératif ou augmenter la limite.",[593,4399,4401],{"id":4400},"pièges-courants","Pièges courants",[1278,4403,4404,4407,4410],{},[647,4405,4406],{},"Oublier le cas de base — récursion infinie.",[647,4408,4409],{},"Faire récursion sur une entrée qui ne diminue pas strictement.",[647,4411,4412],{},"Ne pas mémoïser quand les sous-problèmes se chevauchent.",[593,4414,4416],{"id":4415},"syntaxe-python","Syntaxe Python",[674,4418,4420],{"className":676,"code":4419,"language":678,"meta":679,"style":679},"from functools import lru_cache\n\n@lru_cache(maxsize=None)\ndef fib(n):\n    if n \u003C 2:\n        return n\n    return fib(n - 1) + fib(n - 2)\n",[681,4421,4422,4432,4436,4450,4458,4470,4476],{"__ignoreMap":679},[684,4423,4424,4426,4428,4430],{"class":686,"line":687},[684,4425,3419],{"class":690},[684,4427,3422],{"class":698},[684,4429,3425],{"class":690},[684,4431,3428],{"class":698},[684,4433,4434],{"class":686,"line":714},[684,4435,725],{"emptyLinePlaceholder":724},[684,4437,4438,4440,4442,4444,4446,4448],{"class":686,"line":721},[684,4439,3437],{"class":694},[684,4441,1313],{"class":698},[684,4443,3443],{"class":3442},[684,4445,1023],{"class":690},[684,4447,3448],{"class":702},[684,4449,913],{"class":698},[684,4451,4452,4454,4456],{"class":686,"line":728},[684,4453,691],{"class":690},[684,4455,3457],{"class":694},[684,4457,878],{"class":698},[684,4459,4460,4462,4464,4466,4468],{"class":686,"line":737},[684,4461,787],{"class":690},[684,4463,769],{"class":698},[684,4465,2456],{"class":690},[684,4467,2459],{"class":702},[684,4469,711],{"class":698},[684,4471,4472,4474],{"class":686,"line":743},[684,4473,807],{"class":690},[684,4475,2468],{"class":698},[684,4477,4478,4480,4482,4484,4486,4488,4490,4492,4494,4496],{"class":686,"line":751},[684,4479,816],{"class":690},[684,4481,3492],{"class":698},[684,4483,827],{"class":690},[684,4485,830],{"class":702},[684,4487,2482],{"class":698},[684,4489,1392],{"class":690},[684,4491,3492],{"class":698},[684,4493,827],{"class":690},[684,4495,2459],{"class":702},[684,4497,913],{"class":698},[3662,4499,4500],{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}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":679,"searchDepth":714,"depth":714,"links":4502},[4503,4504,4505,4506,4507],{"id":3696,"depth":714,"text":3697},{"id":3705,"depth":714,"text":3706},{"id":4350,"depth":714,"text":4351},{"id":4400,"depth":714,"text":4401},{"id":4415,"depth":714,"text":4416},"Mémo de révision sur la récursivité, la pile d'appels et la mémoïsation.",{},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Frecursivite.memo",[],{"title":3690,"description":4508},"2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite.memo","JT23HNndr7cjgKLnEzScZSRus_CTtrxkIa2DffbynRA",{"id":4516,"title":4517,"body":4518,"description":5346,"extension":3677,"meta":5347,"navigation":3683,"path":5348,"seo":5349,"stem":5350,"__hash__":5351},"coursSlides\u002F2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite.slides.md","Récursivité — diapositives",{"type":590,"value":4519,"toc":5344},[4520,4524,4530,4533,4536,4539,4556,4561,4563,4565,4612,4621,4623,4626,4632,4638,4640,4644,4786,4883,4886,4888,4891,4944,5212,5219,5221,5225,5232,5238,5240,5244,5324,5329,5331,5335,5341],[4521,4522,549],"h1",{"id":4523},"récursivité",[598,4525,4526,4527,4529],{},"Item ",[607,4528],{"code":588}," — Une fonction qui s'appelle elle-même.",[598,4531,4532],{},"Matriochkas, effet Droste, fractale de Sierpinski : on retrouve l'idée partout.",[4534,4535],"hr",{},[4521,4537,3713],{"id":4538},"deux-ingrédients",[1278,4540,4541,4547],{},[647,4542,4543,4546],{},[602,4544,4545],{},"Cas de base"," : retour direct, sans appel récursif.",[647,4548,4549,4552,4553,609],{},[602,4550,4551],{},"Cas récursif"," : appel sur une entrée ",[602,4554,4555],{},"strictement plus petite",[598,4557,4558,4559,609],{},"Sans cas de base ⇒ récursion infinie ⇒ ",[681,4560,861],{},[4534,4562],{},[4521,4564,3725],{"id":672},[674,4566,4568],{"className":676,"code":4567,"language":678,"meta":679,"style":679},"def factorielle(n):\n    if n == 0:\n        return 1\n    return n * factorielle(n - 1)\n",[681,4569,4570,4578,4590,4596],{"__ignoreMap":679},[684,4571,4572,4574,4576],{"class":686,"line":687},[684,4573,691],{"class":690},[684,4575,695],{"class":694},[684,4577,878],{"class":698},[684,4579,4580,4582,4584,4586,4588],{"class":686,"line":714},[684,4581,787],{"class":690},[684,4583,769],{"class":698},[684,4585,792],{"class":690},[684,4587,775],{"class":702},[684,4589,711],{"class":698},[684,4591,4592,4594],{"class":686,"line":721},[684,4593,807],{"class":690},[684,4595,810],{"class":702},[684,4597,4598,4600,4602,4604,4606,4608,4610],{"class":686,"line":728},[684,4599,816],{"class":690},[684,4601,769],{"class":698},[684,4603,821],{"class":690},[684,4605,824],{"class":698},[684,4607,827],{"class":690},[684,4609,830],{"class":702},[684,4611,913],{"class":698},[598,4613,4614,4615,4617,4618,609],{},"Cas de base : ",[681,4616,853],{},".\nCas récursif : ",[681,4619,4620],{},"n * factorielle(n - 1)",[4534,4622],{},[4521,4624,3719],{"id":4625},"pile-dappels",[674,4627,4630],{"className":4628,"code":4629,"language":978},[976],"factorielle(3)\n  factorielle(2)\n    factorielle(1)\n      factorielle(0)  → 1\n    → 1 * 1 = 1\n  → 2 * 1 = 2\n→ 3 * 2 = 6\n",[681,4631,4629],{"__ignoreMap":679},[598,4633,4634,4635,4637],{},"Profondeur max Python : ~1000.\nImage utile : une ",[602,4636,967],{}," — on n'enlève qu'au sommet.",[4534,4639],{},[4521,4641,4643],{"id":4642},"complexité-de-la-factorielle","Complexité de la factorielle",[598,4645,4646],{},[684,4647,4649,4693],{"className":4648},[991],[684,4650,4652],{"className":4651},[995],[997,4653,4654],{"xmlns":999},[1001,4655,4656,4690],{},[1004,4657,4658,4660,4662,4664,4666,4668,4670,4672,4674,4676,4678,4680,4682,4684,4686,4688],{},[1307,4659,1309],{},[1011,4661,1313],{"stretchy":1312},[1307,4663,849],{},[1011,4665,1318],{"stretchy":1312},[1011,4667,1023],{},[1307,4669,1309],{},[1011,4671,1313],{"stretchy":1312},[1307,4673,849],{},[1011,4675,1385],{},[1007,4677,1009],{},[1011,4679,1318],{"stretchy":1312},[1011,4681,1392],{},[1307,4683,1395],{},[1011,4685,1313],{"stretchy":1312},[1007,4687,1009],{},[1011,4689,1318],{"stretchy":1312},[1054,4691,4692],{"encoding":1056},"T(n) = T(n-1) + O(1)",[684,4694,4696,4723,4747,4768],{"className":4695,"ariaHidden":1062},[1061],[684,4697,4699,4702,4705,4708,4711,4714,4717,4720],{"className":4698},[1066],[684,4700],{"className":4701,"style":1331},[1070],[684,4703,1309],{"className":4704,"style":1336},[1075,1335],[684,4706,1313],{"className":4707},[1340],[684,4709,849],{"className":4710},[1075,1335],[684,4712,1318],{"className":4713},[1347],[684,4715],{"className":4716,"style":1080},[1079],[684,4718,1023],{"className":4719},[1084],[684,4721],{"className":4722,"style":1080},[1079],[684,4724,4726,4729,4732,4735,4738,4741,4744],{"className":4725},[1066],[684,4727],{"className":4728,"style":1331},[1070],[684,4730,1309],{"className":4731,"style":1336},[1075,1335],[684,4733,1313],{"className":4734},[1340],[684,4736,849],{"className":4737},[1075,1335],[684,4739],{"className":4740,"style":1101},[1079],[684,4742,1385],{"className":4743},[1105],[684,4745],{"className":4746,"style":1101},[1079],[684,4748,4750,4753,4756,4759,4762,4765],{"className":4749},[1066],[684,4751],{"className":4752,"style":1331},[1070],[684,4754,1009],{"className":4755},[1075],[684,4757,1318],{"className":4758},[1347],[684,4760],{"className":4761,"style":1101},[1079],[684,4763,1392],{"className":4764},[1105],[684,4766],{"className":4767,"style":1101},[1079],[684,4769,4771,4774,4777,4780,4783],{"className":4770},[1066],[684,4772],{"className":4773,"style":1331},[1070],[684,4775,1395],{"className":4776,"style":1517},[1075,1335],[684,4778,1313],{"className":4779},[1340],[684,4781,1009],{"className":4782},[1075],[684,4784,1318],{"className":4785},[1347],[598,4787,4788],{},[684,4789,4791,4823],{"className":4790},[991],[684,4792,4794],{"className":4793},[995],[997,4795,4796],{"xmlns":999},[1001,4797,4798,4820],{},[1004,4799,4800,4802,4804,4806,4808,4810,4812,4814,4816,4818],{},[1011,4801,3773],{},[1307,4803,1309],{},[1011,4805,1313],{"stretchy":1312},[1307,4807,849],{},[1011,4809,1318],{"stretchy":1312},[1011,4811,1023],{},[1307,4813,1395],{},[1011,4815,1313],{"stretchy":1312},[1307,4817,849],{},[1011,4819,1318],{"stretchy":1312},[1054,4821,4822],{"encoding":1056},"\\Rightarrow T(n) = O(n)",[684,4824,4826,4838,4865],{"className":4825,"ariaHidden":1062},[1061],[684,4827,4829,4832,4835],{"className":4828},[1066],[684,4830],{"className":4831,"style":1831},[1070],[684,4833,3773],{"className":4834},[1084],[684,4836],{"className":4837,"style":1080},[1079],[684,4839,4841,4844,4847,4850,4853,4856,4859,4862],{"className":4840},[1066],[684,4842],{"className":4843,"style":1331},[1070],[684,4845,1309],{"className":4846,"style":1336},[1075,1335],[684,4848,1313],{"className":4849},[1340],[684,4851,849],{"className":4852},[1075,1335],[684,4854,1318],{"className":4855},[1347],[684,4857],{"className":4858,"style":1080},[1079],[684,4860,1023],{"className":4861},[1084],[684,4863],{"className":4864,"style":1080},[1079],[684,4866,4868,4871,4874,4877,4880],{"className":4867},[1066],[684,4869],{"className":4870,"style":1331},[1070],[684,4872,1395],{"className":4873,"style":1517},[1075,1335],[684,4875,1313],{"className":4876},[1340],[684,4878,849],{"className":4879},[1075,1335],[684,4881,1318],{"className":4882},[1347],[598,4884,4885],{},"Linéaire — comme la version itérative.",[4534,4887],{},[4521,4889,3946],{"id":4890},"fibonacci-naïf",[674,4892,4894],{"className":676,"code":4893,"language":678,"meta":679,"style":679},"def fib(n):\n    if n \u003C 2:\n        return n\n    return fib(n - 1) + fib(n - 2)\n",[681,4895,4896,4904,4916,4922],{"__ignoreMap":679},[684,4897,4898,4900,4902],{"class":686,"line":687},[684,4899,691],{"class":690},[684,4901,3457],{"class":694},[684,4903,878],{"class":698},[684,4905,4906,4908,4910,4912,4914],{"class":686,"line":714},[684,4907,787],{"class":690},[684,4909,769],{"class":698},[684,4911,2456],{"class":690},[684,4913,2459],{"class":702},[684,4915,711],{"class":698},[684,4917,4918,4920],{"class":686,"line":721},[684,4919,807],{"class":690},[684,4921,2468],{"class":698},[684,4923,4924,4926,4928,4930,4932,4934,4936,4938,4940,4942],{"class":686,"line":728},[684,4925,816],{"class":690},[684,4927,3492],{"class":698},[684,4929,827],{"class":690},[684,4931,830],{"class":702},[684,4933,2482],{"class":698},[684,4935,1392],{"class":690},[684,4937,3492],{"class":698},[684,4939,827],{"class":690},[684,4941,2459],{"class":702},[684,4943,913],{"class":698},[598,4945,4946],{},[684,4947,4949,5021],{"className":4948},[991],[684,4950,4952],{"className":4951},[995],[997,4953,4954],{"xmlns":999},[1001,4955,4956,5018],{},[1004,4957,4958,4960,4962,4964,4966,4968,4970,4972,4974,4976,4978,4980,4982,4984,4986,4988,4990,4992,4994,4996,4998,5000,5002,5004,5006,5008,5010,5016],{},[1307,4959,1309],{},[1011,4961,1313],{"stretchy":1312},[1307,4963,849],{},[1011,4965,1318],{"stretchy":1312},[1011,4967,1023],{},[1307,4969,1309],{},[1011,4971,1313],{"stretchy":1312},[1307,4973,849],{},[1011,4975,1385],{},[1007,4977,1009],{},[1011,4979,1318],{"stretchy":1312},[1011,4981,1392],{},[1307,4983,1309],{},[1011,4985,1313],{"stretchy":1312},[1307,4987,849],{},[1011,4989,1385],{},[1007,4991,1030],{},[1011,4993,1318],{"stretchy":1312},[1011,4995,1392],{},[1307,4997,1395],{},[1011,4999,1313],{"stretchy":1312},[1007,5001,1009],{},[1011,5003,1318],{"stretchy":1312},[1011,5005,1023],{},[1307,5007,1395],{},[1011,5009,1313],{"stretchy":1312},[2731,5011,5012,5014],{},[1307,5013,2735],{},[1307,5015,849],{},[1011,5017,1318],{"stretchy":1312},[1054,5019,5020],{"encoding":1056},"T(n) = T(n-1) + T(n-2) + O(1) = O(\\varphi^n)",[684,5022,5024,5051,5075,5096,5120,5141,5168],{"className":5023,"ariaHidden":1062},[1061],[684,5025,5027,5030,5033,5036,5039,5042,5045,5048],{"className":5026},[1066],[684,5028],{"className":5029,"style":1331},[1070],[684,5031,1309],{"className":5032,"style":1336},[1075,1335],[684,5034,1313],{"className":5035},[1340],[684,5037,849],{"className":5038},[1075,1335],[684,5040,1318],{"className":5041},[1347],[684,5043],{"className":5044,"style":1080},[1079],[684,5046,1023],{"className":5047},[1084],[684,5049],{"className":5050,"style":1080},[1079],[684,5052,5054,5057,5060,5063,5066,5069,5072],{"className":5053},[1066],[684,5055],{"className":5056,"style":1331},[1070],[684,5058,1309],{"className":5059,"style":1336},[1075,1335],[684,5061,1313],{"className":5062},[1340],[684,5064,849],{"className":5065},[1075,1335],[684,5067],{"className":5068,"style":1101},[1079],[684,5070,1385],{"className":5071},[1105],[684,5073],{"className":5074,"style":1101},[1079],[684,5076,5078,5081,5084,5087,5090,5093],{"className":5077},[1066],[684,5079],{"className":5080,"style":1331},[1070],[684,5082,1009],{"className":5083},[1075],[684,5085,1318],{"className":5086},[1347],[684,5088],{"className":5089,"style":1101},[1079],[684,5091,1392],{"className":5092},[1105],[684,5094],{"className":5095,"style":1101},[1079],[684,5097,5099,5102,5105,5108,5111,5114,5117],{"className":5098},[1066],[684,5100],{"className":5101,"style":1331},[1070],[684,5103,1309],{"className":5104,"style":1336},[1075,1335],[684,5106,1313],{"className":5107},[1340],[684,5109,849],{"className":5110},[1075,1335],[684,5112],{"className":5113,"style":1101},[1079],[684,5115,1385],{"className":5116},[1105],[684,5118],{"className":5119,"style":1101},[1079],[684,5121,5123,5126,5129,5132,5135,5138],{"className":5122},[1066],[684,5124],{"className":5125,"style":1331},[1070],[684,5127,1030],{"className":5128},[1075],[684,5130,1318],{"className":5131},[1347],[684,5133],{"className":5134,"style":1101},[1079],[684,5136,1392],{"className":5137},[1105],[684,5139],{"className":5140,"style":1101},[1079],[684,5142,5144,5147,5150,5153,5156,5159,5162,5165],{"className":5143},[1066],[684,5145],{"className":5146,"style":1331},[1070],[684,5148,1395],{"className":5149,"style":1517},[1075,1335],[684,5151,1313],{"className":5152},[1340],[684,5154,1009],{"className":5155},[1075],[684,5157,1318],{"className":5158},[1347],[684,5160],{"className":5161,"style":1080},[1079],[684,5163,1023],{"className":5164},[1084],[684,5166],{"className":5167,"style":1080},[1079],[684,5169,5171,5174,5177,5180,5209],{"className":5170},[1066],[684,5172],{"className":5173,"style":1331},[1070],[684,5175,1395],{"className":5176,"style":1517},[1075,1335],[684,5178,1313],{"className":5179},[1340],[684,5181,5183,5186],{"className":5182},[1075],[684,5184,2735],{"className":5185},[1075,1335],[684,5187,5189],{"className":5188},[2024],[684,5190,5192],{"className":5191},[2028],[684,5193,5195],{"className":5194},[2033],[684,5196,5198],{"className":5197,"style":2803},[2037],[684,5199,5200,5203],{"style":2806},[684,5201],{"className":5202,"style":2046},[2045],[684,5204,5206],{"className":5205},[2050,2051,2052,2053],[684,5207,849],{"className":5208},[1075,1335,2053],[684,5210,1318],{"className":5211},[1347],[598,5213,5214,5215,5218],{},"Exponentiel. 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recalcul\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cdefs>\u003Cfilter id=\"ill-2-nsi-tale-td-langages-prog-arbre-recursion-fib5-drop-shadow\" height=\"130%\" width=\"130%\">\u003CfeDropShadow dx=\"4\" dy=\"4\" stdDeviation=\"0\" flood-opacity=\"0.06\" flood-color=\"#000000\"\u002F>\u003C\u002Ffilter>\u003C\u002Fdefs>\u003Cdefs>\u003Cfilter id=\"ill-2-nsi-tale-td-langages-prog-arbre-recursion-fib5-drop-shadow-small\" height=\"150%\" width=\"150%\">\u003CfeDropShadow dx=\"2\" dy=\"2\" stdDeviation=\"0\" flood-opacity=\"0.06\" flood-color=\"#000000\"\u002F>\u003C\u002Ffilter>\u003C\u002Fdefs>\u003ClinearGradient id=\"ill-2-nsi-tale-td-langages-prog-arbre-recursion-fib5-gradient\" gradientUnits=\"objectBoundingBox\" x1=\"0%\" y1=\"0%\" x2=\"100%\" y2=\"0%\">\u003Cstop offset=\"0%\" stop-color=\"#4948b8\" stop-opacity=\"1\"\u002F>\u003Cstop offset=\"100%\" stop-color=\"#c4c8cd\" stop-opacity=\"1\"\u002F>\u003C\u002FlinearGradient>\u003C\u002Fsvg>",[598,5233,5234,5235,609],{},"Beaucoup de ",[602,5236,5237],{},"sous-problèmes recalculés",[4534,5239],{},[4521,5241,5243],{"id":5242},"mémoïsation-on","Mémoïsation — O(n)",[674,5245,5246],{"className":676,"code":4419,"language":678,"meta":679,"style":679},[681,5247,5248,5258,5262,5276,5284,5296,5302],{"__ignoreMap":679},[684,5249,5250,5252,5254,5256],{"class":686,"line":687},[684,5251,3419],{"class":690},[684,5253,3422],{"class":698},[684,5255,3425],{"class":690},[684,5257,3428],{"class":698},[684,5259,5260],{"class":686,"line":714},[684,5261,725],{"emptyLinePlaceholder":724},[684,5263,5264,5266,5268,5270,5272,5274],{"class":686,"line":721},[684,5265,3437],{"class":694},[684,5267,1313],{"class":698},[684,5269,3443],{"class":3442},[684,5271,1023],{"class":690},[684,5273,3448],{"class":702},[684,5275,913],{"class":698},[684,5277,5278,5280,5282],{"class":686,"line":728},[684,5279,691],{"class":690},[684,5281,3457],{"class":694},[684,5283,878],{"class":698},[684,5285,5286,5288,5290,5292,5294],{"class":686,"line":737},[684,5287,787],{"class":690},[684,5289,769],{"class":698},[684,5291,2456],{"class":690},[684,5293,2459],{"class":702},[684,5295,711],{"class":698},[684,5297,5298,5300],{"class":686,"line":743},[684,5299,807],{"class":690},[684,5301,2468],{"class":698},[684,5303,5304,5306,5308,5310,5312,5314,5316,5318,5320,5322],{"class":686,"line":751},[684,5305,816],{"class":690},[684,5307,3492],{"class":698},[684,5309,827],{"class":690},[684,5311,830],{"class":702},[684,5313,2482],{"class":698},[684,5315,1392],{"class":690},[684,5317,3492],{"class":698},[684,5319,827],{"class":690},[684,5321,2459],{"class":702},[684,5323,913],{"class":698},[598,5325,5326,5327,609],{},"Chaque valeur calculée ",[602,5328,3227],{},[4534,5330],{},[4521,5332,5334],{"id":5333},"à-retenir","À retenir",[598,5336,5337,5340],{},[602,5338,5339],{},"Récursion = définition mathématique directe, à condition d'analyser la complexité.","\nSous-problèmes qui se répètent ⇒ mémoïser.",[3662,5342,5343],{},"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 pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}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: 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