diff --git a/slides/cours_10.md b/slides/cours_10.md index e52cec76eac0fc28529cc5329abd7333867c839d..31637d6a51c385430f57eab21efa855a8b9211cf 100644 --- a/slides/cours_10.md +++ b/slides/cours_10.md @@ -460,249 +460,5 @@ void stack_pop(stack *s, int *ret); . . . -## Faisons l'implémentation ensemble - -# Le tri à deux piles (1/3) - -## Cas pratique - -{width=70%} - -# Le tri à deux piles (2/3) - -## Exercice: formaliser l'algorithme - -. . . - -## Algorithme de tri nécessitant 2 piles (G, D) - -Soit `tab` le tableau à trier: - -```C -pour i de 0 à N-1 - tant que (tab[i] > que le sommet de G) - dépiler G dans D - tant que (tab[i] < que le sommet de D) - dépiler de D dans G - empiler tab[i] sur G -dépiler tout D dans G -tab est trié dans G -``` - -# Le tri à deux piles (3/3) - -## Exercice: trier le tableau `[2, 10, 5, 20, 15]` - -```C - - - - - - - - - - - - - - - - -``` - -# La calculatrice (1/8) - -## Vocabulaire - -```C -2 + 3 = 2 3 +, -``` - -`2` et `3` sont les *opérandes*, `+` l'*opérateur*. - -. . . - -## La notation infixe - -```C -2 * (3 + 2) - 4 = 6. -``` - -## La notation postfixe - -```C -2 3 2 + * 4 - = 6. -``` - -## Exercice: écrire `2 * 3 * 4 + 2` en notation `postfixe` - -. . . - -```C -2 3 4 * * 2 + = (2 * (3 * 4)) + 2. -``` - -# La calculatrice (2/8) - -## De infixe à post-fixe - -* Une *pile* est utilisée pour stocker *opérateurs* et *parenthèses*. -* Les opérateurs on des *priorités* différentes. - -```C -^ : priorité 3 -* / : priorité 2 -+ - : priorité 1 -( ) : priorité 0 // pas un opérateur mais bon -``` - - -# La calculatrice (3/8) - -## De infixe à post-fixe: algorithme - -* On lit l'expression infixe de gauche à droite. - -* On examine le prochain caractère de l'expression infixe. - * Si opérande, le placer dans l'expression du résultat. - * Si parenthèse le mettre dans la pile (priorité 0). - * Si opérateur, comparer sa priorité avec celui du sommet de la pile: - * Si sa priorité est plus élevée, empiler. - * Sinon dépiler l'opérateur de la pile dans l'expression du résultat et - recommencer jusqu'à apparition d'un opérateur de priorité plus faible - au sommet de la pile (ou pile vide). - * Si parenthèse fermée, dépiler les opérateurs du sommet de la pile et les - placer dans l'expression du résultat, jusqu'à ce qu'une parenthèse - ouverte apparaisse au sommet, dépiler également la parenthèse. - * Si il n'y a pas de caractère dans l'expression dépiler tous les - opérateurs dans le résultat. - -# La calculatrice (4/8) - -## De infixe à post-fixe: exemple - -```C -Infixe Postfixe Pile Priorité -((A*B)/D-F)/(G+H) Vide Vide Néant - (A*B)/D-F)/(G+H) Vide ( 0 - A*B)/D-F)/(G+H) Vide (( 0 - *B)/D-F)/(G+H) A (( 0 - B)/D-F)/(G+H) A ((* 2 - )/D-F)/(G+H) AB ((* 2 - /D-F)/(G+H) AB* ( 0 - D-F)/(G+H) AB* (/ 2 - -F)/(G+H) AB*D (/ 2 - F)/(G+H) AB*D/ (- 1 - )/(G+H) AB*D/F (- 1 - /(G+H) AB*D/F- Vide Néant -``` - -# La calculatrice (5/8) - -## De infixe à post-fixe: exemple - -```C -Infixe Postfixe Pile Priorité -((A*B)/D-F)/(G+H) Vide Vide Néant --------------------------------------------------------- - /(G+H) AB*D/F- Vide Néant - (G+H) AB*D/F- / 2 - G+H) AB*D/F- /( 0 - +H) AB*D/F-G /( 0 - H) AB*D/F-G /(+ 1 - ) AB*D/F-GH /(+ 1 - Vide AB*D/F-GH+ / 2 - Vide AB*D/F-GH+/ Vide Néant -``` - -# La calculatrice (6/8) - -\footnotesize - -## Exercice: écrire le code et le poster sur matrix - -* Quelle est la signature de la fonction? - -. . . - -* Une sorte de corrigé: - -```C -char *infix_to_postfix(char* infix) { // init and alloc stack and postfix - for (size_t i = 0; i < strlen(infix); ++i) { - if (is_operand(infix[i])) { - // we just add operands in the new postfix string - } else if (infix[i] == '(') { - // we push opening parenthesis into the stack - } else if (infix[i] == ')') { - // we pop everything into the postfix - } else if (is_operator(infix[i])) { - // this is an operator. We add it to the postfix based - // on the priority of what is already in the stack and push it - } - } - // pop all the operators from the s at the end of postfix - // and end the postfix with `\0` - return postfix; -} -``` - - -# La calculatrice (7/8) - -## Évaluation d'expression postfixe: algorithme - -* Chaque *opérateur* porte sur les deux opérandes qui le précèdent. -* Le *résultat d'une opération* est un nouvel *opérande* qui est remis au - sommet de la pile. - -## Exemple - -```C -2 3 4 + * 5 - = ? -``` - -* On parcours de gauche à droite: - -```C -Caractère lu Pile opérandes - 2 2 - 3 2, 3 - 4 2, 3, 4 - + 2, (3 + 4) - * 2 * 7 - 5 14, 5 - - 14 - 5 = 9 -``` - -# La calculatrice (8/8) - -## Évaluation d'expression postfixe: algorithme - -1. La valeur d'un opérande est *toujours* empilée. -2. L'opérateur s'applique *toujours* au 2 opérandes au sommet. -3. Le résultat est remis au sommet. - -## Exercice: écrire l'algorithme en C (et poster sur matrix) - -. . . - -```C -double evaluate(char *postfix) { // init stack - for (size_t i = 0; i < strlen(postfix); ++i) { - if (is_operand(postfix[i])) { - stack_push(&s, postfix[i]); - } else if (is_operator(postfix[i])) { - double rhs = stack_pop(&s); - double lhs = stack_pop(&s); - stack_push(&s, op(postfix[i], lhs, rhs)); - } - } - return stack_pop(&s); -} -``` - - +## Faisons s'implémentation ensemble diff --git a/slides/cours_11.md b/slides/cours_11.md new file mode 100644 index 0000000000000000000000000000000000000000..9694de9bca843f2b087943d4af2ff1f5cc5bbc3d --- /dev/null +++ b/slides/cours_11.md @@ -0,0 +1,937 @@ +--- +title: "Calculatrices, liste chaînée, et files d'attente" +date: "2024-12-09" +--- + +# Rappel: les piles + +## Qu'est-ce donc? + +. . . + +* tructure de données abstraite de type LIFO + +## Quelles fonctionnalités? + +. . . + +1. Empiler (push): ajouter un élément sur la pile. +2. Dépiler (pop): retirer l'élément du sommet de la pile et le retourner. +3. Pile vide? (is_empty?). + +# Le tri à deux piles + +\Huge Le tri à deux piles + +# Le tri à deux piles (1/3) + +## Cas pratique + +{width=70%} + +# Le tri à deux piles (2/3) + +## Exercice: formaliser l'algorithme + +. . . + +## Algorithme de tri nécessitant 2 piles (G, D) + +Soit `tab` le tableau à trier: + +```C +pour i de 0 à N-1 + tant que (tab[i] > que le sommet de G) + dépiler G dans D + tant que (tab[i] < que le sommet de D) + dépiler de D dans G + empiler tab[i] sur G +dépiler tout D dans G +tab est trié dans G +``` + +# Le tri à deux piles (3/3) + +## Exercice: trier le tableau `[2, 10, 5, 20, 15]` + +```C + + + + + + + + + + + + + + + + +``` + +# La Calculatrice + +\Huge La Calculatrice + +# La calculatrice (1/8) + +## Vocabulaire + +```C +2 + 3 = 2 3 +, +``` + +`2` et `3` sont les *opérandes*, `+` l'*opérateur*. + +. . . + +## La notation infixe + +```C +2 * (3 + 2) - 4 = 6. +``` + +## La notation postfixe + +```C +2 3 2 + * 4 - = 6. +``` + +## Exercice: écrire `2 * 3 * 4 + 2` en notation `postfixe` + +. . . + +```C +2 3 4 * * 2 + = (2 * (3 * 4)) + 2. +``` + +# La calculatrice (2/8) + +## De infixe à post-fixe + +* Une *pile* est utilisée pour stocker *opérateurs* et *parenthèses*. +* Les opérateurs on des *priorités* différentes. + +```C +^ : priorité 3 +* / : priorité 2 ++ - : priorité 1 +( ) : priorité 0 // pas un opérateur mais bon +``` + + +# La calculatrice (3/8) + +## De infixe à post-fixe: algorithme + +* On lit l'expression infixe de gauche à droite. + +* On examine le prochain caractère de l'expression infixe. + * Si opérande, le placer dans l'expression du résultat. + * Si parenthèse le mettre dans la pile (priorité 0). + * Si opérateur, comparer sa priorité avec celui du sommet de la pile: + * Si sa priorité est plus élevée, empiler. + * Sinon dépiler l'opérateur de la pile dans l'expression du résultat et + recommencer jusqu'à apparition d'un opérateur de priorité plus faible + au sommet de la pile (ou pile vide). + * Si parenthèse fermée, dépiler les opérateurs du sommet de la pile et les + placer dans l'expression du résultat, jusqu'à ce qu'une parenthèse + ouverte apparaisse au sommet, dépiler également la parenthèse. + * Si il n'y a pas de caractère dans l'expression dépiler tous les + opérateurs dans le résultat. + +# La calculatrice (4/8) + +## De infixe à post-fixe: exemple + +```C +Infixe Postfixe Pile Priorité +((A*B)/D-F)/(G+H) Vide Vide Néant + (A*B)/D-F)/(G+H) Vide ( 0 + A*B)/D-F)/(G+H) Vide (( 0 + *B)/D-F)/(G+H) A (( 0 + B)/D-F)/(G+H) A ((* 2 + )/D-F)/(G+H) AB ((* 2 + /D-F)/(G+H) AB* ( 0 + D-F)/(G+H) AB* (/ 2 + -F)/(G+H) AB*D (/ 2 + F)/(G+H) AB*D/ (- 1 + )/(G+H) AB*D/F (- 1 + /(G+H) AB*D/F- Vide Néant +``` + +# La calculatrice (5/8) + +## De infixe à post-fixe: exemple + +```C +Infixe Postfixe Pile Priorité +((A*B)/D-F)/(G+H) Vide Vide Néant +-------------------------------------------------------- + /(G+H) AB*D/F- Vide Néant + (G+H) AB*D/F- / 2 + G+H) AB*D/F- /( 0 + +H) AB*D/F-G /( 0 + H) AB*D/F-G /(+ 1 + ) AB*D/F-GH /(+ 1 + Vide AB*D/F-GH+ / 2 + Vide AB*D/F-GH+/ Vide Néant +``` + +# La calculatrice (6/8) + +\footnotesize + +## Exercice: écrire le code et le poster sur matrix + +* Quelle est la signature de la fonction? + +. . . + +* Une sorte de corrigé: + +```C +char *infix_to_postfix(char* infix) { // init and alloc stack and postfix + for (size_t i = 0; i < strlen(infix); ++i) { + if (is_operand(infix[i])) { + // we just add operands in the new postfix string + } else if (infix[i] == '(') { + // we push opening parenthesis into the stack + } else if (infix[i] == ')') { + // we pop everything into the postfix + } else if (is_operator(infix[i])) { + // this is an operator. We add it to the postfix based + // on the priority of what is already in the stack and push it + } + } + // pop all the operators from the s at the end of postfix + // and end the postfix with `\0` + return postfix; +} +``` + + +# La calculatrice (7/8) + +## Évaluation d'expression postfixe: algorithme + +* Chaque *opérateur* porte sur les deux opérandes qui le précèdent. +* Le *résultat d'une opération* est un nouvel *opérande* qui est remis au + sommet de la pile. + +## Exemple + +```C +2 3 4 + * 5 - = ? +``` + +* On parcours de gauche à droite: + +```C +Caractère lu Pile opérandes + 2 2 + 3 2, 3 + 4 2, 3, 4 + + 2, (3 + 4) + * 2 * 7 + 5 14, 5 + - 14 - 5 = 9 +``` + +# La calculatrice (8/8) + +## Évaluation d'expression postfixe: algorithme + +1. La valeur d'un opérande est *toujours* empilée. +2. L'opérateur s'applique *toujours* au 2 opérandes au sommet. +3. Le résultat est remis au sommet. + +## Exercice: écrire l'algorithme en C (et poster sur matrix) + +. . . + +```C +double evaluate(char *postfix) { // init stack + for (size_t i = 0; i < strlen(postfix); ++i) { + if (is_operand(postfix[i])) { + stack_push(&s, postfix[i]); + } else if (is_operator(postfix[i])) { + double rhs = stack_pop(&s); + double lhs = stack_pop(&s); + stack_push(&s, op(postfix[i], lhs, rhs)); + } + } + return stack_pop(&s); +} +``` + +# Liste chaînée et pile + +\Huge Liste chaînée et pile + +# La liste chaînée et pile (1/6) + +## Structure de données + +* Chaque élément de la liste contient: + 1. une valeur, + 2. un pointeur vers le prochain élément. +* La pile est un pointeur vers le premier élément. + +{width=80%} + +# La liste chaînée et pile (2/6) + +## Une pile-liste-chaînée + +```C +typedef struct _element { + int data; + struct _element *next; +} element; +typedef element* stack; +``` + +## Fonctionnalités? + +. . . + +```C +void stack_create(stack *s); // *s = NULL; +void stack_destroy(stack *s); +void stack_push(stack *s, int val); +void stack_pop(stack *s, int *val); +void stack_peek(stack s, int *val); +bool stack_is_empty(stack s); // reutrn NULL == stack; +``` + +# La liste chaînée et pile (3/6) + +## Empiler? (faire un dessin) + +. . . + +```C + + + + + + + +``` + +## Empiler? (le code ensemble) + +. . . + +```C +void stack_push(stack *s, int val) { + element *elem = malloc(sizeof(*elem)); + elem->data = val; + elem->next = *s; + *s = elem; +} +``` + +# La liste chaînée et pile (4/6) + +## Jeter un oeil? (faire un dessin) + +. . . + +```C + + + + + + + +``` + +## Jeter un oeil? (le code ensemble) + +. . . + +```C +void stack_peek(stack s, int *val) { + *val = s->data; +} +``` + +# La liste chaînée et pile (5/6) + +## Dépiler? (faire un dessin) + +. . . + +```C + + + + + + + +``` + +## Dépiler? (le code ensemble) + +. . . + +```C +void stack_pop(stack *s, int *val) { + stack_peek(*s, val); + element *tmp = *s; + *s = (*s)->next; + free(tmp); +} +``` + +# La liste chaînée et pile (6/6) + +## Détruire? (faire un dessin) + +. . . + +```C + + + + + + + +``` + +## Détruire? (le code ensemble) + +. . . + +```C +void stack_destroy(stack *s) { + while (!stack_is_empty(*s)) { + int val; + stack_pop(s, &val); + } +} +``` + +# La file d'attente + +\Huge La file d'attente + +# La file d'attente (1/2) + +* Structure de données abstraite permettant le stockage d'éléments. +* *FIFO*: First In First Out, ou première entrée première sortie. +* Analogue de la vie "réelle"": + * File à un guichet, + * Serveur d'impressions, + * Mémoire tampon, ... + +## Fonctionnalités + + . . . + +* Enfiler: ajouter un élément à la fin de la file. +* Défiler: extraire un élément au devant de la file. +* Tester si la file est vide. + +. . . + +* Lire l'élément de la fin de la file. +* Lire l'élément du devant de la file. +* Créer une liste vide. +* Détruire une liste vide. + +# La file d'attente (2/2) + +\footnotesize + +## Implémentation possible + +* La structure file, contient un pointeur vers la tête et un vers le début de la file. +* Entre les deux, les éléments sont stockés dans une liste chaînée. + +{width=80%} + +## Structure de données en C? + +. . . + +```C +typedef struct _element { // Elément de liste + int data; + struct _element* next; +} element; +typedef struct _queue { // File d'attente: + element* head; // tête de file d'attente + element* tail; // queue de file d'attente +} queue; +``` + +# Fonctionnalités d'une file d'attente + +## Creation et consultations + +. . . + +```C +void queue_init(queue *fa); // head = tail = NULL +bool queue_is_empty(queue fa); // fa.head == fa.tail == NULL +int queue_tail(queue fa); // return fa.tail->data +int queue_head(queue fa); // return fa.head->data +``` + +## Manipulations et destruction + +. . . + +```C +void queue_enqueue(queue *fa, int val); +// adds an element before the tail +int queue_dequeue(queue *fa); +// removes the head and returns stored value +void queue_destroy(queue *fa); +// dequeues everything into oblivion +``` + +# Enfilage + +## Deux cas différents: + +1. La file est vide (faire un dessin): + +. . . + +{width=40%} + +2. La file n'est pas vide (faire un dessin): + +. . . + +{width=70%} + +# Enfilage + +## Live (implémentation) + +. . . + +```C +void queue_enqueue(queue *fa, int val) { + element* elmt = malloc(sizeof(*elmt)); + elmt->data = val; + elmt->next = NULL; + if (queue_is_empty(*fa)) { + fa->head = elmt; + fa->tail = elmt; + } else { + fa->tail->next = elmt; + fa->tail = elmt; + } +} +``` + +# Défilage + +## Trois cas différents + +1. La file a plus d'un élément (faire un dessin): + +. . . + +{width=80%} + +2. La file un seul élément (faire un dessin): + +. . . + +{width=25%} + + +3. La file est vide (problème) + +# Défilage + +## Live (implémentation) + +. . . + +```C +int queue_dequeue(queue *fa) { + element* elmt = fa->head; + int val = elmt->data; + fa->head = fa->head->next; + free(elmt); + if (NULL == fa->head) { + fa->tail = NULL; + } + return val; +} +``` + +. . . + +## Problème avec cette implémentation? + +# Destruction + +## Comment on faire la désallocation? + +. . . + +On défile jusqu'à ce que la file soit vide! + +# Complexité + +## Quelle sont les complexité de: + +* Enfiler? + +. . . + +* Défiler? + +. . . + +* Détruire? + +. . . + +* Est vide? + + +# Implémentation alternative + +## Comment implémenter la file autrement? + +. . . + +* Données stockées dans un tableau; +* Tableau de taille connue à la compilation ou pas (réallouable); +* `tail` seraient les indices du tableau; +* `capacity` seraient la capacité maximale; +* On *enfile* "au bout" du tableau, au défile au début (indice `0`). + +. . . + +## Structure de données + +```C +typedef struct _queue { + int *data; + int tail, capacity; +} queue; +``` + +# File basée sur un tableau + +* Initialisation? + +. . . + +```C + + + + +``` + +* Est vide? + +. . . + +```C + + + + +``` + + +* Enfiler? + +. . . + +```C + + + + +``` + +* Défiler? + +. . . + +```C + + + + +``` + +# Complexité + +## Quelle sont les complexités de: + +* Initialisation? + +. . . + +```C + + + + +``` + +* Est vide? + +. . . + +```C + +``` + + +* Enfiler? + +. . . + +```C + + + + +``` + +* Défiler? + +. . . + +```C + + + + +``` + +# Une file plus efficace + +## Comment faire une file plus efficace? + +* Où est-ce que ça coince? + +. . . + +* Défiler est particulièrement lent $\mathcal{O}(N)$. + +## Solution? + +. . . + +* Utiliser un indice séparé pour `head`. + +```C +typedef struct _queue { + int *data; + int head, tail, capacity; +} queue; +``` + +# Une file plus efficace (implémentation) + +## Enfilage + +\footnotesize + +```C +void queue_enqueue(queue *fa, int val) { + if ((fa->head == 0 && fa->tail == fa->capacity-1) || + (fa->tail == (fa->head-1) % (fa->capacity-1))) { + return; // queue is full + } + if (fa->head == -1) { // queue was empty + fa->head = fa->tail = 0; + fa->data[fa->tail] = val; + } else if (fa->tail == fa->capacity-1 && fa->head != 0) { + // the tail reached the end of the array + fa->tail = 0; + fa->data[fa->tail] = val; + } else { + // nothing particular + fa->tail += 1; + fa->data[fa->tail] = val; + } +} +``` + +# Une file plus efficace (implémentation) + +## Défilage + +```C +void queue_dequeue(queue *fa, int *val) { + if (queue_is_empty(*fa)) { + return; // queue is empty + } + *val = fa->data[fa->head]; + if (fa->head == fa->tail) { // that was the last element + fa->head = fa->tail = -1; + } else if (fa->head == fa->capacity-1) { + fa->head = 0; + } else { + fa->head += 1; + } +} +``` + + +# Les listes triées + +Une liste chaînée triée est: + +* une liste chaînée +* dont les éléments sont insérés dans l'ordre. + + + +. . . + +* L'insertion est faite telle que l'ordre est maintenu. + +## Quelle structure de données? + +```C + + + + + +``` + +# Les listes triées + +## Quel but? + +* Permet de retrouver rapidement un élément. +* Utile pour la recherche de plus court chemin dans des graphes. +* Ordonnancement de processus par degré de priorité. + +## Comment? + +* Les implémentations les plus efficaces se basent sur les tableaux. +* Possibles aussi avec des listes chaînées. + +# Les listes triées + +\footnotesize + +## Quelle structure de données dans notre cas? + + +Une liste chaînée bien sûr (oui c'est pour vous entraîner)! + +```C +typedef struct _element { // chaque élément + int data; + struct _element *next; +} element; +typedef element* sorted_list; // la liste +``` + +## Fonctionnalités + +```C +// insertion de val +sorted_list sorted_list_push(sorted_list list, int val); +// la liste est-elle vide? +bool is_empty(sorted_list list); // list == NULL +// extraction de val (il disparaît) +sorted_list sorted_list_extract(sorted_list list, int val); + // rechercher un élément et le retourner +element* sorted_list_search(sorted_list list, int val); +``` + +# L'insertion + +## Trois cas + +1. La liste est vide. + +. . . + +{width=30%} + +. . . + +```C +sorted_list sorted_list_push(sorted_list list, int val) { + if (sorted_list_is_empty(list)) { + list = malloc(sizeof(*list)); + list->data = val; + list->next = NULL; + return list; + } +} +``` + +# L'insertion + +2. L'insertion se fait en première position. + +. . . + +{width=80%} + +. . . + +```C +sorted_list sorted_list_push(sorted_list list, int val) { + if (list->data >= val) { + element *tmp = malloc(sizeof(*tmp)); + tmp->data = val; + tmp->next = list; + list = tmp; + return list; + } +} +``` + +# L'insertion + +3. L'insertion se fait sur une autre position que la première. + +. . . + +{width=70%} + +. . . + +\footnotesize + +```C +sorted_list sorted_list_push(sorted_list list, int val) { + element *tmp = malloc(sizeof(*tmp)); + tmp->data = val; + element *crt = list; + while (NULL != crt->next && val > crt->next->data) { + crt = crt->next; + } + tmp->next = crt->next; + crt->next = tmp; + return list; +} +``` + + diff --git a/slides/figs/Singly-linked-list.svg b/slides/figs/Singly-linked-list.svg new file mode 100644 index 0000000000000000000000000000000000000000..dae4ee152b4f617747665651f8dd39e9c2cdbedb --- /dev/null +++ b/slides/figs/Singly-linked-list.svg @@ -0,0 +1,638 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<!-- Created with Inkscape (http://www.inkscape.org/) --> + +<svg + width="271.77063mm" + height="108.99983mm" + viewBox="0 0 271.77063 108.99983" + version="1.1" + id="svg92" + sodipodi:docname="Singly-linked-list.svg" + inkscape:version="1.1.1 (3bf5ae0d25, 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