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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":425,"bo":587,"body":589,"description":3692,"eval":3693,"extension":3694,"labo":3693,"meta":3695,"navigation":995,"path":426,"quizzes":3696,"readingTime":848,"seo":3704,"stem":427,"tp":3693,"__hash__":3705},"cours\u002F1.nsi-1ere\u002Fph-algorithmique\u002F3.tris.md",[588],"PH05",{"type":590,"value":591,"toc":3678},"minimark",[592,597,735,738,742,752,755,773,835,1043,1064,1072,1077,1109,1212,1301,1305,1358,1385,1389,1397,1400,1414,1602,1793,1797,1838,1962,1965,1986,1990,2022,2182,2481,2561,2993,3296,3306,3327,3331,3587,3591,3674],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,604,605,608,609,612,613,616,617,734],"p",{},[601,602,603],"strong",{},"Trier"," un tableau, c'est en ranger les éléments dans l'ordre croissant\n(ou décroissant). C'est l'une des opérations les plus utiles en informatique\n— les annuaires sont triés par nom, les classements par score, les fichiers\npar date. Au-delà de l'usage direct, ",[601,606,607],{},"un tableau trié ouvre des\npossibilités"," : on peut y chercher beaucoup plus vite (vous le verrez au\ncours suivant avec la dichotomie). Ce cours présente deux tris naïfs mais\nfondamentaux, le ",[601,610,611],{},"tri par sélection"," et le ",[601,614,615],{},"tri par insertion",", et\ndémontre qu'ils coûtent tous deux ",[618,619,622,664],"span",{"className":620},[621],"katex",[618,623,626],{"className":624},[625],"katex-mathml",[627,628,630],"math",{"xmlns":629},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[631,632,633,659],"semantics",{},[634,635,636,641,646,656],"mrow",{},[637,638,640],"mi",{"mathvariant":639},"script","O",[642,643,645],"mo",{"stretchy":644},"false","(",[647,648,649,652],"msup",{},[637,650,651],{},"n",[653,654,655],"mn",{},"2",[642,657,658],{"stretchy":644},")",[660,661,663],"annotation",{"encoding":662},"application\u002Fx-tex","\\mathcal{O}(n^2)",[618,665,669],{"className":666,"ariaHidden":668},[667],"katex-html","true",[618,670,673,678,684,688,730],{"className":671},[672],"base",[618,674],{"className":675,"style":677},[676],"strut","height:1.0641em;vertical-align:-0.25em;",[618,679,640],{"className":680,"style":683},[681,682],"mord","mathcal","margin-right:0.0278em;",[618,685,645],{"className":686},[687],"mopen",[618,689,691,695],{"className":690},[681],[618,692,651],{"className":693},[681,694],"mathnormal",[618,696,699],{"className":697},[698],"msupsub",[618,700,703],{"className":701},[702],"vlist-t",[618,704,707],{"className":705},[706],"vlist-r",[618,708,712],{"className":709,"style":711},[710],"vlist","height:0.8141em;",[618,713,715,720],{"style":714},"top:-3.063em;margin-right:0.05em;",[618,716],{"className":717,"style":719},[718],"pstrut","height:2.7em;",[618,721,727],{"className":722},[723,724,725,726],"sizing","reset-size6","size3","mtight",[618,728,655],{"className":729},[681,726],[618,731,658],{"className":732},[733],"mclose",".",[736,737],"bo-ref",{"code":588},[593,739,741],{"id":740},"le-tri-par-sélection","Le tri par sélection",[598,743,744,747,748,751],{},[601,745,746],{},"Idée"," : à chaque étape, on ",[601,749,750],{},"sélectionne"," le plus petit élément non\nencore trié et on le place à sa position définitive.",[598,753,754],{},"Plus précisément, on découpe le tableau en deux zones :",[756,757,758,766],"ul",{},[759,760,761,762,765],"li",{},"à gauche, une zone ",[601,763,764],{},"triée"," (au début vide, puis qui grandit) ;",[759,767,768,769,772],{},"à droite, une zone ",[601,770,771],{},"non triée"," (qui rétrécit).",[598,774,775,776,806,807,734],{},"À l'étape ",[618,777,779,793],{"className":778},[621],[618,780,782],{"className":781},[625],[627,783,784],{"xmlns":629},[631,785,786,791],{},[634,787,788],{},[637,789,790],{},"i",[660,792,790],{"encoding":662},[618,794,796],{"className":795,"ariaHidden":668},[667],[618,797,799,803],{"className":798},[672],[618,800],{"className":801,"style":802},[676],"height:0.6595em;",[618,804,790],{"className":805},[681,694],", on cherche le minimum dans la zone non triée et on l'échange\navec l'élément en position ",[618,808,810,823],{"className":809},[621],[618,811,813],{"className":812},[625],[627,814,815],{"xmlns":629},[631,816,817,821],{},[634,818,819],{},[637,820,790],{},[660,822,790],{"encoding":662},[618,824,826],{"className":825,"ariaHidden":668},[667],[618,827,829,832],{"className":828},[672],[618,830],{"className":831,"style":802},[676],[618,833,790],{"className":834},[681,694],[836,837,842],"pre",{"className":838,"code":839,"language":840,"meta":841,"style":841},"language-python shiki shiki-themes github-light github-light github-dark","def tri_selection(t):\n    n = len(t)\n    for i in range(n):\n        # Cherche l'indice du minimum dans t[i:]\n        i_min = i\n        for j in range(i + 1, n):\n            if t[j] \u003C t[i_min]:\n                i_min = j\n        # Échange t[i] et t[i_min]\n        t[i], t[i_min] = t[i_min], t[i]\n    return t\n\nprint(tri_selection([5, 2, 8, 1, 9, 3]))\n# [1, 2, 3, 5, 8, 9]\n","python","",[843,844,845,861,877,895,902,913,938,953,964,970,981,990,997,1037],"code",{"__ignoreMap":841},[618,846,849,853,857],{"class":847,"line":848},"line",1,[618,850,852],{"class":851},"s8jYJ","def",[618,854,856],{"class":855},"snPdu"," tri_selection",[618,858,860],{"class":859},"sxrX7","(t):\n",[618,862,864,867,870,874],{"class":847,"line":863},2,[618,865,866],{"class":859},"    n ",[618,868,869],{"class":851},"=",[618,871,873],{"class":872},"sBjJW"," len",[618,875,876],{"class":859},"(t)\n",[618,878,880,883,886,889,892],{"class":847,"line":879},3,[618,881,882],{"class":851},"    for",[618,884,885],{"class":859}," i ",[618,887,888],{"class":851},"in",[618,890,891],{"class":872}," range",[618,893,894],{"class":859},"(n):\n",[618,896,898],{"class":847,"line":897},4,[618,899,901],{"class":900},"sCsY4","        # Cherche l'indice du minimum dans t[i:]\n",[618,903,905,908,910],{"class":847,"line":904},5,[618,906,907],{"class":859},"        i_min ",[618,909,869],{"class":851},[618,911,912],{"class":859}," i\n",[618,914,916,919,922,924,926,929,932,935],{"class":847,"line":915},6,[618,917,918],{"class":851},"        for",[618,920,921],{"class":859}," j ",[618,923,888],{"class":851},[618,925,891],{"class":872},[618,927,928],{"class":859},"(i ",[618,930,931],{"class":851},"+",[618,933,934],{"class":872}," 1",[618,936,937],{"class":859},", n):\n",[618,939,941,944,947,950],{"class":847,"line":940},7,[618,942,943],{"class":851},"            if",[618,945,946],{"class":859}," t[j] ",[618,948,949],{"class":851},"\u003C",[618,951,952],{"class":859}," t[i_min]:\n",[618,954,956,959,961],{"class":847,"line":955},8,[618,957,958],{"class":859},"                i_min ",[618,960,869],{"class":851},[618,962,963],{"class":859}," j\n",[618,965,967],{"class":847,"line":966},9,[618,968,969],{"class":900},"        # Échange t[i] et t[i_min]\n",[618,971,973,976,978],{"class":847,"line":972},10,[618,974,975],{"class":859},"        t[i], t[i_min] ",[618,977,869],{"class":851},[618,979,980],{"class":859}," t[i_min], t[i]\n",[618,982,984,987],{"class":847,"line":983},11,[618,985,986],{"class":851},"    return",[618,988,989],{"class":859}," t\n",[618,991,993],{"class":847,"line":992},12,[618,994,996],{"emptyLinePlaceholder":995},true,"\n",[618,998,1000,1003,1006,1009,1012,1014,1016,1019,1021,1024,1026,1029,1031,1034],{"class":847,"line":999},13,[618,1001,1002],{"class":872},"print",[618,1004,1005],{"class":859},"(tri_selection([",[618,1007,1008],{"class":872},"5",[618,1010,1011],{"class":859},", ",[618,1013,655],{"class":872},[618,1015,1011],{"class":859},[618,1017,1018],{"class":872},"8",[618,1020,1011],{"class":859},[618,1022,1023],{"class":872},"1",[618,1025,1011],{"class":859},[618,1027,1028],{"class":872},"9",[618,1030,1011],{"class":859},[618,1032,1033],{"class":872},"3",[618,1035,1036],{"class":859},"]))\n",[618,1038,1040],{"class":847,"line":1039},14,[618,1041,1042],{"class":900},"# [1, 2, 3, 5, 8, 9]\n",[1044,1045,1046],"tip",{},[598,1047,1048,1051,1052,1055,1056,1059,1060,1063],{},[601,1049,1050],{},"Idiome Python — l'échange en une ligne."," L'écriture\n",[843,1053,1054],{},"t[i], t[i_min] = t[i_min], t[i]"," est une ",[601,1057,1058],{},"affectation parallèle"," : Python\ncalcule d'abord le membre de droite (un tuple ",[843,1061,1062],{},"(t[i_min], t[i])","), puis\nl'attribue à gauche. Pas besoin de variable temporaire — c'est lisible et\nsans erreur d'ordre.",[1044,1065,1066],{},[598,1067,1068,1071],{},[601,1069,1070],{},"Analogie"," : trier une main de cartes en cherchant à chaque fois la plus\npetite carte restante pour la mettre à gauche.",[1073,1074,1076],"h3",{"id":1075},"invariant-du-tri-par-sélection","Invariant du tri par sélection",[598,1078,1079,1080,1108],{},"Notons l'invariant suivant, vrai à la fin de chaque tour de la boucle externe\npour l'indice ",[618,1081,1083,1096],{"className":1082},[621],[618,1084,1086],{"className":1085},[625],[627,1087,1088],{"xmlns":629},[631,1089,1090,1094],{},[634,1091,1092],{},[637,1093,790],{},[660,1095,790],{"encoding":662},[618,1097,1099],{"className":1098,"ariaHidden":668},[667],[618,1100,1102,1105],{"className":1101},[672],[618,1103],{"className":1104,"style":802},[676],[618,1106,790],{"className":1107},[681,694]," :",[1110,1111,1112],"blockquote",{},[598,1113,1114,1115,1143,1144,1147,1148,1151,1152,1208,1209,734],{},"Après le tour ",[618,1116,1118,1131],{"className":1117},[621],[618,1119,1121],{"className":1120},[625],[627,1122,1123],{"xmlns":629},[631,1124,1125,1129],{},[634,1126,1127],{},[637,1128,790],{},[660,1130,790],{"encoding":662},[618,1132,1134],{"className":1133,"ariaHidden":668},[667],[618,1135,1137,1140],{"className":1136},[672],[618,1138],{"className":1139,"style":802},[676],[618,1141,790],{"className":1142},[681,694],", les cases ",[843,1145,1146],{},"t[0]",", …, ",[843,1149,1150],{},"t[i]"," contiennent les ",[618,1153,1155,1173],{"className":1154},[621],[618,1156,1158],{"className":1157},[625],[627,1159,1160],{"xmlns":629},[631,1161,1162,1170],{},[634,1163,1164,1166,1168],{},[637,1165,790],{},[642,1167,931],{},[653,1169,1023],{},[660,1171,1172],{"encoding":662},"i+1",[618,1174,1176,1198],{"className":1175,"ariaHidden":668},[667],[618,1177,1179,1183,1186,1191,1195],{"className":1178},[672],[618,1180],{"className":1181,"style":1182},[676],"height:0.7429em;vertical-align:-0.0833em;",[618,1184,790],{"className":1185},[681,694],[618,1187],{"className":1188,"style":1190},[1189],"mspace","margin-right:0.2222em;",[618,1192,931],{"className":1193},[1194],"mbin",[618,1196],{"className":1197,"style":1190},[1189],[618,1199,1201,1205],{"className":1200},[672],[618,1202],{"className":1203,"style":1204},[676],"height:0.6444em;",[618,1206,1023],{"className":1207},[681]," plus\npetits éléments du tableau, ",[601,1210,1211],{},"rangés dans l'ordre croissant",[598,1213,1214,1215,1243,1244,1297,1298],{},"Quand ",[618,1216,1218,1231],{"className":1217},[621],[618,1219,1221],{"className":1220},[625],[627,1222,1223],{"xmlns":629},[631,1224,1225,1229],{},[634,1226,1227],{},[637,1228,790],{},[660,1230,790],{"encoding":662},[618,1232,1234],{"className":1233,"ariaHidden":668},[667],[618,1235,1237,1240],{"className":1236},[672],[618,1238],{"className":1239,"style":802},[676],[618,1241,790],{"className":1242},[681,694]," atteint ",[618,1245,1247,1266],{"className":1246},[621],[618,1248,1250],{"className":1249},[625],[627,1251,1252],{"xmlns":629},[631,1253,1254,1263],{},[634,1255,1256,1258,1261],{},[637,1257,651],{},[642,1259,1260],{},"−",[653,1262,1023],{},[660,1264,1265],{"encoding":662},"n-1",[618,1267,1269,1288],{"className":1268,"ariaHidden":668},[667],[618,1270,1272,1276,1279,1282,1285],{"className":1271},[672],[618,1273],{"className":1274,"style":1275},[676],"height:0.6667em;vertical-align:-0.0833em;",[618,1277,651],{"className":1278},[681,694],[618,1280],{"className":1281,"style":1190},[1189],[618,1283,1260],{"className":1284},[1194],[618,1286],{"className":1287,"style":1190},[1189],[618,1289,1291,1294],{"className":1290},[672],[618,1292],{"className":1293,"style":1204},[676],[618,1295,1023],{"className":1296},[681],", l'invariant dit que tout le tableau est trié.\n",[601,1299,1300],{},"C'est ce qu'on voulait.",[1073,1302,1304],{"id":1303},"terminaison","Terminaison",[598,1306,1307,1308,1311,1312,1315,1316,1319,1320,1349,1350,1353,1354,1357],{},"La boucle externe parcourt ",[843,1309,1310],{},"range(n)"," — c'est un ",[843,1313,1314],{},"for"," sur un intervalle\n",[601,1317,1318],{},"fini",". Elle s'arrête donc forcément après exactement ",[618,1321,1323,1336],{"className":1322},[621],[618,1324,1326],{"className":1325},[625],[627,1327,1328],{"xmlns":629},[631,1329,1330,1334],{},[634,1331,1332],{},[637,1333,651],{},[660,1335,651],{"encoding":662},[618,1337,1339],{"className":1338,"ariaHidden":668},[667],[618,1340,1342,1346],{"className":1341},[672],[618,1343],{"className":1344,"style":1345},[676],"height:0.4306em;",[618,1347,651],{"className":1348},[681,694]," tours. La boucle\ninterne aussi (",[843,1351,1352],{},"range(i+1, n)","). L'algorithme ",[601,1355,1356],{},"termine"," toujours.",[1359,1360,1362,1372],"self-eval-qcm",{"correct":1023,"slug":1361},"tris-c1",[1363,1364,1365],"template",{"v-slot:question":841},[598,1366,1367,1368,1371],{},"Dans le tri par sélection, après le tour i, qu'y a-t-il dans t",[618,1369,1370],{},"0..i"," ?",[1363,1373,1374],{"v-slot:choices":841},[756,1375,1376,1379,1382],{},[759,1377,1378],{},"Les éléments examinés",[759,1380,1381],{},"Les i+1 plus petits éléments, triés",[759,1383,1384],{},"Les i+1 éléments du début du tableau original",[593,1386,1388],{"id":1387},"le-tri-par-insertion","Le tri par insertion",[598,1390,1391,747,1393,1396],{},[601,1392,746],{},[601,1394,1395],{},"insère"," l'élément courant à sa place\ncorrecte dans la portion déjà triée à gauche.",[598,1398,1399],{},"C'est exactement comme on trie une main de cartes en la prenant carte par\ncarte :",[756,1401,1402,1405,1408,1411],{},[759,1403,1404],{},"la première carte forme une « main triée » de taille 1 ;",[759,1406,1407],{},"on prend la 2ᵉ carte et on l'insère à gauche ou à droite de la 1ʳᵉ ;",[759,1409,1410],{},"on prend la 3ᵉ et on l'insère parmi les 2 déjà triées ;",[759,1412,1413],{},"etc.",[836,1415,1417],{"className":838,"code":1416,"language":840,"meta":841,"style":841},"def tri_insertion(t):\n    n = len(t)\n    for i in range(1, n):\n        # On veut insérer t[i] dans t[0..i-1] (déjà trié)\n        valeur = t[i]\n        j = i - 1\n        # On décale vers la droite tant qu'on trouve plus grand\n        while j >= 0 and t[j] > valeur:\n            t[j + 1] = t[j]\n            j -= 1\n        t[j + 1] = valeur\n    return t\n\nprint(tri_insertion([5, 2, 8, 1, 9, 3]))\n# [1, 2, 3, 5, 8, 9]\n",[843,1418,1419,1428,1438,1454,1459,1469,1484,1489,1513,1530,1540,1556,1562,1566,1597],{"__ignoreMap":841},[618,1420,1421,1423,1426],{"class":847,"line":848},[618,1422,852],{"class":851},[618,1424,1425],{"class":855}," tri_insertion",[618,1427,860],{"class":859},[618,1429,1430,1432,1434,1436],{"class":847,"line":863},[618,1431,866],{"class":859},[618,1433,869],{"class":851},[618,1435,873],{"class":872},[618,1437,876],{"class":859},[618,1439,1440,1442,1444,1446,1448,1450,1452],{"class":847,"line":879},[618,1441,882],{"class":851},[618,1443,885],{"class":859},[618,1445,888],{"class":851},[618,1447,891],{"class":872},[618,1449,645],{"class":859},[618,1451,1023],{"class":872},[618,1453,937],{"class":859},[618,1455,1456],{"class":847,"line":897},[618,1457,1458],{"class":900},"        # On veut insérer t[i] dans t[0..i-1] (déjà trié)\n",[618,1460,1461,1464,1466],{"class":847,"line":904},[618,1462,1463],{"class":859},"        valeur ",[618,1465,869],{"class":851},[618,1467,1468],{"class":859}," t[i]\n",[618,1470,1471,1474,1476,1478,1481],{"class":847,"line":915},[618,1472,1473],{"class":859},"        j ",[618,1475,869],{"class":851},[618,1477,885],{"class":859},[618,1479,1480],{"class":851},"-",[618,1482,1483],{"class":872}," 1\n",[618,1485,1486],{"class":847,"line":940},[618,1487,1488],{"class":900},"        # On décale vers la droite tant qu'on trouve plus grand\n",[618,1490,1491,1494,1496,1499,1502,1505,1507,1510],{"class":847,"line":955},[618,1492,1493],{"class":851},"        while",[618,1495,921],{"class":859},[618,1497,1498],{"class":851},">=",[618,1500,1501],{"class":872}," 0",[618,1503,1504],{"class":851}," and",[618,1506,946],{"class":859},[618,1508,1509],{"class":851},">",[618,1511,1512],{"class":859}," valeur:\n",[618,1514,1515,1518,1520,1522,1525,1527],{"class":847,"line":966},[618,1516,1517],{"class":859},"            t[j ",[618,1519,931],{"class":851},[618,1521,934],{"class":872},[618,1523,1524],{"class":859},"] ",[618,1526,869],{"class":851},[618,1528,1529],{"class":859}," t[j]\n",[618,1531,1532,1535,1538],{"class":847,"line":972},[618,1533,1534],{"class":859},"            j ",[618,1536,1537],{"class":851},"-=",[618,1539,1483],{"class":872},[618,1541,1542,1545,1547,1549,1551,1553],{"class":847,"line":983},[618,1543,1544],{"class":859},"        t[j ",[618,1546,931],{"class":851},[618,1548,934],{"class":872},[618,1550,1524],{"class":859},[618,1552,869],{"class":851},[618,1554,1555],{"class":859}," valeur\n",[618,1557,1558,1560],{"class":847,"line":992},[618,1559,986],{"class":851},[618,1561,989],{"class":859},[618,1563,1564],{"class":847,"line":999},[618,1565,996],{"emptyLinePlaceholder":995},[618,1567,1568,1570,1573,1575,1577,1579,1581,1583,1585,1587,1589,1591,1593,1595],{"class":847,"line":1039},[618,1569,1002],{"class":872},[618,1571,1572],{"class":859},"(tri_insertion([",[618,1574,1008],{"class":872},[618,1576,1011],{"class":859},[618,1578,655],{"class":872},[618,1580,1011],{"class":859},[618,1582,1018],{"class":872},[618,1584,1011],{"class":859},[618,1586,1023],{"class":872},[618,1588,1011],{"class":859},[618,1590,1028],{"class":872},[618,1592,1011],{"class":859},[618,1594,1033],{"class":872},[618,1596,1036],{"class":859},[618,1598,1600],{"class":847,"line":1599},15,[618,1601,1042],{"class":900},[1603,1604,1605],"python-snippet",{},[836,1606,1608],{"className":838,"code":1607,"language":840,"meta":841,"style":841},"def tri_insertion(t):\n    n = len(t)\n    for i in range(1, n):\n        valeur = t[i]\n        j = i - 1\n        while j >= 0 and t[j] > valeur:\n            t[j + 1] = t[j]\n            j -= 1\n        t[j + 1] = valeur\n    return t\n\n# Trace pas à pas sur un petit exemple\nexemple = [5, 2, 8, 1, 9, 3]\nprint(\"Avant :\", exemple)\nprint(\"Après :\", tri_insertion(exemple))\n",[843,1609,1610,1618,1628,1644,1652,1664,1682,1696,1704,1718,1724,1728,1733,1768,1781],{"__ignoreMap":841},[618,1611,1612,1614,1616],{"class":847,"line":848},[618,1613,852],{"class":851},[618,1615,1425],{"class":855},[618,1617,860],{"class":859},[618,1619,1620,1622,1624,1626],{"class":847,"line":863},[618,1621,866],{"class":859},[618,1623,869],{"class":851},[618,1625,873],{"class":872},[618,1627,876],{"class":859},[618,1629,1630,1632,1634,1636,1638,1640,1642],{"class":847,"line":879},[618,1631,882],{"class":851},[618,1633,885],{"class":859},[618,1635,888],{"class":851},[618,1637,891],{"class":872},[618,1639,645],{"class":859},[618,1641,1023],{"class":872},[618,1643,937],{"class":859},[618,1645,1646,1648,1650],{"class":847,"line":897},[618,1647,1463],{"class":859},[618,1649,869],{"class":851},[618,1651,1468],{"class":859},[618,1653,1654,1656,1658,1660,1662],{"class":847,"line":904},[618,1655,1473],{"class":859},[618,1657,869],{"class":851},[618,1659,885],{"class":859},[618,1661,1480],{"class":851},[618,1663,1483],{"class":872},[618,1665,1666,1668,1670,1672,1674,1676,1678,1680],{"class":847,"line":915},[618,1667,1493],{"class":851},[618,1669,921],{"class":859},[618,1671,1498],{"class":851},[618,1673,1501],{"class":872},[618,1675,1504],{"class":851},[618,1677,946],{"class":859},[618,1679,1509],{"class":851},[618,1681,1512],{"class":859},[618,1683,1684,1686,1688,1690,1692,1694],{"class":847,"line":940},[618,1685,1517],{"class":859},[618,1687,931],{"class":851},[618,1689,934],{"class":872},[618,1691,1524],{"class":859},[618,1693,869],{"class":851},[618,1695,1529],{"class":859},[618,1697,1698,1700,1702],{"class":847,"line":955},[618,1699,1534],{"class":859},[618,1701,1537],{"class":851},[618,1703,1483],{"class":872},[618,1705,1706,1708,1710,1712,1714,1716],{"class":847,"line":966},[618,1707,1544],{"class":859},[618,1709,931],{"class":851},[618,1711,934],{"class":872},[618,1713,1524],{"class":859},[618,1715,869],{"class":851},[618,1717,1555],{"class":859},[618,1719,1720,1722],{"class":847,"line":972},[618,1721,986],{"class":851},[618,1723,989],{"class":859},[618,1725,1726],{"class":847,"line":983},[618,1727,996],{"emptyLinePlaceholder":995},[618,1729,1730],{"class":847,"line":992},[618,1731,1732],{"class":900},"# Trace pas à pas sur un petit exemple\n",[618,1734,1735,1738,1740,1743,1745,1747,1749,1751,1753,1755,1757,1759,1761,1763,1765],{"class":847,"line":999},[618,1736,1737],{"class":859},"exemple ",[618,1739,869],{"class":851},[618,1741,1742],{"class":859}," [",[618,1744,1008],{"class":872},[618,1746,1011],{"class":859},[618,1748,655],{"class":872},[618,1750,1011],{"class":859},[618,1752,1018],{"class":872},[618,1754,1011],{"class":859},[618,1756,1023],{"class":872},[618,1758,1011],{"class":859},[618,1760,1028],{"class":872},[618,1762,1011],{"class":859},[618,1764,1033],{"class":872},[618,1766,1767],{"class":859},"]\n",[618,1769,1770,1772,1774,1778],{"class":847,"line":1039},[618,1771,1002],{"class":872},[618,1773,645],{"class":859},[618,1775,1777],{"class":1776},"sIIMD","\"Avant :\"",[618,1779,1780],{"class":859},", exemple)\n",[618,1782,1783,1785,1787,1790],{"class":847,"line":1599},[618,1784,1002],{"class":872},[618,1786,645],{"class":859},[618,1788,1789],{"class":1776},"\"Après :\"",[618,1791,1792],{"class":859},", tri_insertion(exemple))\n",[1073,1794,1796],{"id":1795},"invariant-du-tri-par-insertion","Invariant du tri par insertion",[1110,1798,1799],{},[598,1800,1801,1802,1830,1831,1834,1835,734],{},"Au début du tour pour l'indice ",[618,1803,1805,1818],{"className":1804},[621],[618,1806,1808],{"className":1807},[625],[627,1809,1810],{"xmlns":629},[631,1811,1812,1816],{},[634,1813,1814],{},[637,1815,790],{},[660,1817,790],{"encoding":662},[618,1819,1821],{"className":1820,"ariaHidden":668},[667],[618,1822,1824,1827],{"className":1823},[672],[618,1825],{"className":1826,"style":802},[676],[618,1828,790],{"className":1829},[681,694],", la portion ",[843,1832,1833],{},"t[0..i-1]"," contient les\nmêmes éléments qu'au départ — mais ",[601,1836,1837],{},"triés dans l'ordre croissant",[598,1839,1840,1841,1894,1895,1898,1899,1901,1902,1930,1931,1011,1959,734],{},"Au début (",[618,1842,1844,1862],{"className":1843},[621],[618,1845,1847],{"className":1846},[625],[627,1848,1849],{"xmlns":629},[631,1850,1851,1859],{},[634,1852,1853,1855,1857],{},[637,1854,790],{},[642,1856,869],{},[653,1858,1023],{},[660,1860,1861],{"encoding":662},"i = 1",[618,1863,1865,1885],{"className":1864,"ariaHidden":668},[667],[618,1866,1868,1871,1874,1878,1882],{"className":1867},[672],[618,1869],{"className":1870,"style":802},[676],[618,1872,790],{"className":1873},[681,694],[618,1875],{"className":1876,"style":1877},[1189],"margin-right:0.2778em;",[618,1879,869],{"className":1880},[1881],"mrel",[618,1883],{"className":1884,"style":1877},[1189],[618,1886,1888,1891],{"className":1887},[672],[618,1889],{"className":1890,"style":1204},[676],[618,1892,1023],{"className":1893},[681],"), ",[843,1896,1897],{},"t[0..0]"," ne contient qu'un élément, donc trivialement\ntrié. À chaque tour, on insère ",[843,1900,1150],{}," à la bonne place et la portion triée\ns'agrandit d'un élément. Quand ",[618,1903,1905,1918],{"className":1904},[621],[618,1906,1908],{"className":1907},[625],[627,1909,1910],{"xmlns":629},[631,1911,1912,1916],{},[634,1913,1914],{},[637,1915,790],{},[660,1917,790],{"encoding":662},[618,1919,1921],{"className":1920,"ariaHidden":668},[667],[618,1922,1924,1927],{"className":1923},[672],[618,1925],{"className":1926,"style":802},[676],[618,1928,790],{"className":1929},[681,694]," vaut ",[618,1932,1934,1947],{"className":1933},[621],[618,1935,1937],{"className":1936},[625],[627,1938,1939],{"xmlns":629},[631,1940,1941,1945],{},[634,1942,1943],{},[637,1944,651],{},[660,1946,651],{"encoding":662},[618,1948,1950],{"className":1949,"ariaHidden":668},[667],[618,1951,1953,1956],{"className":1952},[672],[618,1954],{"className":1955,"style":1345},[676],[618,1957,651],{"className":1958},[681,694],[601,1960,1961],{},"tout le tableau est trié",[1073,1963,1304],{"id":1964},"terminaison-1",[598,1966,1967,1968,1970,1971,1974,1975,1978,1979,1982,1983,1985],{},"Boucle externe : ",[843,1969,1314],{}," sur ",[843,1972,1973],{},"range(1, n)"," — fini.\nBoucle interne : un ",[843,1976,1977],{},"while"," qui décrémente ",[843,1980,1981],{},"j"," à chaque tour, et qui s'arrête\nau plus tard quand ",[843,1984,1981],{}," devient négatif. Donc terminaison garantie.",[593,1987,1989],{"id":1988},"comparaison-de-coût","Comparaison de coût",[598,1991,1992,1993,2021],{},"Pour un tableau de taille ",[618,1994,1996,2009],{"className":1995},[621],[618,1997,1999],{"className":1998},[625],[627,2000,2001],{"xmlns":629},[631,2002,2003,2007],{},[634,2004,2005],{},[637,2006,651],{},[660,2008,651],{"encoding":662},[618,2010,2012],{"className":2011,"ariaHidden":668},[667],[618,2013,2015,2018],{"className":2014},[672],[618,2016],{"className":2017,"style":1345},[676],[618,2019,651],{"className":2020},[681,694],", comptons les comparaisons :",[598,2023,2024,2027,2028,2056,2057,2107,2108,2181],{},[601,2025,2026],{},"Tri par sélection"," : pour chaque ",[618,2029,2031,2044],{"className":2030},[621],[618,2032,2034],{"className":2033},[625],[627,2035,2036],{"xmlns":629},[631,2037,2038,2042],{},[634,2039,2040],{},[637,2041,790],{},[660,2043,790],{"encoding":662},[618,2045,2047],{"className":2046,"ariaHidden":668},[667],[618,2048,2050,2053],{"className":2049},[672],[618,2051],{"className":2052,"style":802},[676],[618,2054,790],{"className":2055},[681,694]," de 0 à ",[618,2058,2060,2077],{"className":2059},[621],[618,2061,2063],{"className":2062},[625],[627,2064,2065],{"xmlns":629},[631,2066,2067,2075],{},[634,2068,2069,2071,2073],{},[637,2070,651],{},[642,2072,1260],{},[653,2074,1023],{},[660,2076,1265],{"encoding":662},[618,2078,2080,2098],{"className":2079,"ariaHidden":668},[667],[618,2081,2083,2086,2089,2092,2095],{"className":2082},[672],[618,2084],{"className":2085,"style":1275},[676],[618,2087,651],{"className":2088},[681,694],[618,2090],{"className":2091,"style":1190},[1189],[618,2093,1260],{"className":2094},[1194],[618,2096],{"className":2097,"style":1190},[1189],[618,2099,2101,2104],{"className":2100},[672],[618,2102],{"className":2103,"style":1204},[676],[618,2105,1023],{"className":2106},[681],", on parcourt les\n",[618,2109,2111,2133],{"className":2110},[621],[618,2112,2114],{"className":2113},[625],[627,2115,2116],{"xmlns":629},[631,2117,2118,2130],{},[634,2119,2120,2122,2124,2126,2128],{},[637,2121,651],{},[642,2123,1260],{},[637,2125,790],{},[642,2127,1260],{},[653,2129,1023],{},[660,2131,2132],{"encoding":662},"n - i - 1",[618,2134,2136,2154,2172],{"className":2135,"ariaHidden":668},[667],[618,2137,2139,2142,2145,2148,2151],{"className":2138},[672],[618,2140],{"className":2141,"style":1275},[676],[618,2143,651],{"className":2144},[681,694],[618,2146],{"className":2147,"style":1190},[1189],[618,2149,1260],{"className":2150},[1194],[618,2152],{"className":2153,"style":1190},[1189],[618,2155,2157,2160,2163,2166,2169],{"className":2156},[672],[618,2158],{"className":2159,"style":1182},[676],[618,2161,790],{"className":2162},[681,694],[618,2164],{"className":2165,"style":1190},[1189],[618,2167,1260],{"className":2168},[1194],[618,2170],{"className":2171,"style":1190},[1189],[618,2173,2175,2178],{"className":2174},[672],[618,2176],{"className":2177,"style":1204},[676],[618,2179,1023],{"className":2180},[681]," éléments suivants. Total :",[618,2183,2186],{"className":2184},[2185],"katex-display",[618,2187,2189,2254],{"className":2188},[621],[618,2190,2192],{"className":2191},[625],[627,2193,2195],{"xmlns":629,"display":2194},"block",[631,2196,2197,2251],{},[634,2198,2199,2201,2203,2205,2207,2209,2211,2213,2215,2217,2219,2221,2223,2226,2228,2230,2232],{},[642,2200,645],{"stretchy":644},[637,2202,651],{},[642,2204,1260],{},[653,2206,1023],{},[642,2208,658],{"stretchy":644},[642,2210,931],{},[642,2212,645],{"stretchy":644},[637,2214,651],{},[642,2216,1260],{},[653,2218,655],{},[642,2220,658],{"stretchy":644},[642,2222,931],{},[642,2224,2225],{},"⋯",[642,2227,931],{},[653,2229,1023],{},[642,2231,869],{},[2233,2234,2235,2249],"mfrac",{},[634,2236,2237,2239,2241,2243,2245,2247],{},[637,2238,651],{},[642,2240,645],{"stretchy":644},[637,2242,651],{},[642,2244,1260],{},[653,2246,1023],{},[642,2248,658],{"stretchy":644},[653,2250,655],{},[660,2252,2253],{"encoding":662},"(n - 1) + (n - 2) + \\dots + 1 = \\frac{n(n-1)}{2}",[618,2255,2257,2279,2300,2321,2342,2361,2379],{"className":2256,"ariaHidden":668},[667],[618,2258,2260,2264,2267,2270,2273,2276],{"className":2259},[672],[618,2261],{"className":2262,"style":2263},[676],"height:1em;vertical-align:-0.25em;",[618,2265,645],{"className":2266},[687],[618,2268,651],{"className":2269},[681,694],[618,2271],{"className":2272,"style":1190},[1189],[618,2274,1260],{"className":2275},[1194],[618,2277],{"className":2278,"style":1190},[1189],[618,2280,2282,2285,2288,2291,2294,2297],{"className":2281},[672],[618,2283],{"className":2284,"style":2263},[676],[618,2286,1023],{"className":2287},[681],[618,2289,658],{"className":2290},[733],[618,2292],{"className":2293,"style":1190},[1189],[618,2295,931],{"className":2296},[1194],[618,2298],{"className":2299,"style":1190},[1189],[618,2301,2303,2306,2309,2312,2315,2318],{"className":2302},[672],[618,2304],{"className":2305,"style":2263},[676],[618,2307,645],{"className":2308},[687],[618,2310,651],{"className":2311},[681,694],[618,2313],{"className":2314,"style":1190},[1189],[618,2316,1260],{"className":2317},[1194],[618,2319],{"className":2320,"style":1190},[1189],[618,2322,2324,2327,2330,2333,2336,2339],{"className":2323},[672],[618,2325],{"className":2326,"style":2263},[676],[618,2328,655],{"className":2329},[681],[618,2331,658],{"className":2332},[733],[618,2334],{"className":2335,"style":1190},[1189],[618,2337,931],{"className":2338},[1194],[618,2340],{"className":2341,"style":1190},[1189],[618,2343,2345,2348,2352,2355,2358],{"className":2344},[672],[618,2346],{"className":2347,"style":1275},[676],[618,2349,2225],{"className":2350},[2351],"minner",[618,2353],{"className":2354,"style":1190},[1189],[618,2356,931],{"className":2357},[1194],[618,2359],{"className":2360,"style":1190},[1189],[618,2362,2364,2367,2370,2373,2376],{"className":2363},[672],[618,2365],{"className":2366,"style":1204},[676],[618,2368,1023],{"className":2369},[681],[618,2371],{"className":2372,"style":1877},[1189],[618,2374,869],{"className":2375},[1881],[618,2377],{"className":2378,"style":1877},[1189],[618,2380,2382,2386],{"className":2381},[672],[618,2383],{"className":2384,"style":2385},[676],"height:2.113em;vertical-align:-0.686em;",[618,2387,2389,2393,2478],{"className":2388},[681],[618,2390],{"className":2391},[687,2392],"nulldelimiter",[618,2394,2396],{"className":2395},[2233],[618,2397,2400,2469],{"className":2398},[702,2399],"vlist-t2",[618,2401,2403,2464],{"className":2402},[706],[618,2404,2407,2420,2431],{"className":2405,"style":2406},[710],"height:1.427em;",[618,2408,2410,2414],{"style":2409},"top:-2.314em;",[618,2411],{"className":2412,"style":2413},[718],"height:3em;",[618,2415,2417],{"className":2416},[681],[618,2418,655],{"className":2419},[681],[618,2421,2423,2426],{"style":2422},"top:-3.23em;",[618,2424],{"className":2425,"style":2413},[718],[618,2427],{"className":2428,"style":2430},[2429],"frac-line","border-bottom-width:0.04em;",[618,2432,2434,2437],{"style":2433},"top:-3.677em;",[618,2435],{"className":2436,"style":2413},[718],[618,2438,2440,2443,2446,2449,2452,2455,2458,2461],{"className":2439},[681],[618,2441,651],{"className":2442},[681,694],[618,2444,645],{"className":2445},[687],[618,2447,651],{"className":2448},[681,694],[618,2450],{"className":2451,"style":1190},[1189],[618,2453,1260],{"className":2454},[1194],[618,2456],{"className":2457,"style":1190},[1189],[618,2459,1023],{"className":2460},[681],[618,2462,658],{"className":2463},[733],[618,2465,2468],{"className":2466},[2467],"vlist-s","​",[618,2470,2472],{"className":2471},[706],[618,2473,2476],{"className":2474,"style":2475},[710],"height:0.686em;",[618,2477],{},[618,2479],{"className":2480},[733,2392],[598,2482,2483,2484,1011,2557,2560],{},"C'est en ",[618,2485,2487,2510],{"className":2486},[621],[618,2488,2490],{"className":2489},[625],[627,2491,2492],{"xmlns":629},[631,2493,2494,2508],{},[634,2495,2496,2498,2500,2506],{},[637,2497,640],{"mathvariant":639},[642,2499,645],{"stretchy":644},[647,2501,2502,2504],{},[637,2503,651],{},[653,2505,655],{},[642,2507,658],{"stretchy":644},[660,2509,663],{"encoding":662},[618,2511,2513],{"className":2512,"ariaHidden":668},[667],[618,2514,2516,2519,2522,2525,2554],{"className":2515},[672],[618,2517],{"className":2518,"style":677},[676],[618,2520,640],{"className":2521,"style":683},[681,682],[618,2523,645],{"className":2524},[687],[618,2526,2528,2531],{"className":2527},[681],[618,2529,651],{"className":2530},[681,694],[618,2532,2534],{"className":2533},[698],[618,2535,2537],{"className":2536},[702],[618,2538,2540],{"className":2539},[706],[618,2541,2543],{"className":2542,"style":711},[710],[618,2544,2545,2548],{"style":714},[618,2546],{"className":2547,"style":719},[718],[618,2549,2551],{"className":2550},[723,724,725,726],[618,2552,655],{"className":2553},[681,726],[618,2555,658],{"className":2556},[733],[601,2558,2559],{},"quel que soit le contenu"," du tableau (déjà\ntrié ou non, peu importe).",[598,2562,2563,2566,2567,2818,2819,2892,2893,2896,2897,2948,2949,734],{},[601,2564,2565],{},"Tri par insertion"," : dans le pire des cas (tableau trié en sens inverse),\nchaque insertion remonte jusqu'au début — soit ",[618,2568,2570,2624],{"className":2569},[621],[618,2571,2573],{"className":2572},[625],[627,2574,2575],{"xmlns":629},[631,2576,2577,2621],{},[634,2578,2579,2581,2583,2585,2587,2589,2591,2593,2595,2597,2599,2601,2603],{},[653,2580,1023],{},[642,2582,931],{},[653,2584,655],{},[642,2586,931],{},[642,2588,2225],{},[642,2590,931],{},[642,2592,645],{"stretchy":644},[637,2594,651],{},[642,2596,1260],{},[653,2598,1023],{},[642,2600,658],{"stretchy":644},[642,2602,869],{},[2233,2604,2605,2619],{},[634,2606,2607,2609,2611,2613,2615,2617],{},[637,2608,651],{},[642,2610,645],{"stretchy":644},[637,2612,651],{},[642,2614,1260],{},[653,2616,1023],{},[642,2618,658],{"stretchy":644},[653,2620,655],{},[660,2622,2623],{"encoding":662},"1 + 2 + \\dots + (n - 1) =\n\\frac{n(n-1)}{2}",[618,2625,2627,2646,2664,2682,2703,2724],{"className":2626,"ariaHidden":668},[667],[618,2628,2630,2634,2637,2640,2643],{"className":2629},[672],[618,2631],{"className":2632,"style":2633},[676],"height:0.7278em;vertical-align:-0.0833em;",[618,2635,1023],{"className":2636},[681],[618,2638],{"className":2639,"style":1190},[1189],[618,2641,931],{"className":2642},[1194],[618,2644],{"className":2645,"style":1190},[1189],[618,2647,2649,2652,2655,2658,2661],{"className":2648},[672],[618,2650],{"className":2651,"style":2633},[676],[618,2653,655],{"className":2654},[681],[618,2656],{"className":2657,"style":1190},[1189],[618,2659,931],{"className":2660},[1194],[618,2662],{"className":2663,"style":1190},[1189],[618,2665,2667,2670,2673,2676,2679],{"className":2666},[672],[618,2668],{"className":2669,"style":1275},[676],[618,2671,2225],{"className":2672},[2351],[618,2674],{"className":2675,"style":1190},[1189],[618,2677,931],{"className":2678},[1194],[618,2680],{"className":2681,"style":1190},[1189],[618,2683,2685,2688,2691,2694,2697,2700],{"className":2684},[672],[618,2686],{"className":2687,"style":2263},[676],[618,2689,645],{"className":2690},[687],[618,2692,651],{"className":2693},[681,694],[618,2695],{"className":2696,"style":1190},[1189],[618,2698,1260],{"className":2699},[1194],[618,2701],{"className":2702,"style":1190},[1189],[618,2704,2706,2709,2712,2715,2718,2721],{"className":2705},[672],[618,2707],{"className":2708,"style":2263},[676],[618,2710,1023],{"className":2711},[681],[618,2713,658],{"className":2714},[733],[618,2716],{"className":2717,"style":1877},[1189],[618,2719,869],{"className":2720},[1881],[618,2722],{"className":2723,"style":1877},[1189],[618,2725,2727,2731],{"className":2726},[672],[618,2728],{"className":2729,"style":2730},[676],"height:1.355em;vertical-align:-0.345em;",[618,2732,2734,2737,2815],{"className":2733},[681],[618,2735],{"className":2736},[687,2392],[618,2738,2740],{"className":2739},[2233],[618,2741,2743,2806],{"className":2742},[702,2399],[618,2744,2746,2803],{"className":2745},[706],[618,2747,2750,2765,2773],{"className":2748,"style":2749},[710],"height:1.01em;",[618,2751,2753,2756],{"style":2752},"top:-2.655em;",[618,2754],{"className":2755,"style":2413},[718],[618,2757,2759],{"className":2758},[723,724,725,726],[618,2760,2762],{"className":2761},[681,726],[618,2763,655],{"className":2764},[681,726],[618,2766,2767,2770],{"style":2422},[618,2768],{"className":2769,"style":2413},[718],[618,2771],{"className":2772,"style":2430},[2429],[618,2774,2776,2779],{"style":2775},"top:-3.485em;",[618,2777],{"className":2778,"style":2413},[718],[618,2780,2782],{"className":2781},[723,724,725,726],[618,2783,2785,2788,2791,2794,2797,2800],{"className":2784},[681,726],[618,2786,651],{"className":2787},[681,694,726],[618,2789,645],{"className":2790},[687,726],[618,2792,651],{"className":2793},[681,694,726],[618,2795,1260],{"className":2796},[1194,726],[618,2798,1023],{"className":2799},[681,726],[618,2801,658],{"className":2802},[733,726],[618,2804,2468],{"className":2805},[2467],[618,2807,2809],{"className":2808},[706],[618,2810,2813],{"className":2811,"style":2812},[710],"height:0.345em;",[618,2814],{},[618,2816],{"className":2817},[733,2392]," comparaisons. C'est aussi en ",[618,2820,2822,2845],{"className":2821},[621],[618,2823,2825],{"className":2824},[625],[627,2826,2827],{"xmlns":629},[631,2828,2829,2843],{},[634,2830,2831,2833,2835,2841],{},[637,2832,640],{"mathvariant":639},[642,2834,645],{"stretchy":644},[647,2836,2837,2839],{},[637,2838,651],{},[653,2840,655],{},[642,2842,658],{"stretchy":644},[660,2844,663],{"encoding":662},[618,2846,2848],{"className":2847,"ariaHidden":668},[667],[618,2849,2851,2854,2857,2860,2889],{"className":2850},[672],[618,2852],{"className":2853,"style":677},[676],[618,2855,640],{"className":2856,"style":683},[681,682],[618,2858,645],{"className":2859},[687],[618,2861,2863,2866],{"className":2862},[681],[618,2864,651],{"className":2865},[681,694],[618,2867,2869],{"className":2868},[698],[618,2870,2872],{"className":2871},[702],[618,2873,2875],{"className":2874},[706],[618,2876,2878],{"className":2877,"style":711},[710],[618,2879,2880,2883],{"style":714},[618,2881],{"className":2882,"style":719},[718],[618,2884,2886],{"className":2885},[723,724,725,726],[618,2887,655],{"className":2888},[681,726],[618,2890,658],{"className":2891},[733],". En\nrevanche, ",[601,2894,2895],{},"dans le meilleur des cas"," (tableau déjà trié), chaque tour ne\nfait qu'une seule comparaison qui échoue immédiatement — total ",[618,2898,2900,2918],{"className":2899},[621],[618,2901,2903],{"className":2902},[625],[627,2904,2905],{"xmlns":629},[631,2906,2907,2915],{},[634,2908,2909,2911,2913],{},[637,2910,651],{},[642,2912,1260],{},[653,2914,1023],{},[660,2916,2917],{"encoding":662},"n - 1",[618,2919,2921,2939],{"className":2920,"ariaHidden":668},[667],[618,2922,2924,2927,2930,2933,2936],{"className":2923},[672],[618,2925],{"className":2926,"style":1275},[676],[618,2928,651],{"className":2929},[681,694],[618,2931],{"className":2932,"style":1190},[1189],[618,2934,1260],{"className":2935},[1194],[618,2937],{"className":2938,"style":1190},[1189],[618,2940,2942,2945],{"className":2941},[672],[618,2943],{"className":2944,"style":1204},[676],[618,2946,1023],{"className":2947},[681]," et\ncoût ",[618,2950,2952,2972],{"className":2951},[621],[618,2953,2955],{"className":2954},[625],[627,2956,2957],{"xmlns":629},[631,2958,2959,2969],{},[634,2960,2961,2963,2965,2967],{},[637,2962,640],{"mathvariant":639},[642,2964,645],{"stretchy":644},[637,2966,651],{},[642,2968,658],{"stretchy":644},[660,2970,2971],{"encoding":662},"\\mathcal{O}(n)",[618,2973,2975],{"className":2974,"ariaHidden":668},[667],[618,2976,2978,2981,2984,2987,2990],{"className":2977},[672],[618,2979],{"className":2980,"style":2263},[676],[618,2982,640],{"className":2983,"style":683},[681,682],[618,2985,645],{"className":2986},[687],[618,2988,651],{"className":2989},[681,694],[618,2991,658],{"className":2992},[733],[2994,2995,2996,3012],"table",{},[2997,2998,2999],"thead",{},[3000,3001,3002,3006,3009],"tr",{},[3003,3004,3005],"th",{},"Tri",[3003,3007,3008],{},"Pire cas",[3003,3010,3011],{},"Meilleur cas",[3013,3014,3015,3171],"tbody",{},[3000,3016,3017,3021,3096],{},[3018,3019,3020],"td",{},"Sélection",[3018,3022,3023],{},[618,3024,3026,3049],{"className":3025},[621],[618,3027,3029],{"className":3028},[625],[627,3030,3031],{"xmlns":629},[631,3032,3033,3047],{},[634,3034,3035,3037,3039,3045],{},[637,3036,640],{"mathvariant":639},[642,3038,645],{"stretchy":644},[647,3040,3041,3043],{},[637,3042,651],{},[653,3044,655],{},[642,3046,658],{"stretchy":644},[660,3048,663],{"encoding":662},[618,3050,3052],{"className":3051,"ariaHidden":668},[667],[618,3053,3055,3058,3061,3064,3093],{"className":3054},[672],[618,3056],{"className":3057,"style":677},[676],[618,3059,640],{"className":3060,"style":683},[681,682],[618,3062,645],{"className":3063},[687],[618,3065,3067,3070],{"className":3066},[681],[618,3068,651],{"className":3069},[681,694],[618,3071,3073],{"className":3072},[698],[618,3074,3076],{"className":3075},[702],[618,3077,3079],{"className":3078},[706],[618,3080,3082],{"className":3081,"style":711},[710],[618,3083,3084,3087],{"style":714},[618,3085],{"className":3086,"style":719},[718],[618,3088,3090],{"className":3089},[723,724,725,726],[618,3091,655],{"className":3092},[681,726],[618,3094,658],{"className":3095},[733],[3018,3097,3098],{},[618,3099,3101,3124],{"className":3100},[621],[618,3102,3104],{"className":3103},[625],[627,3105,3106],{"xmlns":629},[631,3107,3108,3122],{},[634,3109,3110,3112,3114,3120],{},[637,3111,640],{"mathvariant":639},[642,3113,645],{"stretchy":644},[647,3115,3116,3118],{},[637,3117,651],{},[653,3119,655],{},[642,3121,658],{"stretchy":644},[660,3123,663],{"encoding":662},[618,3125,3127],{"className":3126,"ariaHidden":668},[667],[618,3128,3130,3133,3136,3139,3168],{"className":3129},[672],[618,3131],{"className":3132,"style":677},[676],[618,3134,640],{"className":3135,"style":683},[681,682],[618,3137,645],{"className":3138},[687],[618,3140,3142,3145],{"className":3141},[681],[618,3143,651],{"className":3144},[681,694],[618,3146,3148],{"className":3147},[698],[618,3149,3151],{"className":3150},[702],[618,3152,3154],{"className":3153},[706],[618,3155,3157],{"className":3156,"style":711},[710],[618,3158,3159,3162],{"style":714},[618,3160],{"className":3161,"style":719},[718],[618,3163,3165],{"className":3164},[723,724,725,726],[618,3166,655],{"className":3167},[681,726],[618,3169,658],{"className":3170},[733],[3000,3172,3173,3176,3251],{},[3018,3174,3175],{},"Insertion",[3018,3177,3178],{},[618,3179,3181,3204],{"className":3180},[621],[618,3182,3184],{"className":3183},[625],[627,3185,3186],{"xmlns":629},[631,3187,3188,3202],{},[634,3189,3190,3192,3194,3200],{},[637,3191,640],{"mathvariant":639},[642,3193,645],{"stretchy":644},[647,3195,3196,3198],{},[637,3197,651],{},[653,3199,655],{},[642,3201,658],{"stretchy":644},[660,3203,663],{"encoding":662},[618,3205,3207],{"className":3206,"ariaHidden":668},[667],[618,3208,3210,3213,3216,3219,3248],{"className":3209},[672],[618,3211],{"className":3212,"style":677},[676],[618,3214,640],{"className":3215,"style":683},[681,682],[618,3217,645],{"className":3218},[687],[618,3220,3222,3225],{"className":3221},[681],[618,3223,651],{"className":3224},[681,694],[618,3226,3228],{"className":3227},[698],[618,3229,3231],{"className":3230},[702],[618,3232,3234],{"className":3233},[706],[618,3235,3237],{"className":3236,"style":711},[710],[618,3238,3239,3242],{"style":714},[618,3240],{"className":3241,"style":719},[718],[618,3243,3245],{"className":3244},[723,724,725,726],[618,3246,655],{"className":3247},[681,726],[618,3249,658],{"className":3250},[733],[3018,3252,3253],{},[618,3254,3256,3275],{"className":3255},[621],[618,3257,3259],{"className":3258},[625],[627,3260,3261],{"xmlns":629},[631,3262,3263,3273],{},[634,3264,3265,3267,3269,3271],{},[637,3266,640],{"mathvariant":639},[642,3268,645],{"stretchy":644},[637,3270,651],{},[642,3272,658],{"stretchy":644},[660,3274,2971],{"encoding":662},[618,3276,3278],{"className":3277,"ariaHidden":668},[667],[618,3279,3281,3284,3287,3290,3293],{"className":3280},[672],[618,3282],{"className":3283,"style":2263},[676],[618,3285,640],{"className":3286,"style":683},[681,682],[618,3288,645],{"className":3289},[687],[618,3291,651],{"className":3292},[681,694],[618,3294,658],{"className":3295},[733],[3297,3298,3299],"note",{},[598,3300,3301,3302,3305],{},"Le tri par insertion est ",[601,3303,3304],{},"plus rapide sur des tableaux presque triés",". Le\ntri par sélection ignore le contenu — il fait toujours le même nombre de\ncomparaisons.",[1359,3307,3309,3314],{"correct":1023,"slug":3308},"tris-c2",[1363,3310,3311],{"v-slot:question":841},[598,3312,3313],{},"Le tri par insertion sur un tableau déjà trié coûte :",[1363,3315,3316],{"v-slot:choices":841},[756,3317,3318,3321,3324],{},[759,3319,3320],{},"O(1)",[759,3322,3323],{},"O(n)",[759,3325,3326],{},"O(n²)",[593,3328,3330],{"id":3329},"pièges-courants","Pièges courants",[756,3332,3333,3347,3364],{},[759,3334,3335,3338,3339,3342,3343,3346],{},[601,3336,3337],{},"Confondre les deux"," : la sélection ",[601,3340,3341],{},"cherche"," un minimum puis échange,\nl'insertion ",[601,3344,3345],{},"décale"," des éléments puis pose.",[759,3348,3349,3352,3353,3355,3356,3359,3360,3363],{},[601,3350,3351],{},"Oublier la terminaison"," du ",[843,3354,1977],{}," dans le tri par insertion :\n",[843,3357,3358],{},"j >= 0"," est essentiel — sinon ",[843,3361,3362],{},"t[-1]"," lit la fin du tableau (bug subtil\nen Python).",[759,3365,3366,3443,3444,3498,3499,3586],{},[601,3367,3368,3369,3442],{},"Croire qu'un tri en ",[618,3370,3372,3395],{"className":3371},[621],[618,3373,3375],{"className":3374},[625],[627,3376,3377],{"xmlns":629},[631,3378,3379,3393],{},[634,3380,3381,3383,3385,3391],{},[637,3382,640],{"mathvariant":639},[642,3384,645],{"stretchy":644},[647,3386,3387,3389],{},[637,3388,651],{},[653,3390,655],{},[642,3392,658],{"stretchy":644},[660,3394,663],{"encoding":662},[618,3396,3398],{"className":3397,"ariaHidden":668},[667],[618,3399,3401,3404,3407,3410,3439],{"className":3400},[672],[618,3402],{"className":3403,"style":677},[676],[618,3405,640],{"className":3406,"style":683},[681,682],[618,3408,645],{"className":3409},[687],[618,3411,3413,3416],{"className":3412},[681],[618,3414,651],{"className":3415},[681,694],[618,3417,3419],{"className":3418},[698],[618,3420,3422],{"className":3421},[702],[618,3423,3425],{"className":3424},[706],[618,3426,3428],{"className":3427,"style":711},[710],[618,3429,3430,3433],{"style":714},[618,3431],{"className":3432,"style":719},[718],[618,3434,3436],{"className":3435},[723,724,725,726],[618,3437,655],{"className":3438},[681,726],[618,3440,658],{"className":3441},[733]," est inutilisable"," : pour\n",[618,3445,3447,3467],{"className":3446},[621],[618,3448,3450],{"className":3449},[625],[627,3451,3452],{"xmlns":629},[631,3453,3454,3464],{},[634,3455,3456,3458,3461],{},[637,3457,651],{},[642,3459,3460],{},"≤",[653,3462,3463],{},"100",[660,3465,3466],{"encoding":662},"n \\leq 100",[618,3468,3470,3489],{"className":3469,"ariaHidden":668},[667],[618,3471,3473,3477,3480,3483,3486],{"className":3472},[672],[618,3474],{"className":3475,"style":3476},[676],"height:0.7719em;vertical-align:-0.136em;",[618,3478,651],{"className":3479},[681,694],[618,3481],{"className":3482,"style":1877},[1189],[618,3484,3460],{"className":3485},[1881],[618,3487],{"className":3488,"style":1877},[1189],[618,3490,3492,3495],{"className":3491},[672],[618,3493],{"className":3494,"style":1204},[676],[618,3496,3463],{"className":3497},[681],", c'est très bien. Pour ",[618,3500,3502,3526],{"className":3501},[621],[618,3503,3505],{"className":3504},[625],[627,3506,3507],{"xmlns":629},[631,3508,3509,3523],{},[634,3510,3511,3513,3515],{},[637,3512,651],{},[642,3514,869],{},[647,3516,3517,3520],{},[653,3518,3519],{},"10",[653,3521,3522],{},"6",[660,3524,3525],{"encoding":662},"n = 10^6",[618,3527,3529,3547],{"className":3528,"ariaHidden":668},[667],[618,3530,3532,3535,3538,3541,3544],{"className":3531},[672],[618,3533],{"className":3534,"style":1345},[676],[618,3536,651],{"className":3537},[681,694],[618,3539],{"className":3540,"style":1877},[1189],[618,3542,869],{"className":3543},[1881],[618,3545],{"className":3546,"style":1877},[1189],[618,3548,3550,3553,3556],{"className":3549},[672],[618,3551],{"className":3552,"style":711},[676],[618,3554,1023],{"className":3555},[681],[618,3557,3559,3563],{"className":3558},[681],[618,3560,3562],{"className":3561},[681],"0",[618,3564,3566],{"className":3565},[698],[618,3567,3569],{"className":3568},[702],[618,3570,3572],{"className":3571},[706],[618,3573,3575],{"className":3574,"style":711},[710],[618,3576,3577,3580],{"style":714},[618,3578],{"className":3579,"style":719},[718],[618,3581,3583],{"className":3582},[723,724,725,726],[618,3584,3522],{"className":3585},[681,726],", c'est rédhibitoire.",[593,3588,3590],{"id":3589},"pour-aller-plus-loin","Pour aller plus loin",[598,3592,3593,3594,3665,3666,3669,3670,3673],{},"Il existe des tris bien plus rapides — tri fusion, tri rapide, tri par tas —\ntous en ",[618,3595,3597,3625],{"className":3596},[621],[618,3598,3600],{"className":3599},[625],[627,3601,3602],{"xmlns":629},[631,3603,3604,3622],{},[634,3605,3606,3608,3610,3612,3615,3618,3620],{},[637,3607,640],{"mathvariant":639},[642,3609,645],{"stretchy":644},[637,3611,651],{},[637,3613,3614],{},"log",[642,3616,3617],{},"⁡",[637,3619,651],{},[642,3621,658],{"stretchy":644},[660,3623,3624],{"encoding":662},"\\mathcal{O}(n \\log n)",[618,3626,3628],{"className":3627,"ariaHidden":668},[667],[618,3629,3631,3634,3637,3640,3643,3647,3656,3659,3662],{"className":3630},[672],[618,3632],{"className":3633,"style":2263},[676],[618,3635,640],{"className":3636,"style":683},[681,682],[618,3638,645],{"className":3639},[687],[618,3641,651],{"className":3642},[681,694],[618,3644],{"className":3645,"style":3646},[1189],"margin-right:0.1667em;",[618,3648,3651,3652],{"className":3649},[3650],"mop","lo",[618,3653,3655],{"style":3654},"margin-right:0.0139em;","g",[618,3657],{"className":3658,"style":3646},[1189],[618,3660,651],{"className":3661},[681,694],[618,3663,658],{"className":3664},[733],". Vous les rencontrerez en Terminale. Pour\nl'instant, retenez que ",[601,3667,3668],{},"trier coûte cher"," — d'où l'intérêt d'avoir une\nrecherche efficace dans un tableau ",[601,3671,3672],{},"déjà trié",", ce qui amène au cours\nsuivant.",[3675,3676,3677],"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 .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: 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: ",[601,3794,3795],{},"sélection"," (chercher le min,\néchanger), ",[601,3798,3799],{},"insertion"," (insérer à la bonne place dans la zone triée).",[593,3802,3804],{"id":3803},"à-mémoriser","À mémoriser",[598,3806,3807],{},[601,3808,2026],{},[756,3810,3811,3817,3820,3909],{},[759,3812,3813,3814,734],{},"Découpe le tableau en ",[843,3815,3816],{},"[trié | non trié]",[759,3818,3819],{},"À chaque tour, cherche le minimum à droite et l'échange avec la première\ncase non triée.",[759,3821,3822,3825,3826,1011,3854,3857,3858,3908],{},[601,3823,3824],{},"Invariant"," : après le tour 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chaque tour, prend l'élément courant et l'insère à sa place dans\n",[843,3999,1833],{}," (déjà trié) en décalant.",[759,4002,4003,4005,4006,1011,4034,4036],{},[601,4004,3824],{}," : au début du tour ",[618,4007,4009,4022],{"className":4008},[621],[618,4010,4012],{"className":4011},[625],[627,4013,4014],{"xmlns":629},[631,4015,4016,4020],{},[634,4017,4018],{},[637,4019,790],{},[660,4021,790],{"encoding":662},[618,4023,4025],{"className":4024,"ariaHidden":668},[667],[618,4026,4028,4031],{"className":4027},[672],[618,4029],{"className":4030,"style":802},[676],[618,4032,790],{"className":4033},[681,694],[843,4035,1833],{}," est trié.",[759,4038,4039,4040,4113,4114,734],{},"Coût pire cas : 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var(--shiki-dark-text-decoration);}",{"title":841,"searchDepth":863,"depth":863,"links":4462},[4463,4464,4465,4466,4467],{"id":3714,"depth":863,"text":3715},{"id":3803,"depth":863,"text":3804},{"id":4172,"depth":863,"text":4173},{"id":3329,"depth":863,"text":3330},{"id":4240,"depth":863,"text":4241},"Mémo de révision sur les deux tris quadratiques classiques.",{},"\u002Fnsi-1ere\u002Fph-algorithmique\u002Ftris.memo",[],{"title":3708,"description":4468},"1.nsi-1ere\u002Fph-algorithmique\u002F3.tris.memo","vpfLEyDJV7X_wjOG3lIaAWNoY4r58rJpv5gTUIDeDsY",{"id":4476,"title":4477,"body":4478,"description":5757,"extension":3694,"meta":5758,"navigation":3701,"path":5759,"seo":5760,"stem":5761,"__hash__":5762},"coursSlides\u002F1.nsi-1ere\u002Fph-algorithmique\u002F3.tris.slides.md","Tris par sélection et par insertion — diapositives",{"type":590,"value":4479,"toc":5755},[4480,4485,4488,4491,4495,4521,4523,4527,4620,4622,4626,4711,4713,4717,4731,4733,4737,4856,4858,4862,4898,4900,4902,4921,4923,4927,5219,5222,5224,5228,5260,5611,5613,5617,5753],[4481,4482,4484],"h1",{"id":4483},"trier-un-tableau","Trier un tableau",[598,4486,4487],{},"Deux algorithmes naïfs, deux invariants, un même coût.",[4489,4490],"hr",{},[4481,4492,4494],{"id":4493},"tri-par-sélection-principe","Tri par sélection — principe",[756,4496,4497,4502,4518],{},[759,4498,4499,4500],{},"Découpe : ",[843,4501,3816],{},[759,4503,4504,4505],{},"À chaque étape :\n",[4506,4507,4508,4515],"ol",{},[759,4509,4510,4511,4514],{},"Cherche le ",[601,4512,4513],{},"minimum"," de la zone non triée",[759,4516,4517],{},"Échange avec la première case non triée",[759,4519,4520],{},"Zone triée croît d'un élément par tour",[4489,4522],{},[4481,4524,4526],{"id":4525},"tri-par-sélection-code","Tri par sélection — code",[836,4528,4530],{"className":838,"code":4529,"language":840,"meta":841,"style":841},"def tri_selection(t):\n    n = len(t)\n    for i in range(n):\n        i_min = i\n        for j in range(i + 1, n):\n            if t[j] \u003C t[i_min]:\n                i_min = j\n        t[i], t[i_min] = t[i_min], t[i]\n    return t\n",[843,4531,4532,4540,4550,4562,4570,4588,4598,4606,4614],{"__ignoreMap":841},[618,4533,4534,4536,4538],{"class":847,"line":848},[618,4535,852],{"class":851},[618,4537,856],{"class":855},[618,4539,860],{"class":859},[618,4541,4542,4544,4546,4548],{"class":847,"line":863},[618,4543,866],{"class":859},[618,4545,869],{"class":851},[618,4547,873],{"class":872},[618,4549,876],{"class":859},[618,4551,4552,4554,4556,4558,4560],{"class":847,"line":879},[618,4553,882],{"class":851},[618,4555,885],{"class":859},[618,4557,888],{"class":851},[618,4559,891],{"class":872},[618,4561,894],{"class":859},[618,4563,4564,4566,4568],{"class":847,"line":897},[618,4565,907],{"class":859},[618,4567,869],{"class":851},[618,4569,912],{"class":859},[618,4571,4572,4574,4576,4578,4580,4582,4584,4586],{"class":847,"line":904},[618,4573,918],{"class":851},[618,4575,921],{"class":859},[618,4577,888],{"class":851},[618,4579,891],{"class":872},[618,4581,928],{"class":859},[618,4583,931],{"class":851},[618,4585,934],{"class":872},[618,4587,937],{"class":859},[618,4589,4590,4592,4594,4596],{"class":847,"line":915},[618,4591,943],{"class":851},[618,4593,946],{"class":859},[618,4595,949],{"class":851},[618,4597,952],{"class":859},[618,4599,4600,4602,4604],{"class":847,"line":940},[618,4601,958],{"class":859},[618,4603,869],{"class":851},[618,4605,963],{"class":859},[618,4607,4608,4610,4612],{"class":847,"line":955},[618,4609,975],{"class":859},[618,4611,869],{"class":851},[618,4613,980],{"class":859},[618,4615,4616,4618],{"class":847,"line":966},[618,4617,986],{"class":851},[618,4619,989],{"class":859},[4489,4621],{},[4481,4623,4625],{"id":4624},"tri-par-sélection-invariant","Tri par sélection — invariant",[1110,4627,4628],{},[598,4629,1114,4630,1011,4658,3857,4660,4710],{},[618,4631,4633,4646],{"className":4632},[621],[618,4634,4636],{"className":4635},[625],[627,4637,4638],{"xmlns":629},[631,4639,4640,4644],{},[634,4641,4642],{},[637,4643,790],{},[660,4645,790],{"encoding":662},[618,4647,4649],{"className":4648,"ariaHidden":668},[667],[618,4650,4652,4655],{"className":4651},[672],[618,4653],{"className":4654,"style":802},[676],[618,4656,790],{"className":4657},[681,694],[843,4659,3856],{},[618,4661,4663,4680],{"className":4662},[621],[618,4664,4666],{"className":4665},[625],[627,4667,4668],{"xmlns":629},[631,4669,4670,4678],{},[634,4671,4672,4674,4676],{},[637,4673,790],{},[642,4675,931],{},[653,4677,1023],{},[660,4679,1172],{"encoding":662},[618,4681,4683,4701],{"className":4682,"ariaHidden":668},[667],[618,4684,4686,4689,4692,4695,4698],{"className":4685},[672],[618,4687],{"className":4688,"style":1182},[676],[618,4690,790],{"className":4691},[681,694],[618,4693],{"className":4694,"style":1190},[1189],[618,4696,931],{"className":4697},[1194],[618,4699],{"className":4700,"style":1190},[1189],[618,4702,4704,4707],{"className":4703},[672],[618,4705],{"className":4706,"style":1204},[676],[618,4708,1023],{"className":4709},[681]," plus petits éléments,\nrangés dans l'ordre croissant.",[4489,4712],{},[4481,4714,4716],{"id":4715},"tri-par-insertion-principe","Tri par insertion — principe",[756,4718,4719,4722,4728],{},[759,4720,4721],{},"Comme trier une main de cartes",[759,4723,4724,4725,4727],{},"À chaque étape, on ",[601,4726,1395],{}," la carte courante à sa place dans\nla main déjà triée",[759,4729,4730],{},"On décale les cartes plus grandes d'une position vers la droite",[4489,4732],{},[4481,4734,4736],{"id":4735},"tri-par-insertion-code","Tri par insertion — code",[836,4738,4740],{"className":838,"code":4739,"language":840,"meta":841,"style":841},"def tri_insertion(t):\n    n = len(t)\n    for i in range(1, n):\n        valeur = t[i]\n        j = i - 1\n        while j >= 0 and t[j] > valeur:\n            t[j + 1] = t[j]\n            j -= 1\n        t[j + 1] = valeur\n    return t\n",[843,4741,4742,4750,4760,4776,4784,4796,4814,4828,4836,4850],{"__ignoreMap":841},[618,4743,4744,4746,4748],{"class":847,"line":848},[618,4745,852],{"class":851},[618,4747,1425],{"class":855},[618,4749,860],{"class":859},[618,4751,4752,4754,4756,4758],{"class":847,"line":863},[618,4753,866],{"class":859},[618,4755,869],{"class":851},[618,4757,873],{"class":872},[618,4759,876],{"class":859},[618,4761,4762,4764,4766,4768,4770,4772,4774],{"class":847,"line":879},[618,4763,882],{"class":851},[618,4765,885],{"class":859},[618,4767,888],{"class":851},[618,4769,891],{"class":872},[618,4771,645],{"class":859},[618,4773,1023],{"class":872},[618,4775,937],{"class":859},[618,4777,4778,4780,4782],{"class":847,"line":897},[618,4779,1463],{"class":859},[618,4781,869],{"class":851},[618,4783,1468],{"class":859},[618,4785,4786,4788,4790,4792,4794],{"class":847,"line":904},[618,4787,1473],{"class":859},[618,4789,869],{"class":851},[618,4791,885],{"class":859},[618,4793,1480],{"class":851},[618,4795,1483],{"class":872},[618,4797,4798,4800,4802,4804,4806,4808,4810,4812],{"class":847,"line":915},[618,4799,1493],{"class":851},[618,4801,921],{"class":859},[618,4803,1498],{"class":851},[618,4805,1501],{"class":872},[618,4807,1504],{"class":851},[618,4809,946],{"class":859},[618,4811,1509],{"class":851},[618,4813,1512],{"class":859},[618,4815,4816,4818,4820,4822,4824,4826],{"class":847,"line":940},[618,4817,1517],{"class":859},[618,4819,931],{"class":851},[618,4821,934],{"class":872},[618,4823,1524],{"class":859},[618,4825,869],{"class":851},[618,4827,1529],{"class":859},[618,4829,4830,4832,4834],{"class":847,"line":955},[618,4831,1534],{"class":859},[618,4833,1537],{"class":851},[618,4835,1483],{"class":872},[618,4837,4838,4840,4842,4844,4846,4848],{"class":847,"line":966},[618,4839,1544],{"class":859},[618,4841,931],{"class":851},[618,4843,934],{"class":872},[618,4845,1524],{"class":859},[618,4847,869],{"class":851},[618,4849,1555],{"class":859},[618,4851,4852,4854],{"class":847,"line":972},[618,4853,986],{"class":851},[618,4855,989],{"class":859},[4489,4857],{},[4481,4859,4861],{"id":4860},"tri-par-insertion-invariant","Tri par insertion — invariant",[1110,4863,4864],{},[598,4865,4866,4867,1011,4895,4897],{},"Au début du tour ",[618,4868,4870,4883],{"className":4869},[621],[618,4871,4873],{"className":4872},[625],[627,4874,4875],{"xmlns":629},[631,4876,4877,4881],{},[634,4878,4879],{},[637,4880,790],{},[660,4882,790],{"encoding":662},[618,4884,4886],{"className":4885,"ariaHidden":668},[667],[618,4887,4889,4892],{"className":4888},[672],[618,4890],{"className":4891,"style":802},[676],[618,4893,790],{"className":4894},[681,694],[843,4896,1833],{}," contient les mêmes éléments qu'au\ndépart — mais triés.",[4489,4899],{},[4481,4901,1304],{"id":1303},[756,4903,4904,4910],{},[759,4905,4906,4907,4909],{},"Boucles ",[843,4908,1314],{}," sur intervalles finis → toujours OK",[759,4911,4912,4914,4915,4917,4918],{},[843,4913,1977],{}," du tri insertion : ",[843,4916,1981],{}," décroît à chaque tour, s'arrête au plus\ntard quand ",[843,4919,4920],{},"j \u003C 0",[4489,4922],{},[4481,4924,4926],{"id":4925},"coût","Coût",[2994,4928,4929,4939],{},[2997,4930,4931],{},[3000,4932,4933,4935,4937],{},[3003,4934,3005],{},[3003,4936,3008],{},[3003,4938,3011],{},[3013,4940,4941,5095],{},[3000,4942,4943,4945,5020],{},[3018,4944,3020],{},[3018,4946,4947],{},[618,4948,4950,4973],{"className":4949},[621],[618,4951,4953],{"className":4952},[625],[627,4954,4955],{"xmlns":629},[631,4956,4957,4971],{},[634,4958,4959,4961,4963,4969],{},[637,4960,640],{"mathvariant":639},[642,4962,645],{"stretchy":644},[647,4964,4965,4967],{},[637,4966,651],{},[653,4968,655],{},[642,4970,658],{"stretchy":644},[660,4972,663],{"encoding":662},[618,4974,4976],{"className":4975,"ariaHidden":668},[667],[618,4977,4979,4982,4985,4988,5017],{"className":4978},[672],[618,4980],{"className":4981,"style":677},[676],[618,4983,640],{"className":4984,"style":683},[681,682],[618,4986,645],{"className":4987},[687],[618,4989,4991,4994],{"className":4990},[681],[618,4992,651],{"className":4993},[681,694],[618,4995,4997],{"className":4996},[698],[618,4998,5000],{"className":4999},[702],[618,5001,5003],{"className":5002},[706],[618,5004,5006],{"className":5005,"style":711},[710],[618,5007,5008,5011],{"style":714},[618,5009],{"className":5010,"style":719},[718],[618,5012,5014],{"className":5013},[723,724,725,726],[618,5015,655],{"className":5016},[681,726],[618,5018,658],{"className":5019},[733],[3018,5021,5022],{},[618,5023,5025,5048],{"className":5024},[621],[618,5026,5028],{"className":5027},[625],[627,5029,5030],{"xmlns":629},[631,5031,5032,5046],{},[634,5033,5034,5036,5038,5044],{},[637,5035,640],{"mathvariant":639},[642,5037,645],{"stretchy":644},[647,5039,5040,5042],{},[637,5041,651],{},[653,5043,655],{},[642,5045,658],{"stretchy":644},[660,5047,663],{"encoding":662},[618,5049,5051],{"className":5050,"ariaHidden":668},[667],[618,5052,5054,5057,5060,5063,5092],{"className":5053},[672],[618,5055],{"className":5056,"style":677},[676],[618,5058,640],{"className":5059,"style":683},[681,682],[618,5061,645],{"className":5062},[687],[618,5064,5066,5069],{"className":5065},[681],[618,5067,651],{"className":5068},[681,694],[618,5070,5072],{"className":5071},[698],[618,5073,5075],{"className":5074},[702],[618,5076,5078],{"className":5077},[706],[618,5079,5081],{"className":5080,"style":711},[710],[618,5082,5083,5086],{"style":714},[618,5084],{"className":5085,"style":719},[718],[618,5087,5089],{"className":5088},[723,724,725,726],[618,5090,655],{"className":5091},[681,726],[618,5093,658],{"className":5094},[733],[3000,5096,5097,5099,5174],{},[3018,5098,3175],{},[3018,5100,5101],{},[618,5102,5104,5127],{"className":5103},[621],[618,5105,5107],{"className":5106},[625],[627,5108,5109],{"xmlns":629},[631,5110,5111,5125],{},[634,5112,5113,5115,5117,5123],{},[637,5114,640],{"mathvariant":639},[642,5116,645],{"stretchy":644},[647,5118,5119,5121],{},[637,5120,651],{},[653,5122,655],{},[642,5124,658],{"stretchy":644},[660,5126,663],{"encoding":662},[618,5128,5130],{"className":5129,"ariaHidden":668},[667],[618,5131,5133,5136,5139,5142,5171],{"className":5132},[672],[618,5134],{"className":5135,"style":677},[676],[618,5137,640],{"className":5138,"style":683},[681,682],[618,5140,645],{"className":5141},[687],[618,5143,5145,5148],{"className":5144},[681],[618,5146,651],{"className":5147},[681,694],[618,5149,5151],{"className":5150},[698],[618,5152,5154],{"className":5153},[702],[618,5155,5157],{"className":5156},[706],[618,5158,5160],{"className":5159,"style":711},[710],[618,5161,5162,5165],{"style":714},[618,5163],{"className":5164,"style":719},[718],[618,5166,5168],{"className":5167},[723,724,725,726],[618,5169,655],{"className":5170},[681,726],[618,5172,658],{"className":5173},[733],[3018,5175,5176],{},[618,5177,5179,5198],{"className":5178},[621],[618,5180,5182],{"className":5181},[625],[627,5183,5184],{"xmlns":629},[631,5185,5186,5196],{},[634,5187,5188,5190,5192,5194],{},[637,5189,640],{"mathvariant":639},[642,5191,645],{"stretchy":644},[637,5193,651],{},[642,5195,658],{"stretchy":644},[660,5197,2971],{"encoding":662},[618,5199,5201],{"className":5200,"ariaHidden":668},[667],[618,5202,5204,5207,5210,5213,5216],{"className":5203},[672],[618,5205],{"className":5206,"style":2263},[676],[618,5208,640],{"className":5209,"style":683},[681,682],[618,5211,645],{"className":5212},[687],[618,5214,651],{"className":5215},[681,694],[618,5217,658],{"className":5218},[733],[598,5220,5221],{},"Insertion plus rapide sur tableaux presque triés.",[4489,5223],{},[4481,5225,5227],{"id":5226},"pourquoi-quadratique","Pourquoi quadratique ?",[598,5229,5230,5231,5259],{},"Pour ",[618,5232,5234,5247],{"className":5233},[621],[618,5235,5237],{"className":5236},[625],[627,5238,5239],{"xmlns":629},[631,5240,5241,5245],{},[634,5242,5243],{},[637,5244,651],{},[660,5246,651],{"encoding":662},[618,5248,5250],{"className":5249,"ariaHidden":668},[667],[618,5251,5253,5256],{"className":5252},[672],[618,5254],{"className":5255,"style":1345},[676],[618,5257,651],{"className":5258},[681,694]," éléments :",[618,5261,5263],{"className":5262},[2185],[618,5264,5266,5343],{"className":5265},[621],[618,5267,5269],{"className":5268},[625],[627,5270,5271],{"xmlns":629,"display":2194},[631,5272,5273,5340],{},[634,5274,5275,5277,5279,5281,5283,5285,5287,5289,5291,5293,5295,5297,5299,5301,5303,5305,5307,5325,5328,5330,5332,5338],{},[642,5276,645],{"stretchy":644},[637,5278,651],{},[642,5280,1260],{},[653,5282,1023],{},[642,5284,658],{"stretchy":644},[642,5286,931],{},[642,5288,645],{"stretchy":644},[637,5290,651],{},[642,5292,1260],{},[653,5294,655],{},[642,5296,658],{"stretchy":644},[642,5298,931],{},[642,5300,2225],{},[642,5302,931],{},[653,5304,1023],{},[642,5306,869],{},[2233,5308,5309,5323],{},[634,5310,5311,5313,5315,5317,5319,5321],{},[637,5312,651],{},[642,5314,645],{"stretchy":644},[637,5316,651],{},[642,5318,1260],{},[653,5320,1023],{},[642,5322,658],{"stretchy":644},[653,5324,655],{},[642,5326,5327],{},"∈",[637,5329,640],{"mathvariant":639},[642,5331,645],{"stretchy":644},[647,5333,5334,5336],{},[637,5335,651],{},[653,5337,655],{},[642,5339,658],{"stretchy":644},[660,5341,5342],{"encoding":662},"(n-1) + (n-2) + \\dots + 1 = \\frac{n(n-1)}{2} \\in 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