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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 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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":429,"bo":587,"body":589,"description":2680,"eval":2681,"extension":2682,"labo":2681,"meta":2683,"navigation":1077,"path":430,"quizzes":2684,"readingTime":900,"seo":2691,"stem":431,"tp":2681,"__hash__":2692},"cours\u002F1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie.md",[588],"PH04",{"type":590,"value":591,"toc":2670},"minimark",[592,597,618,621,625,632,648,799,808,812,826,841,877,888,1178,1426,1430,1444,1498,1522,1646,1677,1755,1790,1980,2253,2313,2411,2434,2438,2500,2510,2531,2535,2666],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,605,606,609,610,613,614,617],"p",{},"La recherche linéaire du cours 1 examine ",[602,603,604],"strong",{},"toutes"," les cases d'un tableau au\npire. Pour 1 million d'éléments, c'est un million d'opérations. Si le\ntableau est ",[602,607,608],{},"trié",", on peut faire spectaculairement mieux : c'est l'objet\nde la ",[602,611,612],{},"recherche dichotomique",". L'idée est simple — comparer la cible à\nl'élément du milieu, et selon le résultat, jeter la moitié inutile. À\nchaque étape on ",[602,615,616],{},"divise par deux"," l'espace de recherche.",[619,620],"bo-ref",{"code":588},[593,622,624],{"id":623},"lintuition-plus-chaud-plus-froid","L'intuition « plus chaud, plus froid »",[598,626,627,628,631],{},"Vous connaissez le jeu : « j'ai un nombre entre 1 et 100, devine-le, je te dis\nplus haut ou plus bas ». La stratégie optimale est de toujours proposer la\n",[602,629,630],{},"moitié"," de l'intervalle restant.",[633,634,635,639,642,645],"ul",{},[636,637,638],"li",{},"Au départ : 1 à 100. Vous proposez 50.",[636,640,641],{},"Réponse « plus haut » : 51 à 100. Vous proposez 75.",[636,643,644],{},"Réponse « plus bas » : 51 à 74. Vous proposez 62.",[636,646,647],{},"…",[598,649,650,651,794,795,798],{},"En 7 questions au maximum, vous trouvez. Pourquoi 7 ? Parce que\n",[652,653,656,698],"span",{"className":654},[655],"katex",[652,657,660],{"className":658},[659],"katex-mathml",[661,662,664],"math",{"xmlns":663},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[665,666,667,693],"semantics",{},[668,669,670,680,684,687,690],"mrow",{},[671,672,673,677],"msup",{},[674,675,676],"mn",{},"2",[674,678,679],{},"7",[681,682,683],"mo",{},"=",[674,685,686],{},"128",[681,688,689],{},"≥",[674,691,692],{},"100",[694,695,697],"annotation",{"encoding":696},"application\u002Fx-tex","2^7 = 128 \\geq 100",[652,699,703,765,784],{"className":700,"ariaHidden":702},[701],"katex-html","true",[652,704,707,712,753,758,762],{"className":705},[706],"base",[652,708],{"className":709,"style":711},[710],"strut","height:0.8141em;",[652,713,716,719],{"className":714},[715],"mord",[652,717,676],{"className":718},[715],[652,720,723],{"className":721},[722],"msupsub",[652,724,727],{"className":725},[726],"vlist-t",[652,728,731],{"className":729},[730],"vlist-r",[652,732,735],{"className":733,"style":711},[734],"vlist",[652,736,738,743],{"style":737},"top:-3.063em;margin-right:0.05em;",[652,739],{"className":740,"style":742},[741],"pstrut","height:2.7em;",[652,744,750],{"className":745},[746,747,748,749],"sizing","reset-size6","size3","mtight",[652,751,679],{"className":752},[715,749],[652,754],{"className":755,"style":757},[756],"mspace","margin-right:0.2778em;",[652,759,683],{"className":760},[761],"mrel",[652,763],{"className":764,"style":757},[756],[652,766,768,772,775,778,781],{"className":767},[706],[652,769],{"className":770,"style":771},[710],"height:0.7804em;vertical-align:-0.136em;",[652,773,686],{"className":774},[715],[652,776],{"className":777,"style":757},[756],[652,779,689],{"className":780},[761],[652,782],{"className":783,"style":757},[756],[652,785,787,791],{"className":786},[706],[652,788],{"className":789,"style":790},[710],"height:0.6444em;",[652,792,692],{"className":793},[715],". Sept divisions par deux suffisent à isoler ",[602,796,797],{},"un","\nnombre parmi 100. C'est exactement l'idée de la dichotomie.",[800,801,802],"tip",{},[598,803,804,807],{},[602,805,806],{},"Analogie de l'annuaire papier"," : pour trouver « Dupont », vous n'ouvrez\npas le livre page 1. Vous l'ouvrez au milieu, regardez la lettre, et vous\nallez à gauche ou à droite. Puis encore au milieu de la moitié restante. En\nquelques étapes, vous y êtes.",[593,809,811],{"id":810},"lalgorithme","L'algorithme",[598,813,814,817,818,822,823,825],{},[602,815,816],{},"Prérequis"," : le tableau ",[819,820,821],"code",{},"t"," doit être ",[602,824,608],{}," dans l'ordre croissant.\nSinon l'algorithme ne fonctionne pas.",[598,827,828,829,832,833,836,837,840],{},"On maintient deux indices ",[819,830,831],{},"gauche"," et ",[819,834,835],{},"droite"," qui délimitent la zone où la\ncible ",[602,838,839],{},"pourrait encore se trouver",". Tant que cette zone n'est pas vide :",[842,843,844,851,861,870],"ol",{},[636,845,846,847,850],{},"on calcule l'indice du milieu, ",[819,848,849],{},"m = (gauche + droite) \u002F\u002F 2"," ;",[636,852,853,854,857,858,850],{},"si ",[819,855,856],{},"t[m] == cible",", on a trouvé, on retourne ",[819,859,860],{},"m",[636,862,853,863,866,867,850],{},[819,864,865],{},"t[m] \u003C cible",", la cible est forcément à droite : on pose\n",[819,868,869],{},"gauche = m + 1",[636,871,872,873,876],{},"sinon, la cible est forcément à gauche : on pose ",[819,874,875],{},"droite = m - 1",".",[598,878,879,880,883,884,887],{},"Si la zone devient vide (",[819,881,882],{},"gauche > droite","), la cible ",[602,885,886],{},"n'est pas"," dans le\ntableau.",[889,890,895],"pre",{"className":891,"code":892,"language":893,"meta":894,"style":894},"language-python shiki shiki-themes github-light github-light github-dark","def dichotomie(t, cible):\n    \"\"\"Retourne l'indice de cible dans t (trié), ou -1 si absente.\"\"\"\n    gauche, droite = 0, len(t) - 1\n    while gauche \u003C= droite:\n        m = (gauche + droite) \u002F\u002F 2\n        if t[m] == cible:\n            return m\n        elif t[m] \u003C cible:\n            gauche = m + 1\n        else:\n            droite = m - 1\n    return -1\n\ntrie = [1, 4, 7, 12, 18, 25, 33, 47, 58, 66, 80]\nprint(dichotomie(trie, 25))   # 5\nprint(dichotomie(trie, 10))   # -1\n","python","",[819,896,897,913,920,947,962,985,1000,1009,1022,1037,1046,1060,1072,1079,1145,1163],{"__ignoreMap":894},[652,898,901,905,909],{"class":899,"line":900},"line",1,[652,902,904],{"class":903},"s8jYJ","def",[652,906,908],{"class":907},"snPdu"," dichotomie",[652,910,912],{"class":911},"sxrX7","(t, cible):\n",[652,914,916],{"class":899,"line":915},2,[652,917,919],{"class":918},"sIIMD","    \"\"\"Retourne l'indice de cible dans t (trié), ou -1 si absente.\"\"\"\n",[652,921,923,926,928,932,935,938,941,944],{"class":899,"line":922},3,[652,924,925],{"class":911},"    gauche, droite ",[652,927,683],{"class":903},[652,929,931],{"class":930},"sBjJW"," 0",[652,933,934],{"class":911},", ",[652,936,937],{"class":930},"len",[652,939,940],{"class":911},"(t) ",[652,942,943],{"class":903},"-",[652,945,946],{"class":930}," 1\n",[652,948,950,953,956,959],{"class":899,"line":949},4,[652,951,952],{"class":903},"    while",[652,954,955],{"class":911}," gauche ",[652,957,958],{"class":903},"\u003C=",[652,960,961],{"class":911}," droite:\n",[652,963,965,968,970,973,976,979,982],{"class":899,"line":964},5,[652,966,967],{"class":911},"        m ",[652,969,683],{"class":903},[652,971,972],{"class":911}," (gauche ",[652,974,975],{"class":903},"+",[652,977,978],{"class":911}," droite) ",[652,980,981],{"class":903},"\u002F\u002F",[652,983,984],{"class":930}," 2\n",[652,986,988,991,994,997],{"class":899,"line":987},6,[652,989,990],{"class":903},"        if",[652,992,993],{"class":911}," t[m] ",[652,995,996],{"class":903},"==",[652,998,999],{"class":911}," cible:\n",[652,1001,1003,1006],{"class":899,"line":1002},7,[652,1004,1005],{"class":903},"            return",[652,1007,1008],{"class":911}," m\n",[652,1010,1012,1015,1017,1020],{"class":899,"line":1011},8,[652,1013,1014],{"class":903},"        elif",[652,1016,993],{"class":911},[652,1018,1019],{"class":903},"\u003C",[652,1021,999],{"class":911},[652,1023,1025,1028,1030,1033,1035],{"class":899,"line":1024},9,[652,1026,1027],{"class":911},"            gauche ",[652,1029,683],{"class":903},[652,1031,1032],{"class":911}," m ",[652,1034,975],{"class":903},[652,1036,946],{"class":930},[652,1038,1040,1043],{"class":899,"line":1039},10,[652,1041,1042],{"class":903},"        else",[652,1044,1045],{"class":911},":\n",[652,1047,1049,1052,1054,1056,1058],{"class":899,"line":1048},11,[652,1050,1051],{"class":911},"            droite ",[652,1053,683],{"class":903},[652,1055,1032],{"class":911},[652,1057,943],{"class":903},[652,1059,946],{"class":930},[652,1061,1063,1066,1069],{"class":899,"line":1062},12,[652,1064,1065],{"class":903},"    return",[652,1067,1068],{"class":903}," -",[652,1070,1071],{"class":930},"1\n",[652,1073,1075],{"class":899,"line":1074},13,[652,1076,1078],{"emptyLinePlaceholder":1077},true,"\n",[652,1080,1082,1085,1087,1090,1093,1095,1098,1100,1102,1104,1107,1109,1112,1114,1117,1119,1122,1124,1127,1129,1132,1134,1137,1139,1142],{"class":899,"line":1081},14,[652,1083,1084],{"class":911},"trie ",[652,1086,683],{"class":903},[652,1088,1089],{"class":911}," [",[652,1091,1092],{"class":930},"1",[652,1094,934],{"class":911},[652,1096,1097],{"class":930},"4",[652,1099,934],{"class":911},[652,1101,679],{"class":930},[652,1103,934],{"class":911},[652,1105,1106],{"class":930},"12",[652,1108,934],{"class":911},[652,1110,1111],{"class":930},"18",[652,1113,934],{"class":911},[652,1115,1116],{"class":930},"25",[652,1118,934],{"class":911},[652,1120,1121],{"class":930},"33",[652,1123,934],{"class":911},[652,1125,1126],{"class":930},"47",[652,1128,934],{"class":911},[652,1130,1131],{"class":930},"58",[652,1133,934],{"class":911},[652,1135,1136],{"class":930},"66",[652,1138,934],{"class":911},[652,1140,1141],{"class":930},"80",[652,1143,1144],{"class":911},"]\n",[652,1146,1148,1151,1154,1156,1159],{"class":899,"line":1147},15,[652,1149,1150],{"class":930},"print",[652,1152,1153],{"class":911},"(dichotomie(trie, ",[652,1155,1116],{"class":930},[652,1157,1158],{"class":911},"))   ",[652,1160,1162],{"class":1161},"sCsY4","# 5\n",[652,1164,1166,1168,1170,1173,1175],{"class":899,"line":1165},16,[652,1167,1150],{"class":930},[652,1169,1153],{"class":911},[652,1171,1172],{"class":930},"10",[652,1174,1158],{"class":911},[652,1176,1177],{"class":1161},"# -1\n",[1179,1180,1181],"python-snippet",{},[889,1182,1184],{"className":891,"code":1183,"language":893,"meta":894,"style":894},"def dichotomie(t, cible):\n    gauche, droite = 0, len(t) - 1\n    etapes = 0\n    while gauche \u003C= droite:\n        etapes += 1\n        m = (gauche + droite) \u002F\u002F 2\n        if t[m] == cible:\n            return m, etapes\n        elif t[m] \u003C cible:\n            gauche = m + 1\n        else:\n            droite = m - 1\n    return -1, etapes\n\n# Sur un grand tableau trié, on compte les étapes\nimport math\ntrie = list(range(1, 1001))   # 1, 2, …, 1000\nprint(\"Recherche de 777 :\", dichotomie(trie, 777))\nprint(\"Recherche de 1 :\", dichotomie(trie, 1))\nprint(\"Étapes max théoriques :\", math.ceil(math.log2(1001)))\n",[819,1185,1186,1194,1212,1222,1232,1242,1258,1268,1275,1285,1297,1303,1315,1326,1330,1335,1343,1373,1392,1408],{"__ignoreMap":894},[652,1187,1188,1190,1192],{"class":899,"line":900},[652,1189,904],{"class":903},[652,1191,908],{"class":907},[652,1193,912],{"class":911},[652,1195,1196,1198,1200,1202,1204,1206,1208,1210],{"class":899,"line":915},[652,1197,925],{"class":911},[652,1199,683],{"class":903},[652,1201,931],{"class":930},[652,1203,934],{"class":911},[652,1205,937],{"class":930},[652,1207,940],{"class":911},[652,1209,943],{"class":903},[652,1211,946],{"class":930},[652,1213,1214,1217,1219],{"class":899,"line":922},[652,1215,1216],{"class":911},"    etapes ",[652,1218,683],{"class":903},[652,1220,1221],{"class":930}," 0\n",[652,1223,1224,1226,1228,1230],{"class":899,"line":949},[652,1225,952],{"class":903},[652,1227,955],{"class":911},[652,1229,958],{"class":903},[652,1231,961],{"class":911},[652,1233,1234,1237,1240],{"class":899,"line":964},[652,1235,1236],{"class":911},"        etapes ",[652,1238,1239],{"class":903},"+=",[652,1241,946],{"class":930},[652,1243,1244,1246,1248,1250,1252,1254,1256],{"class":899,"line":987},[652,1245,967],{"class":911},[652,1247,683],{"class":903},[652,1249,972],{"class":911},[652,1251,975],{"class":903},[652,1253,978],{"class":911},[652,1255,981],{"class":903},[652,1257,984],{"class":930},[652,1259,1260,1262,1264,1266],{"class":899,"line":1002},[652,1261,990],{"class":903},[652,1263,993],{"class":911},[652,1265,996],{"class":903},[652,1267,999],{"class":911},[652,1269,1270,1272],{"class":899,"line":1011},[652,1271,1005],{"class":903},[652,1273,1274],{"class":911}," m, etapes\n",[652,1276,1277,1279,1281,1283],{"class":899,"line":1024},[652,1278,1014],{"class":903},[652,1280,993],{"class":911},[652,1282,1019],{"class":903},[652,1284,999],{"class":911},[652,1286,1287,1289,1291,1293,1295],{"class":899,"line":1039},[652,1288,1027],{"class":911},[652,1290,683],{"class":903},[652,1292,1032],{"class":911},[652,1294,975],{"class":903},[652,1296,946],{"class":930},[652,1298,1299,1301],{"class":899,"line":1048},[652,1300,1042],{"class":903},[652,1302,1045],{"class":911},[652,1304,1305,1307,1309,1311,1313],{"class":899,"line":1062},[652,1306,1051],{"class":911},[652,1308,683],{"class":903},[652,1310,1032],{"class":911},[652,1312,943],{"class":903},[652,1314,946],{"class":930},[652,1316,1317,1319,1321,1323],{"class":899,"line":1074},[652,1318,1065],{"class":903},[652,1320,1068],{"class":903},[652,1322,1092],{"class":930},[652,1324,1325],{"class":911},", etapes\n",[652,1327,1328],{"class":899,"line":1081},[652,1329,1078],{"emptyLinePlaceholder":1077},[652,1331,1332],{"class":899,"line":1147},[652,1333,1334],{"class":1161},"# Sur un grand tableau trié, on compte les étapes\n",[652,1336,1337,1340],{"class":899,"line":1165},[652,1338,1339],{"class":903},"import",[652,1341,1342],{"class":911}," math\n",[652,1344,1346,1348,1350,1353,1356,1359,1361,1363,1365,1368,1370],{"class":899,"line":1345},17,[652,1347,1084],{"class":911},[652,1349,683],{"class":903},[652,1351,1352],{"class":930}," list",[652,1354,1355],{"class":911},"(",[652,1357,1358],{"class":930},"range",[652,1360,1355],{"class":911},[652,1362,1092],{"class":930},[652,1364,934],{"class":911},[652,1366,1367],{"class":930},"1001",[652,1369,1158],{"class":911},[652,1371,1372],{"class":1161},"# 1, 2, …, 1000\n",[652,1374,1376,1378,1380,1383,1386,1389],{"class":899,"line":1375},18,[652,1377,1150],{"class":930},[652,1379,1355],{"class":911},[652,1381,1382],{"class":918},"\"Recherche de 777 :\"",[652,1384,1385],{"class":911},", dichotomie(trie, ",[652,1387,1388],{"class":930},"777",[652,1390,1391],{"class":911},"))\n",[652,1393,1395,1397,1399,1402,1404,1406],{"class":899,"line":1394},19,[652,1396,1150],{"class":930},[652,1398,1355],{"class":911},[652,1400,1401],{"class":918},"\"Recherche de 1 :\"",[652,1403,1385],{"class":911},[652,1405,1092],{"class":930},[652,1407,1391],{"class":911},[652,1409,1411,1413,1415,1418,1421,1423],{"class":899,"line":1410},20,[652,1412,1150],{"class":930},[652,1414,1355],{"class":911},[652,1416,1417],{"class":918},"\"Étapes max théoriques :\"",[652,1419,1420],{"class":911},", math.ceil(math.log2(",[652,1422,1367],{"class":930},[652,1424,1425],{"class":911},")))\n",[593,1427,1429],{"id":1428},"le-variant-de-boucle","Le variant de boucle",[598,1431,1432,1433,1436,1437,1440,1441,876],{},"Pour la dichotomie, le bon outil de raisonnement est le ",[602,1434,1435],{},"variant"," :\nune quantité qui ",[602,1438,1439],{},"décroît strictement"," à chaque tour et qui ne peut\n",[602,1442,1443],{},"pas descendre indéfiniment",[598,1445,1446,1447,1489,1490,1493,1494,1497],{},"Posons ",[652,1448,1450,1468],{"className":1449},[655],[652,1451,1453],{"className":1452},[659],[661,1454,1455],{"xmlns":663},[665,1456,1457,1465],{},[668,1458,1459,1463],{},[1460,1461,1462],"mi",{},"v",[681,1464,683],{},[694,1466,1467],{"encoding":696},"v = ",[652,1469,1471],{"className":1470,"ariaHidden":702},[701],[652,1472,1474,1478,1483,1486],{"className":1473},[706],[652,1475],{"className":1476,"style":1477},[710],"height:0.4306em;",[652,1479,1462],{"className":1480,"style":1482},[715,1481],"mathnormal","margin-right:0.0359em;",[652,1484],{"className":1485,"style":757},[756],[652,1487,683],{"className":1488},[761]," ",[819,1491,1492],{},"droite - gauche",". C'est la ",[602,1495,1496],{},"taille de la zone de\nrecherche"," moins 1. À chaque tour :",[633,1499,1500,1511,1517],{},[636,1501,1502,1503,1506,1507,1510],{},"soit on trouve et on s'arrête (le ",[819,1504,1505],{},"while"," se termine par ",[819,1508,1509],{},"return","),",[636,1512,1513,1514,1516],{},"soit on fait ",[819,1515,869],{},", ce qui supprime au moins la moitié de\nl'intervalle ;",[636,1518,1513,1519,1521],{},[819,1520,875],{},", idem.",[598,1523,1524,1525,1489,1553,1556,1557,1585,1586,1639,1640,1642,1643],{},"Dans tous les cas, ",[652,1526,1528,1541],{"className":1527},[655],[652,1529,1531],{"className":1530},[659],[661,1532,1533],{"xmlns":663},[665,1534,1535,1539],{},[668,1536,1537],{},[1460,1538,1462],{},[694,1540,1462],{"encoding":696},[652,1542,1544],{"className":1543,"ariaHidden":702},[701],[652,1545,1547,1550],{"className":1546},[706],[652,1548],{"className":1549,"style":1477},[710],[652,1551,1462],{"className":1552,"style":1482},[715,1481],[602,1554,1555],{},"diminue strictement",". Comme ",[652,1558,1560,1573],{"className":1559},[655],[652,1561,1563],{"className":1562},[659],[661,1564,1565],{"xmlns":663},[665,1566,1567,1571],{},[668,1568,1569],{},[1460,1570,1462],{},[694,1572,1462],{"encoding":696},[652,1574,1576],{"className":1575,"ariaHidden":702},[701],[652,1577,1579,1582],{"className":1578},[706],[652,1580],{"className":1581,"style":1477},[710],[652,1583,1462],{"className":1584,"style":1482},[715,1481]," est un entier,\nil finira par devenir négatif (",[652,1587,1589,1608],{"className":1588},[655],[652,1590,1592],{"className":1591},[659],[661,1593,1594],{"xmlns":663},[665,1595,1596,1605],{},[668,1597,1598,1600,1602],{},[1460,1599,1462],{},[681,1601,1019],{},[674,1603,1604],{},"0",[694,1606,1607],{"encoding":696},"v \u003C 0",[652,1609,1611,1630],{"className":1610,"ariaHidden":702},[701],[652,1612,1614,1618,1621,1624,1627],{"className":1613},[706],[652,1615],{"className":1616,"style":1617},[710],"height:0.5782em;vertical-align:-0.0391em;",[652,1619,1462],{"className":1620,"style":1482},[715,1481],[652,1622],{"className":1623,"style":757},[756],[652,1625,1019],{"className":1626},[761],[652,1628],{"className":1629,"style":757},[756],[652,1631,1633,1636],{"className":1632},[706],[652,1634],{"className":1635,"style":790},[710],[652,1637,1604],{"className":1638},[715],", donc ",[819,1641,882],{},"), ce qui\narrête la boucle. ",[602,1644,1645],{},"L'algorithme termine donc toujours.",[1647,1648,1649,1655],"note",{},[598,1650,1651,1654],{},[602,1652,1653],{},"Variant vs invariant"," :",[633,1656,1657,1668],{},[636,1658,1659,1660,1663,1664,1667],{},"Un ",[602,1661,1662],{},"invariant"," est une propriété ",[602,1665,1666],{},"constante"," vraie d'un tour à l'autre\n(utile à la correction).",[636,1669,1659,1670,1672,1673,1676],{},[602,1671,1435],{}," est une grandeur ",[602,1674,1675],{},"qui décroît"," (utile à la terminaison).",[593,1678,1680,1681,1754],{"id":1679},"pourquoi-olognmathcalolog-nologn","Pourquoi ",[652,1682,1684,1715],{"className":1683},[655],[652,1685,1687],{"className":1686},[659],[661,1688,1689],{"xmlns":663},[665,1690,1691,1712],{},[668,1692,1693,1697,1700,1703,1706,1709],{},[1460,1694,1696],{"mathvariant":1695},"script","O",[681,1698,1355],{"stretchy":1699},"false",[1460,1701,1702],{},"log",[681,1704,1705],{},"⁡",[1460,1707,1708],{},"n",[681,1710,1711],{"stretchy":1699},")",[694,1713,1714],{"encoding":696},"\\mathcal{O}(\\log n)",[652,1716,1718],{"className":1717,"ariaHidden":702},[701],[652,1719,1721,1725,1730,1734,1743,1747,1750],{"className":1720},[706],[652,1722],{"className":1723,"style":1724},[710],"height:1em;vertical-align:-0.25em;",[652,1726,1696],{"className":1727,"style":1729},[715,1728],"mathcal","margin-right:0.0278em;",[652,1731,1355],{"className":1732},[1733],"mopen",[652,1735,1738,1739],{"className":1736},[1737],"mop","lo",[652,1740,1742],{"style":1741},"margin-right:0.0139em;","g",[652,1744],{"className":1745,"style":1746},[756],"margin-right:0.1667em;",[652,1748,1708],{"className":1749},[715,1481],[652,1751,1711],{"className":1752},[1753],"mclose"," ?",[598,1756,1757,1758,1761,1762,1654],{},"À chaque tour, la taille de la zone de recherche est ",[602,1759,1760],{},"divisée par deux",".\nEn partant d'une zone de taille ",[652,1763,1765,1778],{"className":1764},[655],[652,1766,1768],{"className":1767},[659],[661,1769,1770],{"xmlns":663},[665,1771,1772,1776],{},[668,1773,1774],{},[1460,1775,1708],{},[694,1777,1708],{"encoding":696},[652,1779,1781],{"className":1780,"ariaHidden":702},[701],[652,1782,1784,1787],{"className":1783},[706],[652,1785],{"className":1786,"style":1477},[710],[652,1788,1708],{"className":1789},[715,1481],[633,1791,1792,1834,1874],{},[636,1793,1794,1795,850],{},"après 1 tour : taille ",[652,1796,1798,1818],{"className":1797},[655],[652,1799,1801],{"className":1800},[659],[661,1802,1803],{"xmlns":663},[665,1804,1805,1815],{},[668,1806,1807,1809,1813],{},[1460,1808,1708],{},[1460,1810,1812],{"mathvariant":1811},"normal","\u002F",[674,1814,676],{},[694,1816,1817],{"encoding":696},"n \u002F 2",[652,1819,1821],{"className":1820,"ariaHidden":702},[701],[652,1822,1824,1827,1830],{"className":1823},[706],[652,1825],{"className":1826,"style":1724},[710],[652,1828,1708],{"className":1829},[715,1481],[652,1831,1833],{"className":1832},[715],"\u002F2",[636,1835,1836,1837,850],{},"après 2 tours : taille ",[652,1838,1840,1858],{"className":1839},[655],[652,1841,1843],{"className":1842},[659],[661,1844,1845],{"xmlns":663},[665,1846,1847,1855],{},[668,1848,1849,1851,1853],{},[1460,1850,1708],{},[1460,1852,1812],{"mathvariant":1811},[674,1854,1097],{},[694,1856,1857],{"encoding":696},"n \u002F 4",[652,1859,1861],{"className":1860,"ariaHidden":702},[701],[652,1862,1864,1867,1870],{"className":1863},[706],[652,1865],{"className":1866,"style":1724},[710],[652,1868,1708],{"className":1869},[715,1481],[652,1871,1873],{"className":1872},[715],"\u002F4",[636,1875,1876,1877,1908,1909,876],{},"après ",[652,1878,1880,1894],{"className":1879},[655],[652,1881,1883],{"className":1882},[659],[661,1884,1885],{"xmlns":663},[665,1886,1887,1892],{},[668,1888,1889],{},[1460,1890,1891],{},"k",[694,1893,1891],{"encoding":696},[652,1895,1897],{"className":1896,"ariaHidden":702},[701],[652,1898,1900,1904],{"className":1899},[706],[652,1901],{"className":1902,"style":1903},[710],"height:0.6944em;",[652,1905,1891],{"className":1906,"style":1907},[715,1481],"margin-right:0.0315em;"," tours : taille ",[652,1910,1912,1934],{"className":1911},[655],[652,1913,1915],{"className":1914},[659],[661,1916,1917],{"xmlns":663},[665,1918,1919,1931],{},[668,1920,1921,1923,1925],{},[1460,1922,1708],{},[1460,1924,1812],{"mathvariant":1811},[671,1926,1927,1929],{},[674,1928,676],{},[1460,1930,1891],{},[694,1932,1933],{"encoding":696},"n \u002F 2^k",[652,1935,1937],{"className":1936,"ariaHidden":702},[701],[652,1938,1940,1944,1947,1950],{"className":1939},[706],[652,1941],{"className":1942,"style":1943},[710],"height:1.0991em;vertical-align:-0.25em;",[652,1945,1708],{"className":1946},[715,1481],[652,1948,1812],{"className":1949},[715],[652,1951,1953,1956],{"className":1952},[715],[652,1954,676],{"className":1955},[715],[652,1957,1959],{"className":1958},[722],[652,1960,1962],{"className":1961},[726],[652,1963,1965],{"className":1964},[730],[652,1966,1969],{"className":1967,"style":1968},[734],"height:0.8491em;",[652,1970,1971,1974],{"style":737},[652,1972],{"className":1973,"style":742},[741],[652,1975,1977],{"className":1976},[746,747,748,749],[652,1978,1891],{"className":1979,"style":1907},[715,1481,749],[598,1981,1982,1983,2065,2066,2094,2095,2220,2221,2224,2225,1654],{},"On s'arrête quand cette taille atteint 0 ou 1, c'est-à-dire quand\n",[652,1984,1986,2008],{"className":1985},[655],[652,1987,1989],{"className":1988},[659],[661,1990,1991],{"xmlns":663},[665,1992,1993,2005],{},[668,1994,1995,2001,2003],{},[671,1996,1997,1999],{},[674,1998,676],{},[1460,2000,1891],{},[681,2002,689],{},[1460,2004,1708],{},[694,2006,2007],{"encoding":696},"2^k \\geq n",[652,2009,2011,2056],{"className":2010,"ariaHidden":702},[701],[652,2012,2014,2018,2047,2050,2053],{"className":2013},[706],[652,2015],{"className":2016,"style":2017},[710],"height:0.9851em;vertical-align:-0.136em;",[652,2019,2021,2024],{"className":2020},[715],[652,2022,676],{"className":2023},[715],[652,2025,2027],{"className":2026},[722],[652,2028,2030],{"className":2029},[726],[652,2031,2033],{"className":2032},[730],[652,2034,2036],{"className":2035,"style":1968},[734],[652,2037,2038,2041],{"style":737},[652,2039],{"className":2040,"style":742},[741],[652,2042,2044],{"className":2043},[746,747,748,749],[652,2045,1891],{"className":2046,"style":1907},[715,1481,749],[652,2048],{"className":2049,"style":757},[756],[652,2051,689],{"className":2052},[761],[652,2054],{"className":2055,"style":757},[756],[652,2057,2059,2062],{"className":2058},[706],[652,2060],{"className":2061,"style":1477},[710],[652,2063,1708],{"className":2064},[715,1481],". Le plus petit ",[652,2067,2069,2082],{"className":2068},[655],[652,2070,2072],{"className":2071},[659],[661,2073,2074],{"xmlns":663},[665,2075,2076,2080],{},[668,2077,2078],{},[1460,2079,1891],{},[694,2081,1891],{"encoding":696},[652,2083,2085],{"className":2084,"ariaHidden":702},[701],[652,2086,2088,2091],{"className":2087},[706],[652,2089],{"className":2090,"style":1903},[710],[652,2092,1891],{"className":2093,"style":1907},[715,1481]," qui convient est ",[652,2096,2098,2133],{"className":2097},[655],[652,2099,2101],{"className":2100},[659],[661,2102,2103],{"xmlns":663},[665,2104,2105,2130],{},[668,2106,2107,2109,2111,2114,2125,2127],{},[1460,2108,1891],{},[681,2110,683],{},[681,2112,2113],{"stretchy":1699},"⌈",[2115,2116,2117,2123],"msub",{},[668,2118,2119,2121],{},[1460,2120,1702],{},[681,2122,1705],{},[674,2124,676],{},[1460,2126,1708],{},[681,2128,2129],{"stretchy":1699},"⌉",[694,2131,2132],{"encoding":696},"k = \\lceil \\log_2 n\n\\rceil",[652,2134,2136,2154],{"className":2135,"ariaHidden":702},[701],[652,2137,2139,2142,2145,2148,2151],{"className":2138},[706],[652,2140],{"className":2141,"style":1903},[710],[652,2143,1891],{"className":2144,"style":1907},[715,1481],[652,2146],{"className":2147,"style":757},[756],[652,2149,683],{"className":2150},[761],[652,2152],{"className":2153,"style":757},[756],[652,2155,2157,2160,2163,2211,2214,2217],{"className":2156},[706],[652,2158],{"className":2159,"style":1724},[710],[652,2161,2113],{"className":2162},[1733],[652,2164,2166,2171],{"className":2165},[1737],[652,2167,1738,2169],{"className":2168},[1737],[652,2170,1742],{"style":1741},[652,2172,2174],{"className":2173},[722],[652,2175,2178,2202],{"className":2176},[726,2177],"vlist-t2",[652,2179,2181,2197],{"className":2180},[730],[652,2182,2185],{"className":2183,"style":2184},[734],"height:0.207em;",[652,2186,2188,2191],{"style":2187},"top:-2.4559em;margin-right:0.05em;",[652,2189],{"className":2190,"style":742},[741],[652,2192,2194],{"className":2193},[746,747,748,749],[652,2195,676],{"className":2196},[715,749],[652,2198,2201],{"className":2199},[2200],"vlist-s","​",[652,2203,2205],{"className":2204},[730],[652,2206,2209],{"className":2207,"style":2208},[734],"height:0.2441em;",[652,2210],{},[652,2212],{"className":2213,"style":1746},[756],[652,2215,1708],{"className":2216},[715,1481],[652,2218,2129],{"className":2219},[1753],". La complexité est donc ",[602,2222,2223],{},"logarithmique"," en ",[652,2226,2228,2241],{"className":2227},[655],[652,2229,2231],{"className":2230},[659],[661,2232,2233],{"xmlns":663},[665,2234,2235,2239],{},[668,2236,2237],{},[1460,2238,1708],{},[694,2240,1708],{"encoding":696},[652,2242,2244],{"className":2243,"ariaHidden":702},[701],[652,2245,2247,2250],{"className":2246},[706],[652,2248],{"className":2249,"style":1477},[710],[652,2251,1708],{"className":2252},[715,1481],[652,2254,2257],{"className":2255},[2256],"katex-display",[652,2258,2260,2284],{"className":2259},[655],[652,2261,2263],{"className":2262},[659],[661,2264,2266],{"xmlns":663,"display":2265},"block",[665,2267,2268,2282],{},[668,2269,2270,2272,2274,2276,2278,2280],{},[1460,2271,1696],{"mathvariant":1695},[681,2273,1355],{"stretchy":1699},[1460,2275,1702],{},[681,2277,1705],{},[1460,2279,1708],{},[681,2281,1711],{"stretchy":1699},[694,2283,1714],{"encoding":696},[652,2285,2287],{"className":2286,"ariaHidden":702},[701],[652,2288,2290,2293,2296,2299,2304,2307,2310],{"className":2289},[706],[652,2291],{"className":2292,"style":1724},[710],[652,2294,1696],{"className":2295,"style":1729},[715,1728],[652,2297,1355],{"className":2298},[1733],[652,2300,1738,2302],{"className":2301},[1737],[652,2303,1742],{"style":1741},[652,2305],{"className":2306,"style":1746},[756],[652,2308,1708],{"className":2309},[715,1481],[652,2311,1711],{"className":2312},[1753],[800,2314,2315],{},[598,2316,2317,2318,2403,2404,2407,2408,876],{},"Pour vous donner une idée concrète : pour ",[652,2319,2321,2344],{"className":2320},[655],[652,2322,2324],{"className":2323},[659],[661,2325,2326],{"xmlns":663},[665,2327,2328,2341],{},[668,2329,2330,2332,2334],{},[1460,2331,1708],{},[681,2333,683],{},[671,2335,2336,2338],{},[674,2337,1172],{},[674,2339,2340],{},"9",[694,2342,2343],{"encoding":696},"n = 10^9",[652,2345,2347,2365],{"className":2346,"ariaHidden":702},[701],[652,2348,2350,2353,2356,2359,2362],{"className":2349},[706],[652,2351],{"className":2352,"style":1477},[710],[652,2354,1708],{"className":2355},[715,1481],[652,2357],{"className":2358,"style":757},[756],[652,2360,683],{"className":2361},[761],[652,2363],{"className":2364,"style":757},[756],[652,2366,2368,2371,2374],{"className":2367},[706],[652,2369],{"className":2370,"style":711},[710],[652,2372,1092],{"className":2373},[715],[652,2375,2377,2380],{"className":2376},[715],[652,2378,1604],{"className":2379},[715],[652,2381,2383],{"className":2382},[722],[652,2384,2386],{"className":2385},[726],[652,2387,2389],{"className":2388},[730],[652,2390,2392],{"className":2391,"style":711},[734],[652,2393,2394,2397],{"style":737},[652,2395],{"className":2396,"style":742},[741],[652,2398,2400],{"className":2399},[746,747,748,749],[652,2401,2340],{"className":2402},[715,749]," (un milliard !),\nla recherche dichotomique trouve en ",[602,2405,2406],{},"30 étapes"," au maximum. La recherche\nlinéaire en exigerait ",[602,2409,2410],{},"un milliard",[2412,2413,2415,2421],"self-eval-qcm",{"correct":1604,"slug":2414},"dichotomie-c1",[2416,2417,2418],"template",{"v-slot:question":894},[598,2419,2420],{},"Pour un tableau trié de 1 048 576 éléments, combien d'étapes au pire pour la dichotomie ?",[2416,2422,2423],{"v-slot:choices":894},[633,2424,2425,2428,2431],{},[636,2426,2427],{},"20",[636,2429,2430],{},"524 288",[636,2432,2433],{},"1 048 576",[593,2435,2437],{"id":2436},"pièges-courants","Pièges courants",[633,2439,2440,2446,2461,2484],{},[636,2441,2442,2445],{},[602,2443,2444],{},"Oublier que le tableau doit être trié"," : la dichotomie ne marche pas sur\nun tableau quelconque.",[636,2447,2448,2457,2458,2460],{},[602,2449,2450,2453,2454],{},[819,2451,2452],{},"m = (gauche + droite) \u002F 2"," au lieu de ",[819,2455,2456],{},"\u002F\u002F 2"," : la division ",[819,2459,1812],{},"\nretourne un flottant en Python, qui ne peut pas servir d'indice.",[636,2462,2463,2466,2467,2453,2470,2472,2473,2476,2477,2480,2481,2483],{},[602,2464,2465],{},"Bornes mal mises à jour"," : ",[819,2468,2469],{},"gauche = m",[819,2471,869],{},"\ncause des ",[602,2474,2475],{},"boucles infinies"," quand ",[819,2478,2479],{},"gauche == droite",". La case ",[819,2482,860],{},"\nvient d'être testée — il faut l'exclure.",[636,2485,2486,2466,2489,1639,2492,2495,2496,2499],{},[602,2487,2488],{},"Tableau vide",[819,2490,2491],{},"len(t) - 1 = -1",[819,2493,2494],{},"gauche = 0 > droite = -1",",\ndonc on n'entre pas dans la boucle, et on retourne ",[819,2497,2498],{},"-1",". Cas géré sans\ncas particulier.",[2501,2502,2503],"warning",{},[598,2504,2505,2506,2509],{},"Si le tableau ",[602,2507,2508],{},"n'est pas trié",", ne convoquez pas la dichotomie. Soit vous\ntriez d'abord (mais alors le coût total est dominé par le tri), soit vous\nrestez sur une recherche linéaire.",[2412,2511,2513,2518],{"correct":1092,"slug":2512},"dichotomie-c2",[2416,2514,2515],{"v-slot:question":894},[598,2516,2517],{},"Le prérequis indispensable pour appliquer la dichotomie est :",[2416,2519,2520],{"v-slot:choices":894},[633,2521,2522,2525,2528],{},[636,2523,2524],{},"tableau de taille puissance de 2",[636,2526,2527],{},"tableau trié",[636,2529,2530],{},"tableau sans doublons",[593,2532,2534],{"id":2533},"pour-aller-plus-loin","Pour aller plus loin",[598,2536,2537,2538,2541,2542,2545,2546,2601,2602,876],{},"La dichotomie est l'archétype de la ",[602,2539,2540],{},"stratégie diviser pour régner",", que\nvous reverrez en Terminale (tri fusion, tri rapide). À chaque étape, on\nréduit le problème à un sous-problème ",[602,2543,2544],{},"deux fois plus petit",", jusqu'à un\ncas de base. Cette stratégie produit naturellement des complexités en\n",[652,2547,2549,2572],{"className":2548},[655],[652,2550,2552],{"className":2551},[659],[661,2553,2554],{"xmlns":663},[665,2555,2556,2570],{},[668,2557,2558,2560,2562,2564,2566,2568],{},[1460,2559,1696],{"mathvariant":1695},[681,2561,1355],{"stretchy":1699},[1460,2563,1702],{},[681,2565,1705],{},[1460,2567,1708],{},[681,2569,1711],{"stretchy":1699},[694,2571,1714],{"encoding":696},[652,2573,2575],{"className":2574,"ariaHidden":702},[701],[652,2576,2578,2581,2584,2587,2592,2595,2598],{"className":2577},[706],[652,2579],{"className":2580,"style":1724},[710],[652,2582,1696],{"className":2583,"style":1729},[715,1728],[652,2585,1355],{"className":2586},[1733],[652,2588,1738,2590],{"className":2589},[1737],[652,2591,1742],{"style":1741},[652,2593],{"className":2594,"style":1746},[756],[652,2596,1708],{"className":2597},[715,1481],[652,2599,1711],{"className":2600},[1753]," ou ",[652,2603,2605,2631],{"className":2604},[655],[652,2606,2608],{"className":2607},[659],[661,2609,2610],{"xmlns":663},[665,2611,2612,2628],{},[668,2613,2614,2616,2618,2620,2622,2624,2626],{},[1460,2615,1696],{"mathvariant":1695},[681,2617,1355],{"stretchy":1699},[1460,2619,1708],{},[1460,2621,1702],{},[681,2623,1705],{},[1460,2625,1708],{},[681,2627,1711],{"stretchy":1699},[694,2629,2630],{"encoding":696},"\\mathcal{O}(n \\log n)",[652,2632,2634],{"className":2633,"ariaHidden":702},[701],[652,2635,2637,2640,2643,2646,2649,2652,2657,2660,2663],{"className":2636},[706],[652,2638],{"className":2639,"style":1724},[710],[652,2641,1696],{"className":2642,"style":1729},[715,1728],[652,2644,1355],{"className":2645},[1733],[652,2647,1708],{"className":2648},[715,1481],[652,2650],{"className":2651,"style":1746},[756],[652,2653,1738,2655],{"className":2654},[1737],[652,2656,1742],{"style":1741},[652,2658],{"className":2659,"style":1746},[756],[652,2661,1708],{"className":2662},[715,1481],[652,2664,1711],{"className":2665},[1753],[2667,2668,2669],"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 .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}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);}",{"title":894,"searchDepth":915,"depth":915,"links":2671},[2672,2673,2674,2675,2676,2678,2679],{"id":595,"depth":915,"text":596},{"id":623,"depth":915,"text":624},{"id":810,"depth":915,"text":811},{"id":1428,"depth":915,"text":1429},{"id":1679,"depth":915,"text":2677},"Pourquoi O(log⁡n)\\mathcal{O}(\\log n)O(logn) ?",{"id":2436,"depth":915,"text":2437},{"id":2533,"depth":915,"text":2534},"Chercher dans un tableau trié en divisant l'espace de recherche par deux à chaque étape — coût $\\mathcal{O}(\\log n)$.",null,"md",{},[2685,2689],{"slug":2414,"kind":2686,"question":2420,"choices":2687,"correct":1604,"multi":2688,"explanation":894},"qcm",[2427,2430,2433],false,{"slug":2512,"kind":2686,"question":2517,"choices":2690,"correct":1092,"multi":2688,"explanation":894},[2524,2527,2530],{"title":429,"description":2680},"UQ9RkZ7GRIIe4_DWjUkxBNpgxJNEeRNwzJIhWSZ5qAw",{"id":2694,"title":2695,"bo":2696,"body":2697,"description":3288,"eval":2681,"extension":2682,"labo":2681,"meta":3289,"navigation":2688,"path":3290,"quizzes":3291,"readingTime":900,"seo":3292,"stem":3293,"tp":2681,"__hash__":3294},"coursMemo\u002F1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie.memo.md","Recherche dichotomique — l'essentiel",[588],{"type":590,"value":2698,"toc":3280},[2699,2703,2768,2772,2909,2913,2982,2986,3028,3030,3152,3156,3277],[593,2700,2702],{"id":2701},"en-une-phrase","En une phrase",[598,2704,2705,2706,2708,2709,2712,2713,876],{},"Dans un tableau ",[602,2707,608],{},", on trouve un élément en ",[602,2710,2711],{},"divisant par deux","\nl'espace de recherche à chaque étape — coût ",[652,2714,2716,2739],{"className":2715},[655],[652,2717,2719],{"className":2718},[659],[661,2720,2721],{"xmlns":663},[665,2722,2723,2737],{},[668,2724,2725,2727,2729,2731,2733,2735],{},[1460,2726,1696],{"mathvariant":1695},[681,2728,1355],{"stretchy":1699},[1460,2730,1702],{},[681,2732,1705],{},[1460,2734,1708],{},[681,2736,1711],{"stretchy":1699},[694,2738,1714],{"encoding":696},[652,2740,2742],{"className":2741,"ariaHidden":702},[701],[652,2743,2745,2748,2751,2754,2759,2762,2765],{"className":2744},[706],[652,2746],{"className":2747,"style":1724},[710],[652,2749,1696],{"className":2750,"style":1729},[715,1728],[652,2752,1355],{"className":2753},[1733],[652,2755,1738,2757],{"className":2756},[1737],[652,2758,1742],{"style":1741},[652,2760],{"className":2761,"style":1746},[756],[652,2763,1708],{"className":2764},[715,1481],[652,2766,1711],{"className":2767},[1753],[593,2769,2771],{"id":2770},"à-mémoriser","À mémoriser",[633,2773,2774,2779,2789,2801,2830,2840,2848],{},[636,2775,2776,2778],{},[602,2777,816],{}," : le tableau doit être trié dans l'ordre croissant.",[636,2780,2781,1489,2784,832,2786,2788],{},[602,2782,2783],{},"Indices",[819,2785,831],{},[819,2787,835],{}," qui délimitent la zone candidate.",[636,2790,2791,2466,2794,2796,2797,2800],{},[602,2792,2793],{},"Milieu",[819,2795,849],{}," (division ",[602,2798,2799],{},"entière",").",[636,2802,2803,2806,2807],{},[602,2804,2805],{},"Trois cas"," :\n",[633,2808,2809,2816,2823],{},[636,2810,2811,2813,2814,876],{},[819,2812,856],{}," → trouvé, return ",[819,2815,860],{},[636,2817,2818,2820,2821,876],{},[819,2819,865],{}," → ",[819,2822,869],{},[636,2824,2825,2820,2828,876],{},[819,2826,2827],{},"t[m] > cible",[819,2829,875],{},[636,2831,2832,2466,2835,2837,2838,876],{},[602,2833,2834],{},"Échec",[819,2836,882],{},", return ",[819,2839,2498],{},[636,2841,2842,2466,2845,2847],{},[602,2843,2844],{},"Variant de boucle",[819,2846,1492],{}," décroît strictement → terminaison.",[636,2849,2850,2466,2853,2908],{},[602,2851,2852],{},"Complexité",[652,2854,2856,2879],{"className":2855},[655],[652,2857,2859],{"className":2858},[659],[661,2860,2861],{"xmlns":663},[665,2862,2863,2877],{},[668,2864,2865,2867,2869,2871,2873,2875],{},[1460,2866,1696],{"mathvariant":1695},[681,2868,1355],{"stretchy":1699},[1460,2870,1702],{},[681,2872,1705],{},[1460,2874,1708],{},[681,2876,1711],{"stretchy":1699},[694,2878,1714],{"encoding":696},[652,2880,2882],{"className":2881,"ariaHidden":702},[701],[652,2883,2885,2888,2891,2894,2899,2902,2905],{"className":2884},[706],[652,2886],{"className":2887,"style":1724},[710],[652,2889,1696],{"className":2890,"style":1729},[715,1728],[652,2892,1355],{"className":2893},[1733],[652,2895,1738,2897],{"className":2896},[1737],[652,2898,1742],{"style":1741},[652,2900],{"className":2901,"style":1746},[756],[652,2903,1708],{"className":2904},[715,1481],[652,2906,1711],{"className":2907},[1753]," au pire.",[593,2910,2912],{"id":2911},"ordres-de-grandeur","Ordres de grandeur",[2914,2915,2916,2956],"table",{},[2917,2918,2919],"thead",{},[2920,2921,2922,2953],"tr",{},[2923,2924,2925],"th",{},[652,2926,2928,2941],{"className":2927},[655],[652,2929,2931],{"className":2930},[659],[661,2932,2933],{"xmlns":663},[665,2934,2935,2939],{},[668,2936,2937],{},[1460,2938,1708],{},[694,2940,1708],{"encoding":696},[652,2942,2944],{"className":2943,"ariaHidden":702},[701],[652,2945,2947,2950],{"className":2946},[706],[652,2948],{"className":2949,"style":1477},[710],[652,2951,1708],{"className":2952},[715,1481],[2923,2954,2955],{},"Étapes max",[2957,2958,2959,2967,2974],"tbody",{},[2920,2960,2961,2965],{},[2962,2963,2964],"td",{},"1 000",[2962,2966,1172],{},[2920,2968,2969,2972],{},[2962,2970,2971],{},"1 000 000",[2962,2973,2427],{},[2920,2975,2976,2979],{},[2962,2977,2978],{},"1 000 000 000",[2962,2980,2981],{},"30",[593,2983,2985],{"id":2984},"à-savoir-reconnaître","À savoir reconnaître",[2914,2987,2988,2998],{},[2917,2989,2990],{},[2920,2991,2992,2995],{},[2923,2993,2994],{},"Situation",[2923,2996,2997],{},"À utiliser",[2957,2999,3000,3012,3020],{},[2920,3001,3002,3009],{},[2962,3003,3004,3005,3008],{},"Tableau ",[602,3006,3007],{},"non trié",", recherche ponctuelle",[2962,3010,3011],{},"Recherche linéaire",[2920,3013,3014,3017],{},[2962,3015,3016],{},"Tableau trié, recherche ponctuelle",[2962,3018,3019],{},"Dichotomie",[2920,3021,3022,3025],{},[2962,3023,3024],{},"Beaucoup de recherches sur les mêmes données",[2962,3026,3027],{},"Trier une fois, dichotomie ensuite",[593,3029,2437],{"id":2436},[633,3031,3032,3035,3042,3049],{},[636,3033,3034],{},"Tableau non trié → résultat erroné.",[636,3036,3037,2453,3039,3041],{},[819,3038,1812],{},[819,3040,981],{}," → indice flottant, plantage.",[636,3043,3044,2453,3046,3048],{},[819,3045,2469],{},[819,3047,869],{}," → boucle infinie.",[636,3050,3051,3052,3096,3097,876],{},"Penser que la complexité est ",[652,3053,3055,3075],{"className":3054},[655],[652,3056,3058],{"className":3057},[659],[661,3059,3060],{"xmlns":663},[665,3061,3062,3072],{},[668,3063,3064,3066,3068,3070],{},[1460,3065,1696],{"mathvariant":1695},[681,3067,1355],{"stretchy":1699},[1460,3069,1708],{},[681,3071,1711],{"stretchy":1699},[694,3073,3074],{"encoding":696},"\\mathcal{O}(n)",[652,3076,3078],{"className":3077,"ariaHidden":702},[701],[652,3079,3081,3084,3087,3090,3093],{"className":3080},[706],[652,3082],{"className":3083,"style":1724},[710],[652,3085,1696],{"className":3086,"style":1729},[715,1728],[652,3088,1355],{"className":3089},[1733],[652,3091,1708],{"className":3092},[715,1481],[652,3094,1711],{"className":3095},[1753]," — non, ",[652,3098,3100,3123],{"className":3099},[655],[652,3101,3103],{"className":3102},[659],[661,3104,3105],{"xmlns":663},[665,3106,3107,3121],{},[668,3108,3109,3111,3113,3115,3117,3119],{},[1460,3110,1696],{"mathvariant":1695},[681,3112,1355],{"stretchy":1699},[1460,3114,1702],{},[681,3116,1705],{},[1460,3118,1708],{},[681,3120,1711],{"stretchy":1699},[694,3122,1714],{"encoding":696},[652,3124,3126],{"className":3125,"ariaHidden":702},[701],[652,3127,3129,3132,3135,3138,3143,3146,3149],{"className":3128},[706],[652,3130],{"className":3131,"style":1724},[710],[652,3133,1696],{"className":3134,"style":1729},[715,1728],[652,3136,1355],{"className":3137},[1733],[652,3139,1738,3141],{"className":3140},[1737],[652,3142,1742],{"style":1741},[652,3144],{"className":3145,"style":1746},[756],[652,3147,1708],{"className":3148},[715,1481],[652,3150,1711],{"className":3151},[1753],[593,3153,3155],{"id":3154},"syntaxe-python","Syntaxe Python",[889,3157,3159],{"className":891,"code":3158,"language":893,"meta":894,"style":894},"def dichotomie(t, cible):\n    gauche, droite = 0, len(t) - 1\n    while gauche \u003C= droite:\n        m = (gauche + droite) \u002F\u002F 2\n        if t[m] == cible:\n            return m\n        elif t[m] \u003C cible:\n            gauche = m + 1\n        else:\n            droite = m - 1\n    return -1\n",[819,3160,3161,3169,3187,3197,3213,3223,3229,3239,3251,3257,3269],{"__ignoreMap":894},[652,3162,3163,3165,3167],{"class":899,"line":900},[652,3164,904],{"class":903},[652,3166,908],{"class":907},[652,3168,912],{"class":911},[652,3170,3171,3173,3175,3177,3179,3181,3183,3185],{"class":899,"line":915},[652,3172,925],{"class":911},[652,3174,683],{"class":903},[652,3176,931],{"class":930},[652,3178,934],{"class":911},[652,3180,937],{"class":930},[652,3182,940],{"class":911},[652,3184,943],{"class":903},[652,3186,946],{"class":930},[652,3188,3189,3191,3193,3195],{"class":899,"line":922},[652,3190,952],{"class":903},[652,3192,955],{"class":911},[652,3194,958],{"class":903},[652,3196,961],{"class":911},[652,3198,3199,3201,3203,3205,3207,3209,3211],{"class":899,"line":949},[652,3200,967],{"class":911},[652,3202,683],{"class":903},[652,3204,972],{"class":911},[652,3206,975],{"class":903},[652,3208,978],{"class":911},[652,3210,981],{"class":903},[652,3212,984],{"class":930},[652,3214,3215,3217,3219,3221],{"class":899,"line":964},[652,3216,990],{"class":903},[652,3218,993],{"class":911},[652,3220,996],{"class":903},[652,3222,999],{"class":911},[652,3224,3225,3227],{"class":899,"line":987},[652,3226,1005],{"class":903},[652,3228,1008],{"class":911},[652,3230,3231,3233,3235,3237],{"class":899,"line":1002},[652,3232,1014],{"class":903},[652,3234,993],{"class":911},[652,3236,1019],{"class":903},[652,3238,999],{"class":911},[652,3240,3241,3243,3245,3247,3249],{"class":899,"line":1011},[652,3242,1027],{"class":911},[652,3244,683],{"class":903},[652,3246,1032],{"class":911},[652,3248,975],{"class":903},[652,3250,946],{"class":930},[652,3252,3253,3255],{"class":899,"line":1024},[652,3254,1042],{"class":903},[652,3256,1045],{"class":911},[652,3258,3259,3261,3263,3265,3267],{"class":899,"line":1039},[652,3260,1051],{"class":911},[652,3262,683],{"class":903},[652,3264,1032],{"class":911},[652,3266,943],{"class":903},[652,3268,946],{"class":930},[652,3270,3271,3273,3275],{"class":899,"line":1048},[652,3272,1065],{"class":903},[652,3274,1068],{"class":903},[652,3276,1071],{"class":930},[2667,3278,3279],{},"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: 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":894,"searchDepth":915,"depth":915,"links":3281},[3282,3283,3284,3285,3286,3287],{"id":2701,"depth":915,"text":2702},{"id":2770,"depth":915,"text":2771},{"id":2911,"depth":915,"text":2912},{"id":2984,"depth":915,"text":2985},{"id":2436,"depth":915,"text":2437},{"id":3154,"depth":915,"text":3155},"Mémo de révision sur la recherche dans un tableau trié en $\\mathcal{O}(\\log n)$.",{},"\u002Fnsi-1ere\u002Fph-algorithmique\u002Fdichotomie.memo",[],{"title":2695,"description":3288},"1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie.memo","vpKDQYbQ18_qLdLFDM5B-rCK4k78ZXIhWF6vN_5M5vI",{"id":3296,"title":3297,"body":3298,"description":4307,"extension":2682,"meta":4308,"navigation":2688,"path":4309,"seo":4310,"stem":4311,"__hash__":4312},"coursSlides\u002F1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie.slides.md","Recherche dichotomique — diapositives",{"type":590,"value":3299,"toc":4305},[3300,3305,3308,3311,3315,3318,3432,3434,3437,3443,3446,3448,3450,3492,3494,3498,3618,3620,3623,3632,3638,3640,3697,3785,3988,3990,3994,4063,4065,4069,4089,4091,4094,4097,4099,4103,4303],[3301,3302,3304],"h1",{"id":3303},"chercher-en-divisant-par-deux","Chercher en divisant par deux",[598,3306,3307],{},"Recherche dichotomique dans un tableau trié.",[3309,3310],"hr",{},[3301,3312,3314],{"id":3313},"lintuition","L'intuition",[598,3316,3317],{},"Jeu « plus chaud, plus froid » entre 1 et 100.",[633,3319,3320,3327],{},[636,3321,3322,3323,3326],{},"Vous proposez ",[602,3324,3325],{},"toujours le milieu"," de l'intervalle restant",[636,3328,3329,3330,1711],{},"En 7 questions au max, vous trouvez (",[652,3331,3333,3358],{"className":3332},[655],[652,3334,3336],{"className":3335},[659],[661,3337,3338],{"xmlns":663},[665,3339,3340,3356],{},[668,3341,3342,3348,3350,3352,3354],{},[671,3343,3344,3346],{},[674,3345,676],{},[674,3347,679],{},[681,3349,683],{},[674,3351,686],{},[681,3353,689],{},[674,3355,692],{},[694,3357,697],{"encoding":696},[652,3359,3361,3405,3423],{"className":3360,"ariaHidden":702},[701],[652,3362,3364,3367,3396,3399,3402],{"className":3363},[706],[652,3365],{"className":3366,"style":711},[710],[652,3368,3370,3373],{"className":3369},[715],[652,3371,676],{"className":3372},[715],[652,3374,3376],{"className":3375},[722],[652,3377,3379],{"className":3378},[726],[652,3380,3382],{"className":3381},[730],[652,3383,3385],{"className":3384,"style":711},[734],[652,3386,3387,3390],{"style":737},[652,3388],{"className":3389,"style":742},[741],[652,3391,3393],{"className":3392},[746,747,748,749],[652,3394,679],{"className":3395},[715,749],[652,3397],{"className":3398,"style":757},[756],[652,3400,683],{"className":3401},[761],[652,3403],{"className":3404,"style":757},[756],[652,3406,3408,3411,3414,3417,3420],{"className":3407},[706],[652,3409],{"className":3410,"style":771},[710],[652,3412,686],{"className":3413},[715],[652,3415],{"className":3416,"style":757},[756],[652,3418,689],{"className":3419},[761],[652,3421],{"className":3422,"style":757},[756],[652,3424,3426,3429],{"className":3425},[706],[652,3427],{"className":3428,"style":790},[710],[652,3430,692],{"className":3431},[715],[3309,3433],{},[3301,3435,816],{"id":3436},"prérequis",[598,3438,3439,3440,3442],{},"Le tableau doit être ",[602,3441,608],{}," dans l'ordre croissant.",[598,3444,3445],{},"Sinon : la dichotomie ne fonctionne pas.",[3309,3447],{},[3301,3449,811],{"id":810},[842,3451,3452,3460,3489],{},[636,3453,3454,934,3457],{},[819,3455,3456],{},"gauche = 0",[819,3458,3459],{},"droite = n - 1",[636,3461,3462,3463,2806,3466],{},"Tant que ",[819,3464,3465],{},"gauche \u003C= droite",[633,3467,3468,3472,3478,3484],{},[636,3469,3470],{},[819,3471,849],{},[636,3473,3474,3475,3477],{},"Si ",[819,3476,856],{}," → trouvé !",[636,3479,3474,3480,2820,3482],{},[819,3481,865],{},[819,3483,869],{},[636,3485,3486,3487],{},"Sinon → ",[819,3488,875],{},[636,3490,3491],{},"Sinon : pas trouvé",[3309,3493],{},[3301,3495,3497],{"id":3496},"le-code","Le code",[889,3499,3500],{"className":891,"code":3158,"language":893,"meta":894,"style":894},[819,3501,3502,3510,3528,3538,3554,3564,3570,3580,3592,3598,3610],{"__ignoreMap":894},[652,3503,3504,3506,3508],{"class":899,"line":900},[652,3505,904],{"class":903},[652,3507,908],{"class":907},[652,3509,912],{"class":911},[652,3511,3512,3514,3516,3518,3520,3522,3524,3526],{"class":899,"line":915},[652,3513,925],{"class":911},[652,3515,683],{"class":903},[652,3517,931],{"class":930},[652,3519,934],{"class":911},[652,3521,937],{"class":930},[652,3523,940],{"class":911},[652,3525,943],{"class":903},[652,3527,946],{"class":930},[652,3529,3530,3532,3534,3536],{"class":899,"line":922},[652,3531,952],{"class":903},[652,3533,955],{"class":911},[652,3535,958],{"class":903},[652,3537,961],{"class":911},[652,3539,3540,3542,3544,3546,3548,3550,3552],{"class":899,"line":949},[652,3541,967],{"class":911},[652,3543,683],{"class":903},[652,3545,972],{"class":911},[652,3547,975],{"class":903},[652,3549,978],{"class":911},[652,3551,981],{"class":903},[652,3553,984],{"class":930},[652,3555,3556,3558,3560,3562],{"class":899,"line":964},[652,3557,990],{"class":903},[652,3559,993],{"class":911},[652,3561,996],{"class":903},[652,3563,999],{"class":911},[652,3565,3566,3568],{"class":899,"line":987},[652,3567,1005],{"class":903},[652,3569,1008],{"class":911},[652,3571,3572,3574,3576,3578],{"class":899,"line":1002},[652,3573,1014],{"class":903},[652,3575,993],{"class":911},[652,3577,1019],{"class":903},[652,3579,999],{"class":911},[652,3581,3582,3584,3586,3588,3590],{"class":899,"line":1011},[652,3583,1027],{"class":911},[652,3585,683],{"class":903},[652,3587,1032],{"class":911},[652,3589,975],{"class":903},[652,3591,946],{"class":930},[652,3593,3594,3596],{"class":899,"line":1024},[652,3595,1042],{"class":903},[652,3597,1045],{"class":911},[652,3599,3600,3602,3604,3606,3608],{"class":899,"line":1039},[652,3601,1051],{"class":911},[652,3603,683],{"class":903},[652,3605,1032],{"class":911},[652,3607,943],{"class":903},[652,3609,946],{"class":930},[652,3611,3612,3614,3616],{"class":899,"line":1048},[652,3613,1065],{"class":903},[652,3615,1068],{"class":903},[652,3617,1071],{"class":930},[3309,3619],{},[3301,3621,2844],{"id":3622},"variant-de-boucle",[3624,3625,3626],"blockquote",{},[598,3627,3628,3631],{},[819,3629,3630],{},"v = droite - gauche"," décroît strictement à chaque tour",[598,3633,3634,3635],{},"→ ne peut pas descendre indéfiniment → ",[602,3636,3637],{},"terminaison",[3309,3639],{},[3301,3641,1680,3642,1754],{"id":1679},[652,3643,3645,3668],{"className":3644},[655],[652,3646,3648],{"className":3647},[659],[661,3649,3650],{"xmlns":663},[665,3651,3652,3666],{},[668,3653,3654,3656,3658,3660,3662,3664],{},[1460,3655,1696],{"mathvariant":1695},[681,3657,1355],{"stretchy":1699},[1460,3659,1702],{},[681,3661,1705],{},[1460,3663,1708],{},[681,3665,1711],{"stretchy":1699},[694,3667,1714],{"encoding":696},[652,3669,3671],{"className":3670,"ariaHidden":702},[701],[652,3672,3674,3677,3680,3683,3688,3691,3694],{"className":3673},[706],[652,3675],{"className":3676,"style":1724},[710],[652,3678,1696],{"className":3679,"style":1729},[715,1728],[652,3681,1355],{"className":3682},[1733],[652,3684,1738,3686],{"className":3685},[1737],[652,3687,1742],{"style":1741},[652,3689],{"className":3690,"style":1746},[756],[652,3692,1708],{"className":3693},[715,1481],[652,3695,1711],{"className":3696},[1753],[598,3698,3699,3700],{},"Taille de la zone : ",[652,3701,3703,3738],{"className":3702},[655],[652,3704,3706],{"className":3705},[659],[661,3707,3708],{"xmlns":663},[665,3709,3710,3735],{},[668,3711,3712,3714,3717,3719,3721,3723,3725,3727,3729,3731,3733],{},[1460,3713,1708],{},[681,3715,3716],{"separator":702},",",[1460,3718,1708],{},[1460,3720,1812],{"mathvariant":1811},[674,3722,676],{},[681,3724,3716],{"separator":702},[1460,3726,1708],{},[1460,3728,1812],{"mathvariant":1811},[674,3730,1097],{},[681,3732,3716],{"separator":702},[681,3734,647],{},[694,3736,3737],{"encoding":696},"n, n\u002F2, n\u002F4, \\dots",[652,3739,3741],{"className":3740,"ariaHidden":702},[701],[652,3742,3744,3747,3750,3754,3757,3760,3763,3766,3769,3772,3775,3778,3781],{"className":3743},[706],[652,3745],{"className":3746,"style":1724},[710],[652,3748,1708],{"className":3749},[715,1481],[652,3751,3716],{"className":3752},[3753],"mpunct",[652,3755],{"className":3756,"style":1746},[756],[652,3758,1708],{"className":3759},[715,1481],[652,3761,1833],{"className":3762},[715],[652,3764,3716],{"className":3765},[3753],[652,3767],{"className":3768,"style":1746},[756],[652,3770,1708],{"className":3771},[715,1481],[652,3773,1873],{"className":3774},[715],[652,3776,3716],{"className":3777},[3753],[652,3779],{"className":3780,"style":1746},[756],[652,3782,647],{"className":3783},[3784],"minner",[598,3786,3787,3788,3879,3880,3987],{},"On s'arrête quand ",[652,3789,3791,3817],{"className":3790},[655],[652,3792,3794],{"className":3793},[659],[661,3795,3796],{"xmlns":663},[665,3797,3798,3814],{},[668,3799,3800,3802,3804,3810,3812],{},[1460,3801,1708],{},[1460,3803,1812],{"mathvariant":1811},[671,3805,3806,3808],{},[674,3807,676],{},[1460,3809,1891],{},[681,3811,683],{},[674,3813,1092],{},[694,3815,3816],{"encoding":696},"n \u002F 2^k = 1",[652,3818,3820,3870],{"className":3819,"ariaHidden":702},[701],[652,3821,3823,3826,3829,3832,3861,3864,3867],{"className":3822},[706],[652,3824],{"className":3825,"style":1943},[710],[652,3827,1708],{"className":3828},[715,1481],[652,3830,1812],{"className":3831},[715],[652,3833,3835,3838],{"className":3834},[715],[652,3836,676],{"className":3837},[715],[652,3839,3841],{"className":3840},[722],[652,3842,3844],{"className":3843},[726],[652,3845,3847],{"className":3846},[730],[652,3848,3850],{"className":3849,"style":1968},[734],[652,3851,3852,3855],{"style":737},[652,3853],{"className":3854,"style":742},[741],[652,3856,3858],{"className":3857},[746,747,748,749],[652,3859,1891],{"className":3860,"style":1907},[715,1481,749],[652,3862],{"className":3863,"style":757},[756],[652,3865,683],{"className":3866},[761],[652,3868],{"className":3869,"style":757},[756],[652,3871,3873,3876],{"className":3872},[706],[652,3874],{"className":3875,"style":790},[710],[652,3877,1092],{"className":3878},[715],", soit ",[652,3881,3883,3911],{"className":3882},[655],[652,3884,3886],{"className":3885},[659],[661,3887,3888],{"xmlns":663},[665,3889,3890,3908],{},[668,3891,3892,3894,3896,3906],{},[1460,3893,1891],{},[681,3895,683],{},[2115,3897,3898,3904],{},[668,3899,3900,3902],{},[1460,3901,1702],{},[681,3903,1705],{},[674,3905,676],{},[1460,3907,1708],{},[694,3909,3910],{"encoding":696},"k = \\log_2 n",[652,3912,3914,3932],{"className":3913,"ariaHidden":702},[701],[652,3915,3917,3920,3923,3926,3929],{"className":3916},[706],[652,3918],{"className":3919,"style":1903},[710],[652,3921,1891],{"className":3922,"style":1907},[715,1481],[652,3924],{"className":3925,"style":757},[756],[652,3927,683],{"className":3928},[761],[652,3930],{"className":3931,"style":757},[756],[652,3933,3935,3939,3981,3984],{"className":3934},[706],[652,3936],{"className":3937,"style":3938},[710],"height:0.9386em;vertical-align:-0.2441em;",[652,3940,3942,3947],{"className":3941},[1737],[652,3943,1738,3945],{"className":3944},[1737],[652,3946,1742],{"style":1741},[652,3948,3950],{"className":3949},[722],[652,3951,3953,3973],{"className":3952},[726,2177],[652,3954,3956,3970],{"className":3955},[730],[652,3957,3959],{"className":3958,"style":2184},[734],[652,3960,3961,3964],{"style":2187},[652,3962],{"className":3963,"style":742},[741],[652,3965,3967],{"className":3966},[746,747,748,749],[652,3968,676],{"className":3969},[715,749],[652,3971,2201],{"className":3972},[2200],[652,3974,3976],{"className":3975},[730],[652,3977,3979],{"className":3978,"style":2208},[734],[652,3980],{},[652,3982],{"className":3983,"style":1746},[756],[652,3985,1708],{"className":3986},[715,1481]," tours.",[3309,3989],{},[3301,3991,3993],{"id":3992},"comparaison-spectaculaire","Comparaison spectaculaire",[2914,3995,3996,4036],{},[2917,3997,3998],{},[2920,3999,4000,4030,4033],{},[2923,4001,4002],{},[652,4003,4005,4018],{"className":4004},[655],[652,4006,4008],{"className":4007},[659],[661,4009,4010],{"xmlns":663},[665,4011,4012,4016],{},[668,4013,4014],{},[1460,4015,1708],{},[694,4017,1708],{"encoding":696},[652,4019,4021],{"className":4020,"ariaHidden":702},[701],[652,4022,4024,4027],{"className":4023},[706],[652,4025],{"className":4026,"style":1477},[710],[652,4028,1708],{"className":4029},[715,1481],[2923,4031,4032],{},"Linéaire",[2923,4034,4035],{},"Dichotomique",[2957,4037,4038,4046,4054],{},[2920,4039,4040,4042,4044],{},[2962,4041,2964],{},[2962,4043,2964],{},[2962,4045,1172],{},[2920,4047,4048,4050,4052],{},[2962,4049,2971],{},[2962,4051,2971],{},[2962,4053,2427],{},[2920,4055,4056,4059,4061],{},[2962,4057,4058],{},"1 milliard",[2962,4060,4058],{},[2962,4062,2981],{},[3309,4064],{},[3301,4066,4068],{"id":4067},"pièges","Pièges",[633,4070,4071,4074,4081],{},[636,4072,4073],{},"Tableau non trié → résultat faux",[636,4075,4076,2453,4078,4080],{},[819,4077,1812],{},[819,4079,981],{}," → indice flottant",[636,4082,4083,2453,4085,4088],{},[819,4084,2469],{},[819,4086,4087],{},"m + 1"," → boucle infinie",[3309,4090],{},[3301,4092,574],{"id":4093},"diviser-pour-régner",[598,4095,4096],{},"La dichotomie est l'archétype de cette stratégie : réduire à un sous-problème\ndeux fois plus petit. Vous reverrez ce schéma en Terminale (tri fusion).",[3309,4098],{},[3301,4100,4102],{"id":4101},"à-retenir","À retenir",[598,4104,4105,4106,2601,4181,4244,4245,4302],{},"Trier coûte ",[652,4107,4109,4133],{"className":4108},[655],[652,4110,4112],{"className":4111},[659],[661,4113,4114],{"xmlns":663},[665,4115,4116,4130],{},[668,4117,4118,4120,4122,4128],{},[1460,4119,1696],{"mathvariant":1695},[681,4121,1355],{"stretchy":1699},[671,4123,4124,4126],{},[1460,4125,1708],{},[674,4127,676],{},[681,4129,1711],{"stretchy":1699},[694,4131,4132],{"encoding":696},"\\mathcal{O}(n^2)",[652,4134,4136],{"className":4135,"ariaHidden":702},[701],[652,4137,4139,4143,4146,4149,4178],{"className":4138},[706],[652,4140],{"className":4141,"style":4142},[710],"height:1.0641em;vertical-align:-0.25em;",[652,4144,1696],{"className":4145,"style":1729},[715,1728],[652,4147,1355],{"className":4148},[1733],[652,4150,4152,4155],{"className":4151},[715],[652,4153,1708],{"className":4154},[715,1481],[652,4156,4158],{"className":4157},[722],[652,4159,4161],{"className":4160},[726],[652,4162,4164],{"className":4163},[730],[652,4165,4167],{"className":4166,"style":711},[734],[652,4168,4169,4172],{"style":737},[652,4170],{"className":4171,"style":742},[741],[652,4173,4175],{"className":4174},[746,747,748,749],[652,4176,676],{"className":4177},[715,749],[652,4179,1711],{"className":4180},[1753],[652,4182,4184,4209],{"className":4183},[655],[652,4185,4187],{"className":4186},[659],[661,4188,4189],{"xmlns":663},[665,4190,4191,4207],{},[668,4192,4193,4195,4197,4199,4201,4203,4205],{},[1460,4194,1696],{"mathvariant":1695},[681,4196,1355],{"stretchy":1699},[1460,4198,1708],{},[1460,4200,1702],{},[681,4202,1705],{},[1460,4204,1708],{},[681,4206,1711],{"stretchy":1699},[694,4208,2630],{"encoding":696},[652,4210,4212],{"className":4211,"ariaHidden":702},[701],[652,4213,4215,4218,4221,4224,4227,4230,4235,4238,4241],{"className":4214},[706],[652,4216],{"className":4217,"style":1724},[710],[652,4219,1696],{"className":4220,"style":1729},[715,1728],[652,4222,1355],{"className":4223},[1733],[652,4225,1708],{"className":4226},[715,1481],[652,4228],{"className":4229,"style":1746},[756],[652,4231,1738,4233],{"className":4232},[1737],[652,4234,1742],{"style":1741},[652,4236],{"className":4237,"style":1746},[756],[652,4239,1708],{"className":4240},[715,1481],[652,4242,1711],{"className":4243},[1753],", mais ensuite on\npeut chercher en ",[602,4246,4247],{},[652,4248,4250,4273],{"className":4249},[655],[652,4251,4253],{"className":4252},[659],[661,4254,4255],{"xmlns":663},[665,4256,4257,4271],{},[668,4258,4259,4261,4263,4265,4267,4269],{},[1460,4260,1696],{"mathvariant":1695},[681,4262,1355],{"stretchy":1699},[1460,4264,1702],{},[681,4266,1705],{},[1460,4268,1708],{},[681,4270,1711],{"stretchy":1699},[694,4272,1714],{"encoding":696},[652,4274,4276],{"className":4275,"ariaHidden":702},[701],[652,4277,4279,4282,4285,4288,4293,4296,4299],{"className":4278},[706],[652,4280],{"className":4281,"style":1724},[710],[652,4283,1696],{"className":4284,"style":1729},[715,1728],[652,4286,1355],{"className":4287},[1733],[652,4289,1738,4291],{"className":4290},[1737],[652,4292,1742],{"style":1741},[652,4294],{"className":4295,"style":1746},[756],[652,4297,1708],{"className":4298},[715,1481],[652,4300,1711],{"className":4301},[1753],". Rentable dès qu'on cherche\nplusieurs fois.",[2667,4304,3279],{},{"title":894,"searchDepth":915,"depth":915,"links":4306},[],"Deck pour la séance sur la recherche 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