From 52d9f72cdaf34823a62a89017e9656d12c41416c Mon Sep 17 00:00:00 2001 From: Alexandre Agostinho Date: Fri, 24 Mar 2023 23:49:44 +0100 Subject: [PATCH] fin --- code/p1e4.c | 2 +- main.pdf | Bin 248108 -> 263409 bytes main.tex | 7 ++++++ sections/concl.tex | 16 +++++++++++++ sections/intro.tex | 13 +++++++++++ sections/partie1.tex | 53 +++++++++++++++++++++++++++++++++++-------- sections/partie2.tex | 42 ++++++++++++++++++---------------- 7 files changed, 103 insertions(+), 30 deletions(-) create mode 100644 sections/concl.tex create mode 100644 sections/intro.tex diff --git a/code/p1e4.c b/code/p1e4.c index 9ddd865..e701150 100644 --- a/code/p1e4.c +++ b/code/p1e4.c @@ -50,7 +50,7 @@ int isAccessREC(PointCaracteristique *A, PointCaracteristique *B) return 0; } -// version iterative +// version iterative (ebauche de travail) int isAccessITE(PointCaracteristique *A, PointCaracteristique *B) { int i = 0; diff --git a/main.pdf b/main.pdf index 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z&tY9t1{SQei17htl1YTPPAP*z#{ar~FP@uTbMGbaJ@Y1graP=RuIGOQ_#btQ&c_Gmk*fLr&RrG5+>_#u zO#O?FvK{l853*X)%H5i>S_ZwFxSGksqnc`Cvh1apnni0R&mXTbQUqmoP2bh;Y6S;^ zj|J9sB6ljPR0l!lLFcCX|F<~J53MLj{I7hc9OC)S{izfaxvLrD!iRXtRkmkIp>#xe n>Y}dcoltRI$@UJNqJ!j+G7JGh0MEwB#KsLvPA;w}0sDUdtFc8! diff --git a/main.tex b/main.tex index 9c259a9..88ee476 100644 --- a/main.tex +++ b/main.tex @@ -103,6 +103,10 @@ \tableofcontents \newpage + % Intro + \import{sections/}{intro} + \newpage + % Other pages (sections) \import{sections/}{partie1} \newpage @@ -110,4 +114,7 @@ \import{sections/}{partie2} \newpage + % Conclu + \import{sections/}{concl} + \end{document} \ No newline at end of file diff --git a/sections/concl.tex b/sections/concl.tex new file mode 100644 index 0000000..8cacd4f --- /dev/null +++ b/sections/concl.tex @@ -0,0 +1,16 @@ +% Section Conclusion - SAE 2.02 + +\documentclass[class=article, crop=false]{standalone} + +\begin{document} + \section*{Conclusion} + + \par\quad Nous avons donc pu analyser et comprendre par ces differantes études algorithmiques de problèmes l'importance et l'impact que cela pouvait avoir dans un programme, notamment lorsque il est nécessaire d'effectuer de gros calculs. + + + \section*{Liens annexes} + + Lien vers le dépot git du projet : \href{https://codefirst.iut.uca.fr/git/alexandre.agostinho/SAE-2.02}{https://codefirst.iut.uca.fr/git/alexandre.agostinho/SAE-2.02} + + +\end{document} diff --git a/sections/intro.tex b/sections/intro.tex new file mode 100644 index 0000000..77a1bb0 --- /dev/null +++ b/sections/intro.tex @@ -0,0 +1,13 @@ +% Section Introduction - SAE 2.02 + +\documentclass[class=article, crop=false]{standalone} + +\begin{document} + \section*{Introduction} + + \par\quad Dans ce rapport nous allons voir differentes études sur l'exploration et la complexité algorithmique d'un problème. Le but étant d'analyser un problème avec méthode et de comparer des algorithmes pour des problèmes classiques. \\ + \quad Nous verrons donc dans un premier temps, l'étude d'un réseau de points caractéristiques et de rues d'une ville, et dans un second temps l'étude de 3 algorithmes de tri avec leur différances, leurs avantages et inconvéniants. + + \par\quad Je tenais aussi à préciser que, par manque de temps, je n'ai pas pu réaliser l'exercice 2.1 et 2.3. Par ailleurs, la question 2.2 et 2.4 ont été répondu ensemble. + +\end{document} \ No newline at end of file diff --git a/sections/partie1.tex b/sections/partie1.tex index 37ed4f2..6ff58ca 100644 --- a/sections/partie1.tex +++ b/sections/partie1.tex @@ -11,7 +11,7 @@ \quad Pour représenter ce réseaux, je propose la structure de données suivante : \lstinputlisting[language=C,firstline=8,lastline=17]{code/p1e1.c} - \par\quad La structure du point se compose de son nom et d'un tableau de pointeurs sur d'autre points. Le réseau est ainsi composé d'un tableau de pointeur sur des structures `pointsCaracteristique', elles-même pointant sur une un ou plusieurs point de la liste. + \par\quad La structure du point se compose de son nom et d'un tableau de pointeurs sur d'autre points. Le réseau est ainsi composé d'un tableau de pointeur sur des structures $pointsCaracteristique$, elles-même pointant sur un ou plusieurs point de la liste. \par\quad Le réseau est caractérisé par ses points caractéristiques. Chaque point connaît les points accessibles de manière direct. Cela permet de représenter les rues et leur sens. Prenons par exemple, une rue qui part du point A et qui a pour destination le point B. Elle sera caractérisée par la présence du point B dans la liste des points accessibles du point A. De cette manière on peut facilement dessiner le réseau et calculer les trajets à suivre. En somme, les pointeurs servent à caractériser les rues et leurs sens. @@ -19,7 +19,7 @@ \subsection{Question 2} \par\quad Le choix d'une telle structure est fait sur la base de la simplicité d'implémentation. En effet, cette structure est relativement simple. Elle ne travaille quasiment que sur de la manipulation de pointeurs. Le parcours des points caractéristiques est donc facile. - \par\quad De plus, si on se penche sur la complexité de l'implémentation, le nombre d'opérations requise pour le parcours du réseau est proportionnel au nombre de points que l'on parcours. Au pire, tout les points seront parcouru : c'est-à-dire que la complexité maximale s'élève au nombre de point qui compose le réseaux. + \par\quad De plus, si on se penche sur la complexité de l'implémentation, le nombre d'opérations requise pour le parcours du réseau est proportionnel au nombre de points que l'on parcourt. Au pire, tout les points seront parcouru : c'est-à-dire que la complexité maximale s'élève au nombre de point qui compose le réseaux. \subsection{Question 3} @@ -30,24 +30,59 @@ \subsection{Question 4} - \par\quad Pour savoir si B est accessible depuis A, directement ou non (par une ou plusieurs rues), il est possible de répéter récursivement la fonction précédante. Ainsi, elle va tester chaque point independamment et si un chemain direct est trouvé, c'est que il y à forcément un chemain qui rend B accessible. + \par\quad Pour savoir si B est accessible depuis A, directement ou non (par une ou plusieurs rues), il est possible de répéter récursivement la fonction précédante. Ainsi, elle va tester chaque point independamment et si un chemin direct est trouvé, c'est que il y a forcément un chemain qui rend B accessible. \par\quad Voici le code correspondant : \lstinputlisting[language=C,firstline=36,lastline=51]{code/p1e4.c} \par\quad Voici maintenant une ébauche de travail qui visait à créé une version itérative de cette fonction : - \lstinputlisting[language=C,firstline=54,lastline=89]{code/p1e4.c} + \lstinputlisting[language=C,firstline=53,lastline=89]{code/p1e4.c} - \par\quad Je me suis rendu compte que je n'arrivait pas à pensé l'algorithme autrement que récursivement, c'est pourquoi j'ai décidé de mettre cette ébauche de côté. \\ - \quad Neanmoin, il est quand même possible de réutiliser la partie de code qui teste si un point à déjà été testé. Ceci diviserait, par chaque points testé, la complexité de cette algorithme. + \par\quad Je me suis rendu compte que je n'arrivait pas à penser l'algorithme autrement que récursivement, c'est pourquoi j'ai décidé de mettre cette ébauche de côté. + + \par\quad Néanmoin, il est quand même possible de réutiliser la partie de code qui teste si un point à déjà été testé. Ceci diviserait, par chaque points testé, la complexité de cette algorithme. - \par\quad Voici maintenant une ébauche de travail qui visait à créé une version itérative de cette fonction : - \lstinputlisting[language=C,firstline=91,lastline=123]{code/p1e4.c} + \par\quad Voici ce que donne l'algorithme avec les modifications apportés : + \lstinputlisting[language=C,firstline=91,lastline=122]{code/p1e4.c} \subsection{Question 5} - \par\quad + \subsubsection{Analyse de l'algorithme de test de l'accès direct} + + \par\quad Analysons le premier algorithme, c'est-à-dire celui de la question n°3. + + \par\quad Cet algorithme se base sur un parcours de tableau grâce à une boucle $for$. Dans le principe de ce parcours, on commence du début du tableau, passe un par un les éléments et termine la boucle au moment où l'on trouve l'élèment que l'on cherche. + + \par\quad La boucle for est prévue pour boucler $N$ fois, $N$ étant le nombre d'élèments du tableau dans lequel on effectue la recherche. Cependant, cette boucle est capable de se terminer prématurément si la condition à l'intérieur se valide. + + \par\quad La complexité maximale de ce code est donc linéaire : $o(N)$, où $N$ est le nombre d'élèments maximums à parourir. Le nombre total d'opérations se calcul par la formule : + \[3N+1\] + \quad avec l'initialisation de la variable $i$, le test et l'incrémentation de la boucle $for$, et le test d'égalité de valeur dans la boucle. + + \newpage + + \subsubsection{Analyse de l'algorithme de test de l'accès indirect} + + \par\quad Analysons le second algorithme, c'est-à-dire celui de la question n°4. + + \par\quad Cet algorithme se base sur un parcours récursif sur l'ensemble des points accessibles depuis le point de départ, à la recherche du point d'arrivé. Nous allons étudier la version dite "améliorée" car elle se trouve être plus pertinente. En effet, elle propose de garder en mémoire les points sur lequels l'algorithme est déjà passé et donc déjà testé, de manière à omettre un ènième passage récursif sur ce point, et donc d'éviter une complexité exponentielle et potentiellement une boucle interminable. + + \par\quad Commençons par analyser la fonction $isInTab$, servant de fonction utilitaire afin de rechercher si un $PointCaracteristique$ se trouve dans le tableau des points déjà passé. Cette fonction, est en réalité identique à la fonction de l'exercice n°3, à la différance qu'elle ne recherche pas les valeurs au même endroit. Sa complexité est donc d'ordre linéaire($o(N)$), avec $3N+1$ instructions effectuées par appel. + + \par\quad Attaquons-nous maintenant au gros de l'algorithme. Comme expliqué plus haut, le principe est d'appliquer cet algorithme sur tout les points en accès direct d'un point. Alors on y retrouve une boucle $for$ qui boucle $N$ fois, $N$ étant ce nombre de points directement accessibles. Ensuite, on utilise la fonction $isInTab$ pour savoir si l'on doit lancer une recherche sur ce point où si cela a déjà été fait. On est donc à une complexité calculée par l'expression : + \[N*3P+1\] + \quad où $P$ est le nombre de point déjà testés, et $P$ tendant vers $N$. Sa complexité maximale est donc donnée par la formule : + \[N\frac{1}{2}N(N+1)\] + + \par\quad Ensuite, après avoir rajouté le point courant à la liste des points déjà testés, on utilise la fonction $isAccessDirect$ afin de savoir si parmi la liste de point en accès direct on retrouve notre destination. On rajoute donc $3N+1$ à la complexité, ce qui nous donne l'expression : + \[N*2(3P+1)\] + \quad où $N$ est le nombre de points directement acessible et $P$ même variable que le pour le calcul précédant. Cela nous donne donc l'expression de complexité "au pire" : + \[2N\frac{1}{2}N(N+1)\] + + \par\quad Enfin, et il s'agit ici de la partie la plus délicate de l'analyse, on étudie l'appel récursif de cette fonction. On doit donc imaginer combien de fois cette fonction sera appelée pour multiplier ce nombre par le reste de la complexité de ce code. Mais en réalité, comme on fait attention à ne pas revenir en arrière sur notre chemin, au plus nous parcourerons tout les points du plans. Appelons ce nombre $Q$. Dans ce cas la complexité maximale, dite "au pire" est donnée par la formule : + \[2Q\frac{1}{2}Q(Q+1)\] + \quad avec $Q$ le nombre total de points du plan de la ville. La compléxité maximal de cet algorithme est donc d'ordre quadratique ($o(N^{2})$). \end{document} \ No newline at end of file diff --git a/sections/partie2.tex b/sections/partie2.tex index 71dd170..3c9689f 100644 --- a/sections/partie2.tex +++ b/sections/partie2.tex @@ -8,68 +8,70 @@ \section{Partie 2} \subsection{Question 1} - \par\quad + \par\quad \dots \subsection{Question 2 et 4} - \par\quad Interressons-nous maintenant au calcul de la complexité de chaque algorithmes. Un calcul de omplexité s'effectue en comptabilisant le nombre d'action que le processeur doit effectuer. Cependant, pour effectuer un tel calcul, il faudrait comptabiliser la totalité des actions effectuées par l'ensemble des bibliothèques utilisées dans les algorithmes. Nous n'avons pas besoin d'aller aussi loin, notamment car dans ce cas, ce calcul changerait en fonction d'un language à un autre, d'un compilateur à un autre, ou encore d'un interpréteur à un autre. Nous allons donc nous intéresser uniquement au nombre de lignes executés par les programmes en fonction de parmamêtres donnés. + \par\quad Interressons-nous maintenant au calcul de la complexité de chaque algorithmes. Un calcul de omplexité s'effectue en comptabilisant le nombre d'actions que le processeur doit effectuer. Cependant, pour effectuer un tel calcul, il faudrait comptabiliser la totalité des actions effectuées par l'ensemble des bibliothèques utilisées dans les algorithmes. Nous n'avons pas besoin d'aller aussi loin, notamment car dans ce cas, ce calcul changerait en fonction d'un language à un autre, d'un compilateur à un autre, ou encore d'un interpréteur à un autre. Nous allons donc nous intéresser uniquement au nombre de lignes executés par les programmes en fonction de parmamêtres donnés. \subsubsection{Analyse de l'algorithme n°1} - \par\quad Pour ce qui est du premier algorithme proposé, nous pouvons déjà comptabiliser les 4 actions effectuées par la fonction annexe "echanger". Ajoutons-y les 4 création de variables en début d'algorithme. On remarque que les instructions d'après sont bouclées en fonction de paramètres variables. On en déduit donc que nous avons déjà 4 instructions fixes et 4 autres que l'on multiplira par le nombre de fois que la fonction "echanger" est appelée. + \par\quad Pour ce qui est du premier algorithme proposé, nous pouvons déjà comptabiliser les 4 actions effectuées par la fonction annexe $echanger$. Ajoutons-y les 4 créations de variables en début d'algorithme. On remarque que les instructions d'après sont bouclées en fonction de paramètres variables. On en déduit donc que nous avons déjà 4 instructions fixes et 4 autres que l'on multiplira par le nombre de fois que la fonction $echanger$ est appelée. - \par\quad La première boucle (while), qui est la boucle principale tourne tant que la variable échange est à 1. cette variable comptabilise le nombre d'échanges de valeurs qui est effectué pour ordonner le tableau. Notre equation de complexité se construit donc avec la forme : $4+N*(4+P)$, où $N$ est le nombre de fois que la boucle tourne et $P$ le nombre d'actions effectuées dans cette boucle. + \par\quad La première boucle $while$, qui est la boucle principale, tourne tant que la variable $echange$ est à $1$. Cette variable comptabilise le nombre d'échanges de valeurs qui est effectué pour ordonner le tableau. Notre équation de complexité se construit donc avec la forme : + \[4+N*(4+P)\] + \quad où $N$ est le nombre de fois que la boucle tourne et $P$ le nombre d'actions effectuées dans cette boucle. - \par\quad Dans cette boucle while, on trouve deux boucles "for" pour trier les valeurs du début et de la fin. En analysant l'algorithme caractérisant ces boucles, on en déduit que celui-ci tri le tableau en mettant, pour chaque tour de boucle "while", l'élement le plus petit en premier, et le plus grand à la fin. $P$ est donc de complexité $N*2$. On peut alors en déduire que au pire, la complexité de cet algorithme est de : + \par\quad Dans cette boucle $while$, on trouve deux boucles $for$ pour trier les valeurs du début et de la fin. En analysant l'algorithme caractérisant ces boucles, on en déduit que celui-ci tri le tableau en mettant, pour chaque tours de boucle $while$, l'élement le plus petit en premier, et le plus grand à la fin. $P$ est donc de complexité $N*2$. On peut alors en déduire que au pire, la complexité de cet algorithme est de : \[4+N*(4+(N*2))\]. - \par\quad En conclusion, on trouve que cette fonction est de complexité quadratique $(o(N^{2}))$. Cette algorithme ne comporte pas vraiment d'avantages et son gros inconvéniant est que sa complexité est exponentielle : elle augmente de plus en plus vite. Il serait donc très compliqué de trier un tres grand nombre de valeurs avec. Cependant, elle peut fonctionner sur des petit nombres de valeurs. + \par\quad En conclusion, on trouve que cette fonction est de complexité quadratique $(o(N^{2}))$. Cette algorithme ne comporte pas vraiment d'avantages et son gros inconvéniant est que sa complexité est exponentielle : elle augmente de plus en plus vite. Il serait donc très compliqué de trier un très grand nombre de valeurs avec. Cependant, elle peut fonctionner sur de petits nombres de valeurs. \subsubsection{Analyse de l'algorithme n°2} - \par\quad Maintenant, analysons le second algorithme. Pour celui-ci, partons nous ommettrons l'analyse des instructions fixes, car la réelle compléxité des algorithme se base notamment sur les tours de boucles au sein du programme. + \par\quad Maintenant, analysons le second algorithme. Pour celui-ci, nous omettrons l'analyse des instructions fixes, car la réelle compléxité des algorithme se base notamment sur les tours de boucles au sein du programme. \par\quad La première boucle sert à récupérer les valeurs maximales du tableau. Le nombre de tours que celle-ci effectue est donc égale à la taille du tableau. Appelons cette taille $N$. - \par\quad L'instruction "calloc" qui suit cette boucle initialise un tableau de pointeur en initialisant toute les valeurs par defaut à "NULL". Ce tableau est composé de toute les valeurs possible entre la valeur minimale du tableau, et la valeur maximale. C'est-à-dire, si la valeur maximal est 7 et la valeur minimale est 4, ce tableau sera composé de 3 cases. La complexité de cette instruction est donc dépendante de l'écart entre ces deux valeurs. Appelons cette écart $P$. + \par\quad L'instruction $calloc$ qui suit cette boucle initialise un tableau de pointeur en initialisant toute les valeurs par defaut à $NULL$. Ce tableau est composé de toutes les valeurs possibles entre la valeur minimale du tableau, et la valeur maximale. C'est-à-dire, si la valeur maximale est 7 et la valeur minimale est 4, ce tableau sera composé de 3 cases. La complexité de cette instruction est donc dépendante de l'écart entre ces deux valeurs. Appelons cette écart $P$. - \par\quad La seconde boucle "for" compte le nombre de valeurs identiques et stocke ce nombre dans le tableau compteur décrit juste avant. Cette boucle tourne autant de fois qu'il y a de valeurs dans le tableau, c'est à dire : $N$ fois. + \par\quad La seconde boucle $for$ compte le nombre de valeurs identiques et stocke ce nombre dans le tableau compteur décrit juste avant. Cette boucle tourne autant de fois qu'il y a de valeurs dans le tableau, c'est à dire : $N$ fois. - \par\quad Viens enfin la dernière boucle "for". On retrouve à l'intérieur de celle-ci une boucle "while", qui tourne tant que le nombre sur lequel pointe de compteur ne vaut pas 0. Le compteur compte le nombre de récurence d'un nombre dans le tableau donc au maximum cette boucle fera $N+P$ tours au total, en comptant les tours de la boucle "for" dans laquelle elle se trouve. + \par\quad Viens enfin la dernière boucle $for$. On retrouve à l'intérieur de celle-ci une boucle $while$, qui tourne tant que le nombre sur lequel pointe le compteur ne vaut pas 0. Le compteur compte le nombre de récurences d'un nombre dans le tableau donc au maximum cette boucle fera $N+P$ tours au total, en comptant les tours de la boucle $for$ dans laquelle elle se trouve. \par\quad L'équation final se présente comme : \[N+P+N+N+P=3N+2P\] - La complexité de l'algorithme est donc d'ordre : $o(N)$, c'est à dire une complexité linéaire. Le gros avantage de cet algorithme est que son nombre d'opération augmente proportionnellement en fonction ddu nombre d'élèments à trier. Cependant, son point faible se situe au niveau de l'écart de valeur qu'il peut y avoir dans la plage de données. Un trop grand écart de valeur resulterait un très grand nombre $P$, nombre qui finalement peut avoir plus d'importance dans le calcul de cette complexité. + La complexité de l'algorithme est donc d'ordre : $o(N)$, c'est à dire une complexité linéaire. Le gros avantage de cet algorithme est que son nombre d'opération augmente proportionnellement en fonction du nombre d'élèments à trier. Cependant, son point faible se situe au niveau de l'écart de valeur qu'il peut y avoir dans la plage de données. Un trop grand écart de valeur resulterait à un très grand nombre $P$, nombre qui finalement peut avoir plus d'importance dans le calcul de cette complexité. \subsubsection{Analyse de l'algorithme n°3} - \par\quad Interessons nous enfin au troisième algorithme proposé. Tout comme pour la seconde analyse, nous omettrons l'analyse du nombre d'actions fixe. + \par\quad Interessons nous enfin au troisième algorithme proposé. Tout comme pour la seconde analyse, nous omettrons l'analyse du nombre d'actions fixes. - \par\quad Regardons tout d'abord la complexité de la fonction de recherche de position : "recherchePos". Cette fonction contient une boucle "for" qui, au pire effectue $P$ instructions, $P$ étant passé en second paramètre. En effet, la boucle est prévue pour tournée $P$ fois, mais suivant la réussite de la condition placé à l'intérieur, elle peut se terminée prématurémant. + \par\quad Regardons tout d'abord la complexité de la fonction de recherche de position : $recherchePos$. Cette fonction contient une boucle $for$ qui, au pire effectue $P$ instructions, $P$ étant passé en second paramètre. En effet, la boucle est prévue pour tourner $P$ fois, mais suivant la réussite de la condition placé à l'intérieur, elle peut se terminer prématurémant. - \par\quad Maintenant, si on regarde la fonction principale de cet algorithme, on remarque une boucle "for" majeure, paramètrée pour boucler $N$ fois, $N$ étant le nombre d'éléments du tableau à trier. Dans cette boucle, la première chose qui est faite est d'appeler la fonction "recherchePos", avec en second paramètre une valeur incrémentée de $0$ à $N$ à chaque passage de boucle. - \quad On déduit donc que le calcul de complexité maximal à ce moment est de : + \par\quad Maintenant, si l'on regarde la fonction principale de cet algorithme, on remarque une boucle $for$ majeure, paramètrée pour boucler $N$ fois, $N$ étant le nombre d'éléments du tableau à trier. Dans cette boucle, la première chose qui est faite est d'appeler la fonction $recherchePos$, avec en second paramètre une valeur incrémentée de $0$ à $N$ à chaque passage de boucle. + \quad On déduit donc que le calcul de complexité maximale à ce moment est de : \[\frac{1}{2}N(N+1)\] \quad Le principe de l'algorithme à ce moment est de parcourir tout les éléments du tableau et de rechercher leur position idéale. - \par\quad Ensuite, nous avons la présence d'une condition. Celle-ci test simplement si l'élèment est ou non dans la position idéale considérant que ce tableau est trié. Elle renvoie sur une boucle le cas échéant. Cela veut dire que dans le pire des cas, c'est-à-dire le cas où aucun élément n'est dans sa position idéale, elle renvoie $N$ fois sur cette boucle. + \par\quad Ensuite, nous avons la présence d'une condition. Celle-ci teste simplement si l'élèment est oui ou non dans la position idéale considérant que ce tableau est trié. Elle renvoie sur une boucle le cas échéant. Cela veut dire que dans le pire des cas, c'est-à-dire le cas où aucun élément n'est dans sa position idéale, elle renvoie $N$ fois sur cette boucle. - \par\quad Analysons enfin cette dernière boucle. Celle-ci boucle autant de fois qu'il y a d'écart entre le énième élément du tabeau et sa position idéale. Au maximum, cette valeur peut donc s'élever à $N-1$ et décroit au fûr et à mesure que l'on avance dans la boucle principale. + \par\quad Analysons enfin cette dernière boucle. Celle-ci boucle autant de fois qu'il y a d'écart entre le énième élément du tableau et sa position idéale. Au maximum, cette valeur peut donc s'élever à $N-1$ et décroit au fûr et à mesure que l'on avance dans la boucle principale. \quad On a donc pour cette boucle, une complexité maximale s'exprimant par : \[\frac{1}{2}N(N+1)-1\] - \quad Ici, le principe est de décaler toute les valeurs pour insérer à l'emplacement idéal l'élèment courrant. + \quad Ici, le principe est de décaler toutes les valeurs pour insérer à l'emplacement idéal l'élèment courrant. \par\quad Nous pouvons donc maintemant déduire l'équation générale de la compléxité maximale de ce troisième algorithme : \[(\frac{1}{2}N(N+1))(\frac{1}{2}N(N+1)-1)=\frac{1}{4}N(N+1)(N^{2}+N-2)\] - \quad On remarque donc la complexité exponantielle de cette algorithme, $o(N^{4})$ qui est la plus importante complexité des trois analyses. Cependant, ce n'est pas forcément un algorithme mauvais car il ne s'agit là que de la complexité "au pire". En effet, dans la majeure partie des cas, plusieurs valeurs sont à leur place, et donc cette complexité à tendance à baisser aussi vite qu'a monter. L'avantage est donc que, avec un tableau quasiment trié, cette fonction sera plus rapide que les autre. Dans d'autre cas, elle sera largement plus lente. Donc si on ne connaît pas les arrangement de valeur au préalable du tableau qu'on lui passe, c'est un peu comme si on pariait sur la rapidité de cet algorithme. + \quad On remarque donc la complexité exponantielle de cet algorithme, $o(N^{4})$ qui est la plus importante complexité des trois analyses. Cependant, ce n'est pas forcément un algorithme mauvais car il ne s'agit là que de la complexité "au pire". En effet, dans la majeure partie des cas, plusieurs valeurs sont à leur place, et donc cette complexité à tendance à baisser aussi vite qu'à monter. L'avantage est donc que, avec un tableau quasiment trié, cette fonction sera plus rapide que les autre. Dans d'autre cas, elle sera largement plus lente. Donc si l'on ne connaît pas les arrangements de valeurs au préalable du tableau qu'on lui passe, c'est un peu comme si on pariait sur la rapidité de cet algorithme. \subsection{Question 3} - \par\quad + \par\quad \dots \end{document} \ No newline at end of file