diff --git a/slides_2022/Makefile b/slides_2022/Makefile
new file mode 100644
index 0000000000000000000000000000000000000000..0f4b15f3bfea5871d5b614d85534f38a9d42710c
--- /dev/null
+++ b/slides_2022/Makefile
@@ -0,0 +1,61 @@
+PDFOPTIONS = -t beamer
+# PDFOPTIONS += -F pantable
+PDFOPTIONS += -F mermaid-filter
+PDFOPTIONS += --highlight-style my_highlight.theme
+PDFOPTIONS += --pdf-engine xelatex
+PDFOPTIONS += -V theme:metropolis
+PDFOPTIONS += -V themeoptions:numbering=none -V themeoptions:progressbar=foot
+PDFOPTIONS += -V fontsize=smaller
+PDFOPTIONS += -V urlcolor=blue
+
+MD=$(wildcard *.md) # Tous les fichiers .md
+PDF=$(MD:%.md=%.pdf) # Pour les fichier pdf on transforme .md -> .pdf
+HTML=$(MD:%.md=%.html) # Pour les fichier html on transforme .md -> .html
+MARKDOWN=$(MD:%.md=%.markdown) # Pour les fichier markdown on transforme .md -> .markdown
+CHROMIUM:=$(shell which chromium || which chromium-browser)
+
+all: puppeteer $(PDF)
+# all: puppeteer $(PDF) $(HTML) # La cible par défaut (all) exécute les cibles %.pdf
+
+docker: docker-compose.yml
+ docker-compose run slides
+
+docker_clean: docker-compose.yml
+ docker-compose run slides clean
+
+puppeteer:
+ @echo "Setting chromium to $(CHROMIUM) for puppeteer"
+ @echo -e "{\n\"executablePath\":" \"$(CHROMIUM)\" ",\n\"args\": [\"--no-sandbox\"]\n}" > .puppeteer.json
+
+index.md: gen_index.sh
+ $(shell ./gen_index.sh)
+
+index.html: index.md
+ pandoc -s $(OPTIONS) --css ../css/tufte-css/tufte.css -o $@ $^
+
+markdown: $(MARKDOWN) # La markdown les cibles %.markdown
+
+%.pdf: %.md metadata.yaml # %.pdf (chaque fichier %.md génère un fichier avec le même nom mais l'extension .pdf et la dépendance metadata.yaml)
+ pandoc -s $(OPTIONS) $(PDFOPTIONS) -o $@ $^
+
+%.markdown: %.md metadata.yaml yq
+ sed '1 { /^---/ { :a N; /\n---/! ba; d} }' $< > no_header
+ grep -v -F -x -f no_header $< > header.yaml
+ echo "---" > tmp.yaml
+ ./yq_linux_amd64 merge metadata.yaml header.yaml >> tmp.yaml
+ cat tmp.yaml no_header > $@
+ rm no_header header.yaml tmp.yaml
+
+yq: # On peut même télécharger un petit programme avec notre makefile
+ wget -nc https://github.com/mikefarah/yq/releases/download/3.4.1/yq_linux_amd64
+ chmod "u+x" yq_linux_amd64
+
+deploy: all index.html
+ mkdir -p algo_cours
+ cp *.pdf algo_cours
+ cp index.html algo_cours
+
+clean:
+ rm -rf *.html *.pdf *.markdown yq_linux_amd64* index.md .puppeteer.json algo_cours *.err
+
+.PHONY: clean index.md puppeteer yq
diff --git a/slides_2022/cours_1.md b/slides_2022/cours_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..4044bafc4fa4648d0b225b825c5ae7c6c319a321
--- /dev/null
+++ b/slides_2022/cours_1.md
@@ -0,0 +1,558 @@
+---
+title: "Introduction aux algorithmes"
+date: "2022-09-21"
+---
+
+# Qu'est-ce qu'un algorithme?
+
+## Définition informelle (recette)
+
+* des entrées (les ingrédients, le matériel utilisé) ;
+* des instructions élémentaires simples (frire, flamber, etc.), dont les
+ exécutions dans un ordre précis amènent au résultat voulu ;
+* un résultat : le plat préparé.
+
+. . .
+
+## Histoire et étymologie
+
+- Existent depuis 4500 ans au moins (algorithme de division, crible
+ d'Eratosthène).
+- Le mot algorithme est dérivé du nom du mathématicien perse
+ *Muḥammad ibn MusÄ al-KhwÄrizmÄ«*, qui a été "latinisé" comme
+ *Algoritmi*.
+
+. . .
+
+## Définition formelle
+
+En partant d'un état initial et d'entrées (peut-être vides), une séquence finie
+d'instruction bien définies (ordonnées) implémentables sur un ordinateur, afin
+de résoudre typiquement une classe de problèmes ou effectuer un calcul.
+
+# Notions de base d'algorithmique
+
+## Variable
+
+. . .
+
+* Paire: identifiant - valeur (assignation);
+
+## Séquence d'instructions / expressions
+
+. . .
+
+* Opérateurs (arthimétiques / booléens)
+* Boucles;
+* Structures de contrôle;
+* Fonctions;
+
+
+# Algorithme de vérification qu'un nombre est premier (1/3)
+
+Nombre premier: nombre possédant deux diviseurs entiers distincts.
+
+. . .
+
+## Algorithme naïf (problème)
+
+```C
+booléen est_premier(nombre)
+ si
+ pour tout i, t.q. 1 < i < nombre
+ i ne divise pas nombre
+ alors vrai
+ sinon faux
+```
+
+. . .
+
+## Pas vraiment un algorithme: pas une séquence ordonnée et bien définie
+
+. . .
+
+## Problème: Comment écrire ça sous une forme algorithmique?
+
+# Algorithme de vérification qu'un nombre est premier (2/3)
+
+## Algorithme naïf (une solution)
+
+```C
+booléen est_premier(nombre) // fonction
+ soit i = 2; // variable, type, assignation
+ tant que i < nombre // boucle
+ si nombre modulo i == 0 // expression typée
+ retourne faux // expression typée
+ i = i + 1
+ retourne vrai // expression typée
+```
+
+# Algorithme de vérification qu'un nombre est premier (3/3)
+
+## Algorithme naïf (une solution en C)
+
+```C
+bool est_premier(int nombre) {
+ int i; // i est un entier
+ i = 2; // assignation i à 2
+ while (i < nombre) { // boucle avec condition
+ if (0 == nombre % i) { // is i divise nombre
+ return false; // i n'est pas premier
+ }
+ i += 1; // sinon on incrémente i
+ }
+ return true;
+}
+```
+
+. . .
+
+## Exercice: Comment faire plus rapide?
+
+# Génération d'un exécutable
+
+- Pour pouvoir être exécuté un code C doit être d'abord compilé (avec `gcc` ou `clang`).
+- Pour un code `prog.c` la compilation "minimale" est
+
+ ```bash
+ $ gcc prog.c
+ $ ./a.out # exécutable par défaut
+ ```
+
+- Il existe une multitude d'options de compilation:
+
+ ```bash
+ $ gcc -O1 -std=c11 -Wall -Wextra -g porg.c -o prog
+ -fsanitize=address
+ ```
+ 1. `-std=c11` utilisation de C11.
+ 2. `-Wall et -Wextra` activation des warnings.
+ 3. `-fsanitize=…` contrôles d’erreurs à l’exécution (coût en performance).
+ 4. `-g` symboles de débogages sont gardés.
+ 5. `-o` défini le fichier exécutable à produire en sortie.
+ 6. `-O1`, `-O2`, `-O3`: activation de divers degrés d'optimisation
+
+
+
+# La simplicité de C?
+
+## 32 mots-clé et c'est tout
+
+---------------- -------------- ---------------- ---------------
+`auto`{.C} `double`{.C} `int`{.C} `struct`{.C}
+`break`{.C} `else`{.C} `long`{.C} `switch`{.C}
+`case`{.C} `enum`{.C} `register`{.C} `typedef`{.C}
+`char`{.C} `extern`{.C} `return`{.C} `union`{.C}
+`const`{.C} `float`{.C} `short`{.C} `unsigned`{.C}
+`continue`{.C} `for`{.C} `signed`{.C} `void`{.C}
+`default`{.C} `goto`{.C} `sizeof`{.C} `volatile`{.C}
+`do`{.C} `if`{.C} `static`{.C} `while`{.C}
+---------------- -------------- ---------------- ---------------
+
+# Déclaration et typage
+
+En C lorsqu'on veut utiliser une variable (ou une constante), on doit déclarer son type
+
+```C
+const double two = 2.0; // déclaration et init.
+int x; // déclaration (instruction)
+char c; // déclaration (instruction)
+x = 1; // affectation (expression)
+c = 'a'; // affectation (expression)
+int y = x; // déclaration et initialisation en même temps
+int a, b, c; // déclarations multiples
+a = b = c = 1; // init. multiples
+```
+
+# Les variables (1/2)
+
+## Variables et portée
+
+- Une variable est un identifiant, qui peut être liée à une valeur (un expression).
+- Une variable a une **portée** qui définit où elle est *visible* (où elle peut être accédée).
+- La portée est **globale** ou **locale**.
+- Une variable est **globale** est accessible à tout endroit d'un programme et doit être déclarée en dehors de toute fonction.
+- Une variable est **locale** lorsqu'elle est déclarée dans un **bloc**, `{...}`{.C}.
+- Une variable est dans la portée **après** avoir été déclarée.
+
+# Les variables (2/2)
+
+## Exemple
+
+```C
+float max; // variable globale accessible partout
+int foo() {
+ // max est visible ici
+ float a = max; // valide
+ // par contre les varibles du main() ne sont pas visibles
+}
+int main() {
+ // max est visible ici
+ int x = 1; // x est locale à main
+ {
+ // x est visible ici, y pas encore
+ // on peut par exemple pas faire x = y;
+ int y = 2;
+ } // y est détruite à la sortie du bloc
+} // x est à la sortie de main
+
+```
+
+<!-- TODO: quiz, compile, compile pas -->
+<!-- ```C
+int main() {
+ global = 1;
+} // COMPILE PAS
+```
+
+```C
+int main() {
+ int global = 1;
+ {
+ printf("global = %d", global);
+ }
+} // COMPILE
+```
+
+```C
+int local;
+
+int main() {
+ local = 1;
+ {
+ printf("local = %d", local);
+ }
+} // COMPILE
+```
+
+```C
+#include <stdio.h>
+int local = 0;
+
+int main() {
+ int local = -1;
+ {
+ int local = 1;
+ printf("local = %d\n", local);
+ }
+} // COMPILE
+``` -->
+
+# Quiz: compile ou compile pas?
+
+## [Quiz: compile ou compile pas](https://cyberlearn.hes-so.ch/mod/evoting/view.php?id=1033948)
+
+# Types de base (1/4)
+
+## Numériques
+
+Type Signification (**gcc pour x86-64**)
+---------------------------------- ---------------------------------------------
+`char`{.C}, `unsigned char`{.C} Entier signé/non-signé 8-bit
+`short`{.C}, `unsigned short`{.C} Entier signé/non-signé 16-bit
+`int`{.C}, `unsigned int`{.C} Entier signé/non-signé 32-bit
+`long`{.C}, `unsigned long`{.C} Entier signé/non-signé 64-bit
+`float`{.C} Nombre à virgule flottante, simple précision
+`double`{.C} Nombre à virgule flottante, double précision
+---------------------------------- ---------------------------------------------
+
+**La signification de `short`{.C}, `int`{.C}, ... dépend du compilateur et de l'architecture.**
+
+# Types de base (2/4)
+
+Voir `<stdint.h>` pour des représentations **portables**
+
+Type Signification
+---------------------------------- ---------------------------------------------
+`int8_t`{.C}, `uint8_t`{.C} Entier signé/non-signé 8-bit
+`int16_t`{.C}, `uint16_t`{.C} Entier signé/non-signé 16-bit
+`int32_t`{.C}, `uint32_t`{.C} Entier signé/non-signé 32-bit
+`int64_t`{.C}, `uint64_t`{.C} Entier signé/non-signé 64-bit
+---------------------------------- ---------------------------------------------
+
+. . .
+
+## Prenez l'habitude d'utiliser ces types-là !
+
+# Types de base (3/4)
+
+## Booléens
+
+- Le ANSI C n'offre pas de booléens.
+- L'entier `0`{.C} signifie *faux*, tout le reste *vrai*.
+- Depuis C99, la librairie `stdbool` met à disposition un type `bool`{.C}.
+- En réalité c'est un entier:
+ - $1 \Rightarrow$ `true`{.C}
+ - $0 \Rightarrow$ `false`{.C}
+- On peut les manipuler comme des entier (les sommer, les multiplier, ...).
+
+# Quiz: booléens
+
+## [Quiz: booléens](https://cyberlearn.hes-so.ch/mod/evoting/view.php?id=1032492)
+
+<!-- TODO Quiz en ligne -->
+<!-- ```C
+if (42) { /* vrai */ }
+
+int x = 100;
+if (x == 4) { /* faux */ }
+if (x) { /* vrai */ }
+
+int x = 100;
+while (x−−) { /* répète tant que x est différent de 0 */ }
+
+if (0) { /* faux */ }
+if (i = 4) { /* vrai */ }
+if (i = 0) { /* faux */ }
+
+#include <stdbool.h>
+
+bool x = true;
+if (x) { /* vrai */ }
+``` -->
+
+# Types de base (4/4)
+
+## Conversions
+
+- Les conversions se font de manière:
+ - Explicite:
+ ```C
+ int a = (int)2.8;
+ double b = (double)a;
+ int c = (int)(2.8+0.5);
+ ```
+ - Implicite:
+ ```C
+ int a = 2.8; // warning, si activés, avec clang
+ double b = a + 0.5;
+ char c = b; // pas de warning...
+ int d = 'c';
+ ```
+
+# Quiz: conversions
+
+## [Quiz: conversions](https://cyberlearn.hes-so.ch/mod/evoting/view.php?id=1033446)
+
+<!-- TODO Quiz en ligne -->
+<!-- ```C
+int a = (int)2.8; // 2
+
+double b = 2.85;
+int c = b + 0.5; // 3
+
+int d = a + 0.5; // 2
+
+bool d = 2.78; // 1
+bool e = 1.0; // 1
+``` -->
+
+# Expressions et opérateurs (1/6)
+
+Une expression est tout bout de code qui est **évalué**.
+
+## Expressions simples
+
+- Pas d'opérateurs impliqués.
+- Les littéraux, les variables, et les constantes.
+
+```C
+const int L = -1; // 'L' est une constante, -1 un littéral
+int x = 0; // '0' est un litéral
+int y = x; // 'x' est une variable
+int z = L; // 'L' est une constante
+```
+
+## Expressions complexes
+
+- Obtenues en combinant des *opérandes* avec des *opérateurs*
+
+```C
+int x; // pas une expression (une instruction)
+x = 4 + 5; // 4 + 5 est une expression
+ // dont le résultat est affecté à 'x'
+```
+
+# Expressions et opérateurs (2/6)
+
+## Opérateurs relationnels
+
+Opérateurs testant la relation entre deux *expressions*:
+
+ - `(a opérateur b)` retourne `1`{.C} si l'expression s'évalue à `true`{.C}, `0`{.C} si l'expression s'évalue à `false`{.C}.
+
+| Opérateur | Syntaxe | Résultat |
+|-----------|--------------|----------------------|
+| `<`{.C} | `a < b`{.C} | 1 si a < b; 0 sinon |
+| `>`{.C} | `a > b`{.C} | 1 si a > b; 0 sinon |
+| `<=`{.C} | `a <= b`{.C} | 1 si a <= b; 0 sinon |
+| `>=`{.C} | `a >= b`{.C} | 1 si a >= b; 0 sinon |
+| `==`{.C} | `a == b`{.C} | 1 si a == b; 0 sinon |
+| `!=`{.C} | `a != b`{.C} | 1 si a != b; 0 sinon |
+
+# Expressions et opérateurs (3/6)
+
+## Opérateurs logiques
+
+| Opérateur | Syntaxe | Signification |
+|-----------|--------------|----------------------|
+| `&&`{.C} | `a && b`{.C} | ET logique |
+| `||`{.C} | `a || b`{.C} | OU logique |
+| `!`{.C} | `!a`{.C} | NON logique |
+
+# Quiz: opérateurs logiques
+
+## [Quiz: opérateurs logiques](https://cyberlearn.hes-so.ch/mod/evoting/view.php?id=1033629)
+
+<!-- TODO: Quiz -->
+<!-- ```C
+1 && 0 == 0
+7 && 3 == 1
+4 || 3 == 1
+!34 == 0
+!0 == 1
+
+Soit n un unsigned char initialisé à 127:
+!n == 0
+``` -->
+
+# Expressions et opérateurs (4/6)
+
+## Opérateurs arithmétiques
+
+| Opérateur | Syntaxe | Signification |
+|-----------|--------------|----------------------|
+| `+`{.C} | `a + b`{.C} | Addition |
+| `-`{.C} | `a - b`{.C} | Soustraction |
+| `*`{.C} | `a * b`{.C} | Multiplication |
+| `/`{.C} | `a / b`{.C} | Division |
+| `%`{.C} | `a % b`{.C} | Modulo |
+
+# Expressions et opérateurs (5/6)
+
+## Opérateurs d'assignation
+
+| Opérateur | Syntaxe | Signification |
+|-----------|--------------|---------------------------------------------|
+| `=`{.C} | `a = b`{.C} | Affecte la valeur `b` Ã la variable `a` |
+| | | et retourne la valeur de `b` |
+| `+=`{.C} | `a += b`{.C} | Additionne la valeur de `b` Ã `a` et |
+| | | assigne le résultat à `a`. |
+| `-=`{.C} | `a -= b`{.C} | Soustrait la valeur de `b` Ã `a` et |
+| | | assigne le résultat à `a`. |
+| `*=`{.C} | `a *= b`{.C} | Multiplie la valeur de `b` Ã `a` et |
+| | | assigne le résultat à `a`. |
+| `/=`{.C} | `a /= b`{.C} | Divise la valeur de `b` Ã `a` et |
+| | | assigne le résultat à `a`. |
+| `%=`{.C} | `a %= b`{.C} | Calcule le modulo la valeur de `b` Ã `a` et |
+| | | assigne le résultat à `a`. |
+
+# Expressions et opérateurs (6/6)
+
+## Opérateurs d'assignation (suite)
+
+| Opérateur | Syntaxe | Signification |
+|-----------|--------------|---------------------------------------------|
+| `++`{.C} | `++a`{.C} | Incrémente la valeur de `a` de 1 et |
+| | | retourne le résultat (`a += 1`). |
+| `--`{.C} | `--a`{.C} | Décrémente la valeur de `a` de 1 et |
+| | | retourne le résultat (`a -= 1`). |
+| `++`{.C} | `a++`{.C} | Retourne `a`{.C} et incrémente `a` de 1. |
+| `--`{.C} | `a--`{.C} | Retourne `a`{.C} et décrémente `a` de 1. |
+
+
+# Structures de contrôle: `if`{.C} .. `else if`{.C} .. `else`{.C} (1/2)
+
+## Syntaxe
+
+```C
+if (expression) {
+ instructions;
+} else if (expression) { // optionnel
+ // il peut y en avoir plusieurs
+ instructions;
+} else {
+ instructions; // optionnel
+}
+```
+
+```C
+if (x) { // si x s'évalue à `vrai`
+ printf("x s'évalue à vrai.\n");
+} else if (y == 8) { // si y vaut 8
+ printf("y vaut 8.\n");
+} else {
+ printf("Ni l'un ni l'autre.\n");
+}
+```
+
+# Structures de contrôle: `if`{.C} .. `else if`{.C} .. `else`{.C} (2/2)
+
+## Pièges
+
+```C
+int x, y;
+x = y = 3;
+if (x = 2)
+ printf("x = 2 est vrai.\n");
+else if (y < 8)
+ printf("y < 8.\n");
+else if (y == 3)
+ printf("y vaut 3 mais cela ne sera jamais affiché.\n");
+else
+ printf("Ni l'un ni l'autre.\n");
+ x = -1; // toujours évalué
+```
+
+# Quiz: `if ... else`{.C}
+
+## [Quiz: `if ... else`{.C}](https://cyberlearn.hes-so.ch/mod/evoting/view.php?id=1033916)
+
+
+# Structures de contrôle: `while`{.C}
+
+## La boucle `while`{.C}
+
+```C
+while (condition) {
+ instructions;
+}
+do {
+ instructions;
+} while (condition);
+```
+
+## La boucle `while`{.C}, un exemple
+
+```C
+int sum = 0; // syntaxe C99
+while (sum < 10) {
+ sum += 1;
+}
+do {
+ sum += 10;
+} while (sum < 100)
+```
+
+# Structures de contrôle: `for`{.C}
+
+## La boucle `for`{.C}
+
+```C
+for (expression1; expression2; expression3) {
+ instructions;
+}
+```
+
+## La boucle `for`{.C}
+
+```C
+int sum = 0; // syntaxe C99
+for (int i = 0; i < 10; i++) {
+ sum += i;
+}
+
+for (int i = 0; i != 1; i = rand() % 4) { // ésotérique
+ printf("C'est plus ésotérique.\n");
+}
+```
diff --git a/slides_2022/cours_10.md b/slides_2022/cours_10.md
new file mode 100644
index 0000000000000000000000000000000000000000..5af82c0207b68f25a5e79c363eb5cdf2511acdbc
--- /dev/null
+++ b/slides_2022/cours_10.md
@@ -0,0 +1,402 @@
+---
+title: "Piles"
+date: "2022-12-14"
+patat:
+ eval:
+ tai:
+ command: fish
+ fragment: false
+ replace: true
+ ccc:
+ command: fish
+ fragment: false
+ replace: true
+ images:
+ backend: auto
+---
+
+# Rappel
+
+## Qu'est-ce qu'une pile?
+
+. . .
+
+* Structure de données 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 retrouner.
+3. Liste vide? (is_empty?).
+4. Jeter un oeil (peek): retourner l'élément du sommet de la pile (sans le dépiler).
+5. Nombre d'éléments (length).
+
+# 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
+bool evaluate(char *postfix, double *val) { // 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);
+}
+```
+
+
+
+# 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);
+ return val;
+}
+```
+
+# 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);
+ }
+}
+```
+
+
diff --git a/slides_2022/cours_11.md b/slides_2022/cours_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..13e253640056118af67f1443a073dcc8e84573b3
--- /dev/null
+++ b/slides_2022/cours_11.md
@@ -0,0 +1,514 @@
+---
+title: "Files d'attente et listes triées"
+date: "2022-12-21"
+---
+
+# La file d'attente (1/N)
+
+* 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/N)
+
+\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.head->data
+int queue_head(queue fa); // return fa.tail->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_2022/cours_12.md b/slides_2022/cours_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..c9646d2edda6c00afba9a4fa9a8d5e59728ddcc9
--- /dev/null
+++ b/slides_2022/cours_12.md
@@ -0,0 +1,409 @@
+---
+title: "Listes triées et listes doublement chaînées"
+date: "2023-01-11"
+---
+
+# 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
+
+## 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;
+}
+```
+
+
+# L'extraction
+
+## Trois cas
+
+1. L'élément à extraire n'est **pas** le premier élément de la liste
+
+. . .
+
+{width=70%}
+
+. . .
+
+\scriptsize
+
+```C
+sorted_list sorted_list_extract(sorted_list list, int val) {
+ element *prec = *crt = list; // needed to glue elements together
+ while (NULL != crt && val > crt->data) {
+ prec = crt;
+ crt = crt->next;
+ }
+ if (NULL != crt && prec != crt && crt->data == val) { // glue things together
+ prec->next = crt->next;
+ free(crt);
+ }
+ return list;
+}
+```
+
+
+# L'extraction
+
+2. L'élément à extraire est le premier élément de la liste
+
+. . .
+
+{width=70%}
+
+. . .
+
+\footnotesize
+
+```C
+sorted_list sorted_list_extract(sorted_list list, int val) {
+ element *prec = *crt = list; // needed to glue elements together
+ while (NULL != crt && val > crt->data) {
+ prec = crt;
+ crt = crt->next;
+ }
+ if (NULL != crt && crt->data == val && prec == crt) { // glue things together
+ list = list->next;
+ free(crt);
+ }
+ return list;
+}
+```
+
+# L'extraction
+
+3. L'élément à extraire n'est **pas** dans la liste.
+ * La liste est vide.
+ * La valeur est plus grande que le dernier élément de la liste.
+ * La valeur est plus petite que la valeur de `crt`.
+
+. . .
+
+On retourne la liste inchangée.
+
+. . .
+
+\footnotesize
+
+```C
+sorted_list sorted_list_extract(sorted_list list, int val) {
+ element *prec = *crt = list; // needed to glue elements together
+ while (NULL != crt && val > crt->data) {
+ prec = crt;
+ crt = crt->next;
+ }
+ if (NULL == crt || crt->data != val) { // val not present
+ return list;
+ }
+}
+```
+
+# La recherche
+
+
+
+```C
+element* sorted_list_search(sorted_list list, int val);
+```
+
+* Retourne `NULL` si la valeur n'est pas présente (ou la liste vide).
+* Retourne un pointeur vers l'élément si la valeur est présente.
+
+. . .
+
+```C
+element* sorted_list_search(sorted_list list, int val) {
+ // search for element smaller than val
+ element* pos = sorted_list_position(list, val);
+ if (NULL == pos && val == list->data) {
+ return list; // first element contains val
+ } else if (NULL != pos && NULL != pos->next && val == pos->next->data) {
+ return pos->next; // non-first element contains val
+ } else {
+ return NULL; // well... val's not here
+ }
+}
+```
+
+# La recherche
+
+## La fonction `sorted_list_position`
+
+```C
+element* sorted_list_position(sorted_list list, int val);
+```
+
+
+
+# La recherche
+
+## Exercice: implémenter
+
+```C
+element* sorted_list_position(sorted_list list, int val);
+```
+
+. . .
+
+```C
+element* sorted_list_position(sorted_list list, int val) {
+ element* pos = list;
+ if (sorted_list_is_empty(list) || val <= list->data) {
+ pos = NULL;
+ } else {
+ while (NULL != pos->next && val > pos->next->data) {
+ pos = pos->next;
+ }
+ }
+ return pos;
+}
+```
+
+# Complexité de la liste chaînée triée
+
+## L'insertion?
+
+. . .
+
+$$
+\mathcal{O}(N).
+$$
+
+## L'extraction?
+
+. . .
+
+$$
+\mathcal{O}(N).
+$$
+
+## La recherche?
+
+. . .
+
+$$
+\mathcal{O}(N).
+$$
+
+
+# Liste doublement chaînée
+
+## Application: navigateur ou éditeur de texte
+
+* Avec une liste chaînée:
+ * Comment implémenter les fonctions `back` et `forward` d'un navigateur?
+ * Comment implémenter les fonctions `undo` et `redo` d'un éditeur de texte?
+
+. . .
+
+Pas possible.
+
+## Solution?
+
+. . .
+
+* Garder un pointeur supplémentaire sur l'élément précédent et pas seulement le
+ suivant.
+
+. . .
+
+* Cette structure de donnée est la **liste doublement chaînée** ou **doubly
+ linked list**.
+
+# Liste doublement chaînée
+
+## Exercices
+
+* Partir du dessin suivant et par **groupe de 5**
+
+
+
+1. Écrire les structures de données pour représenter la liste doublement
+ chaînée dont le type sera `dll` (pour
+ `doubly_linked_list`)
+
+# Liste doublement chaînée
+
+2. Écrire les fonctionnalités de création et consultation
+
+```C
+// crée la liste doublement chaînée
+dll dll_create();
+// retourne la valeur à la position actuelle dans la liste
+int dll_value(dll list);
+// la liste est-elle vide?
+bool dll_is_empty(dll list);
+// Est-ce que pos est le 1er élément?
+bool dll_is_head(dll list);
+// Est-ce que pos est le dernier élément?
+bool dll_is_tail(dll list);
+// data est-elle dans la liste?
+bool dll_is_present(dll list, int data);
+// affiche la liste
+void dll_print(dll list);
+```
+
+# Liste doublement chaînée
+
+3. Écrire les fonctionnalités de manipulation
+
+```C
+// déplace pos au début de la liste
+dll dll_move_to_head(dll list);
+// déplace pos à la position suivante dans la liste
+dll dll_next(dll list);
+// déplace pos à la position précédente dans la liste
+dll dll_prev(dll list);
+```
+
+# Liste doublement chaînée
+
+4. Écrire les fonctionnalités d'insertion
+
+```C
+// insertion de data dans l'élément après pos
+dll dll_insert_after(dll list, int data);
+// insertion de data en tête de liste
+dll dll_push(dll list, int data);
+```
+
+5. Écrire les fonctionnalités d'extraction
+
+```C
+// extraction de la valeur se trouvant dans l'élément pos
+// l'élément pos est libéré
+int dll_extract(dll *list);
+// extrait la donnée en tête de liste
+int dll_pop(dll *list);
+// vide la liste
+void dll_destroy(dll *list)
+```
+
diff --git a/slides_2022/cours_13.md b/slides_2022/cours_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..d6d3d138e232715775a661f28dd8486d267a61a2
--- /dev/null
+++ b/slides_2022/cours_13.md
@@ -0,0 +1,605 @@
+---
+title: "Tables de hachage"
+date: "2023-01-18"
+---
+
+# Tableau vs Table
+
+## Tableau
+
+* Chaque élément (ou valeur) est lié à un indice (la case du tableau).
+
+```C
+annuaire tab[2] = {
+ "+41 22 123 45 67", "+41 22 234 56 78", ...
+};
+tab[1] == "+41 22 123 45 67";
+```
+
+## Table
+
+* Chaque élément (ou valeur) est lié à une clé.
+
+```C
+annuaire tab = {
+// Clé , Valeur
+ "Paul", "+41 22 123 45 67",
+ "Orestis", "+41 22 234 56 78",
+};
+tab["Paul"] == "+41 22 123 45 67";
+tab["Orestis"] == "+41 22 234 56 78";
+```
+
+# Table
+
+## Définition
+
+Structure de données abstraite où chaque *valeur* (ou élément) est associée à une *clé* (ou
+argument).
+
+On parle de paires *clé-valeur* (*key-value pairs*).
+
+## Donnez des exemples de telles paires
+
+. . .
+
+* Annuaire (nom-téléphone),
+* Catalogue (objet-prix),
+* Table de valeur fonctions (nombre-nombre),
+* Index (nombre-page)
+* ...
+
+# Table
+
+## Opérations principales sur les tables
+
+* Insertion d'élément (`insert(clé, valeur)`{.C}), insère la paire `clé-valeur`
+* Consultation (`get(clé)`{.C}), retourne la `valeur` correspondant à `clé`
+* Suppression (`remove(clé)`{.C}), supprime la paire `clé-valeur`
+
+## Structure de données / implémentation
+
+Efficacité dépend de différents paramètres:
+
+* taille (nombre de clé-valeurs maximal),
+* fréquence d'utilisation (insertion, consultation, suppression),
+* données triées/non-triées,
+* ...
+
+# Consultation séquentielle (`sequential_get`)
+
+## Séquentielle
+
+* table représentée par un (petit) tableau ou liste chaînée,
+* types: `key_t` et `value_t` quelconques, et `key_value_t`
+
+ ```C
+ typedef struct {
+ key_t key;
+ value_t value;
+ } key_value_t;
+ ```
+* on recherche l'existence de la clé séquentiellement dans le tableau, on
+ retourne la valeur.
+
+# Consultation séquentielle (`sequential_get`)
+
+## Implémentation? Une idée?
+
+. . .
+
+```C
+bool sequential_get(int n, key_value_t table[n], key_t key,
+ value_t *value)
+{
+ int pos = n - 1;
+ while (pos >= 0) {
+ if (key == table[pos].key) {
+ *value = table[pos].value;
+ return true;
+ }
+ pos--;
+ }
+ return false;
+}
+```
+
+. . .
+
+## Inconvénient?
+
+# Consultation séquentielle (`sequential_get`)
+
+## Exercice: implémenter la même fonction avec une liste chaînée
+
+Poster le résultat sur matrix.
+
+# Consultation dichotomique (`binary_get`)
+
+## Dichotomique
+
+* table représentée par un (petit) tableau trié par les clés,
+* types: `key_t` et `value_t` quelconques, et `key_value_t`
+* on recherche l'existence de la clé par dichotomie dans le tableau, on
+ retourne la valeur,
+* les clés possèdent la notion d'ordre (`<, >, =` sont définis).
+
+# Consultation dichotomique (`binary_get`)
+
+\footnotesize
+
+## Implémentation? Une idée?
+
+. . .
+
+```C
+bool binary_get1(int n, value_key_t table[n], key_t key, value_t *value) {
+ int top = n - 1, bottom = 0;
+ while (top > bottom) {
+ int middle = (top + bottom) / 2;
+ if (key > table[middle].key) {
+ bottom = middle+1;
+ } else {
+ top = middle;
+ }
+ }
+ if (key == table[top].key) {
+ *value = table[top].value;
+ return true;
+ } else {
+ return false;
+ }
+}
+```
+
+# Consultation dichotomique (`binary_get`)
+
+\footnotesize
+
+## Autre implémentation
+
+```C
+bool binary_get2(int n, key_value_t table[n], key_t key, value_t *value) {
+ int top = n - 1, bottom = 0;
+ while (true) {
+ int middle = (top + bottom) / 2;
+ if (key > table[middle].key) {
+ bottom = middle + 1;
+ } else if (key < table[middle].key) {
+ top = middle;
+ } else {
+ *value = table[middle].value;
+ return true;
+ }
+ if (top < bottom) {
+ break;
+ }
+ }
+ return false;
+}
+```
+
+## Quelle est la différence avec le code précédent?
+
+# Transformation de clé (hashing)
+
+## Problématique: Numéro AVS (13 chiffres)
+
+* Format: 106.3123.8492.13
+
+ ```
+ Numéro AVS | Nom
+ 0000000000000 | -------
+ ... | ...
+ 1063123849213 | Paul
+ ... | ...
+ 3066713878328 | Orestis
+ ... | ...
+ 9999999999999 | -------
+ ```
+
+## Quelle est la clé? Quelle est la valeur?
+
+. . .
+
+* Clé: Numéro AVS, Valeur: Nom.
+
+## Nombre de clés? Nombre de citoyens? Rapport?
+
+. . .
+
+* $10^{13}$ clés, $10^7$ citoyens, $10^{-5}$ ($10^{-3}\%$ de la table est
+ occupée) $\Rightarrow$ *inefficace*.
+* Pire: $10^{13}$ entrées ne rentre pas dans la mémoire d'un
+ ordinateur.
+
+# Transformation de clé (hashing)
+
+## Problématique 2: Identificateurs d'un programme
+
+* Format: 8 caractères (simplification)
+
+ ```
+ Identificateur | Adresse
+ aaaaaaaa | -------
+ ... | ...
+ resultat | 3aeff
+ compteur | 4fedc
+ ... | ...
+ zzzzzzzz | -------
+ ```
+
+## Quelle est la clé? Quelle est la valeur?
+
+. . .
+
+* Clé: Identificateur, Valeur: Adresse.
+
+## Nombre de clés? Nombre d'identificateur d'un programme? Rapport?
+
+. . .
+
+* $26^{8}\sim 2\cdot 10^{11}$ clés, $2000$ identificateurs, $10^{-8}$ ($10^{-6}\%$ de la table est
+ occupée) $\Rightarrow$ *un peu inefficace*.
+
+# Fonctions de transformation de clé (hash functions)
+
+* La table est représentée avec un tableau.
+* La taille du tableau est beaucoup plus petit que le nombre de clés.
+* On produit un indice du tableau à partir d'une clé:
+$$
+h(key) = n,\quad n\in\mathbb{N}.
+$$
+En français: on transforme `key` en nombre entier qui sera l'indice dans le
+tableau correspondant à `key`.
+
+## La fonction de hash
+
+* La taille du domaine des clés est beaucoup plus grand que le domaine des
+ indices.
+* Plusieurs indices peuvent correspondre à la **même clé**:
+ * Il faut traiter les **collisions**.
+* L'ensemble des indices doit être plus petit ou égal à la taille de la table.
+
+## Une bonne fonction de hash
+
+* Distribue uniformément les clés sur l'ensemble des indices.
+
+# Fonctions de transformation de clés: exemples
+
+## Méthode par troncature
+
+\begin{align*}
+&h: [0,9999]\rightarrow [0,9]\\
+&h(key)=\mbox{troisième chiffre du nombre.}
+\end{align*}
+
+```
+Key | Index
+0003 | 0
+1123 | 2 \
+1234 | 3 |-> collision.
+1224 | 2 /
+1264 | 6
+```
+
+## Quelle est la taille de la table?
+
+. . .
+
+C'est bien dix oui.
+
+# Fonctions de transformation de clés: exemples
+
+## Méthode par découpage
+
+Taille de l'index: 3 chiffres.
+
+```
+key = 321 991 24 -> 321
+ 991
+ + 24
+ ----
+ 1336 -> index = 336
+```
+
+## Devinez l'algorithme?
+
+. . .
+
+On part de la gauche:
+
+1. On découpe la clé en tranche de longueur égale à celle de l'index.
+2. On somme les nombres obtenus.
+3. On tronque à la longueur de l'index.
+
+# Fonctions de transformation de clés: exemples
+
+## Méthode multiplicative
+
+Taille de l'index: 2 chiffres.
+
+```
+key = 5486 -> key^2 = 30096196 -> index = 96
+```
+
+On prend le carré de la clé et on garde les chiffres du milieu du résultat.
+
+# Fonctions de transformation de clés: exemples
+
+## Méthode par division modulo
+
+Taille de l'index: `N` chiffres.
+
+```
+h(key) = key % N.
+```
+
+## Quelle doit être la taille de la table?
+
+. . .
+
+Oui comme vous le pensiez au moins `N`.
+
+# Traitement des collisions
+
+## La collision
+
+```
+key1 != key2, h(key1) == h(key2)
+```
+
+## Traitement (une idée?)
+
+. . .
+
+* La première clé occupe la place prévue dans le tableau.
+* La deuxième (troisième, etc.) est placée ailleurs de façon **déterministe**.
+
+Dans ce qui suit la taille de la table est `table_size`.
+
+# La méthode séquentielle
+
+\footnotesize
+
+## Comment ça marche?
+
+* Quand l'index est déjà occupé on regarde sur la position suivante, jusqu'à en
+ trouver une libre.
+
+```C
+index = h(key);
+while (table[index].state == OCCUPIED && table[index].key != key) {
+ index = (index + 1) % table_size; // attention à pas dépasser
+}
+table[index].key = key;
+table[index].state = OCCUPIED;
+```
+
+## Problème?
+
+. . .
+
+* Regroupement d'éléments (clustering).
+
+# Méthode linéaire
+
+\footnotesize
+
+## Comment ça marche?
+
+* Comme la méthode séquentielle mais on "saute" de `k`.
+
+```C
+index = h(key);
+while (table[index].state == OCCUPIED && table[index].key != key) {
+ index = (index + k) % table_size; // attention à pas dépasser
+}
+table[index].key = key;
+table[index].state = OCCUPIED;
+```
+
+## Quelle valeur de `k` éviter?
+
+. . .
+
+* Une valeur où `table_size` est multiple de `k`.
+
+Cette méthode répartit mieux les regroupements au travers de la table.
+
+# Méthode du double hashing
+
+\footnotesize
+
+## Comment ça marche?
+
+* Comme la méthode linéaire, mais `k = h2(key)` (variable).
+
+```C
+index = h(key);
+while (table[index].state == OCCUPIED && table[index].key != key) {
+ index = (index + h2(k)) % table_size; // attention à pas dépasser
+}
+table[index].key = key;
+table[index].state = OCCUPIED;
+```
+
+## Quelle propriété doit avoir `h2`?
+
+## Exemple
+
+```C
+h2(key) = (table_size - 2) - key % (table_size -2)
+```
+
+# Méthode pseudo-aléatoire
+
+\footnotesize
+
+## Comment ça marche?
+
+* Comme la méthode linéaire mais on génère `k` pseudo-aléatoirement.
+
+ ```C
+ index = h(key);
+ while (table[index].state == OCCUPIED && table[index].key != key) {
+ index = (index + random_number) % table_size;
+ }
+ table[index].key = key;
+ table[index].state = OCCUPIED;
+ ```
+
+## Comment s'assurer qu'on va bien retrouver la bonne clé?
+
+. . .
+
+* Le germe (seed) de la séquence pseudo-aléatoire doit être le même.
+* Le germe à choisir est l'index retourné par `h(key)`.
+
+ ```C
+ srand(h(key));
+ while {
+ random_number = rand();
+ }
+ ```
+
+# Méthode quadratique
+
+* La fonction des indices de collision est de degré 2.
+* Soit $J_0=h(key)$, les indices de collision se construisent comme:
+
+ ```C
+ J_i = J_0 + i^2 % table_size, i > 0,
+ J_0 = 100, J_1 = 101, J_2 = 104, J_3 = 109, ...
+ ```
+
+## Problème possible?
+
+. . .
+
+* Calculer le carré peut-être "lent".
+* En fait on peut ruser un peu.
+
+# Méthode quadratique
+
+\footnotesize
+
+```C
+J_i = J_0 + i^2 % table_size, i > 0,
+J_0 = 100
+ \
+ d_0 = 1
+ / \
+J_1 = 101 Delta = 2
+ \ /
+ d_1 = 3
+ / \
+J_2 = 104 Delta = 2
+ \ /
+ d_2 = 5
+ / \
+J_3 = 109 Delta = 2
+ \ /
+ d_3 = 7
+ /
+J_4 = 116
+--------------------------------------
+J_{i+1} = J_i + d_i,
+d_{i+1} = d_i + Delta, d_0 = 1, i > 0.
+```
+
+# Méthode de chaînage
+
+## Comment ça marche?
+
+* Chaque index de la table contient un pointeur vers une liste chaînée
+ contenant les paires clés-valeurs.
+
+## Un petit dessin
+
+```
+
+
+
+
+
+
+
+
+
+
+
+```
+
+# Méthode de chaînage
+
+## Exemple
+
+On hash avec la fonction `h(key) = key % 11` (`key` est le numéro de la lettre
+de l'alphabet)
+
+```
+ U | N | E | X | E | M | P | L | E | D | E | T | A | B | L | E
+ 10 | 3 | 5 | 2 | 5 | 2 | 5 | 1 | 5 | 4 | 5 | 9 | 1 | 2 | 1 | 5
+```
+
+## Comment on représente ça? (à vous)
+
+. . .
+
+{width=80%}
+
+# Méthode de chaînage
+
+Avantages:
+
+* Si les clés sont grandes l'économie de place est importante (les places vides
+ sont `NULL`).
+* La gestion des collisions est conceptuellement simple.
+* Pas de problème de regroupement (clustering).
+
+# Exercice 1
+
+* Construire une table à partir de la liste de clés suivante:
+ ```
+ R, E, C, O, U, P, A, N, T
+ ```
+
+* On suppose que la table est initialement vide, de taille $n = 13$.
+* Utiliser la fonction $h1(k)= k \mod 13$ où k est la $k$-ème lettre de l'alphabet et un traitement séquentiel des collisions.
+
+# Exercice 2
+
+* Reprendre l'exercice 1 et utiliser la technique de double hachage pour traiter
+ les collisions avec
+
+\begin{align*}
+h_1(k)&=k\mod 13,\\
+h_2(k)&=1+(k\mod 11).
+\end{align*}
+* La fonction de hachage est donc $h(k)=(h(k)+h_2(k)) \% 13$ en cas de
+ collision.
+
+
+# Exercice 3
+
+* Stocker les numéros de téléphones internes d'une entreprise suivants dans un
+tableau de 10 positions.
+* Les numéros sont compris entre 100 et 299.
+* Soit $N$ le numéro de téléphone, la fonction de hachage est
+$$
+h(N)=N\mod 10.
+$$
+* La fonction de gestion des collisions est
+$$
+C_1(N,i)=(h(N)+3\cdot i)\mod 10.
+$$
+* Placer 145, 167, 110, 175, 210, 215 (mettre son état à occupé).
+* Supprimer 175 (rechercher 175, et mettre son état à supprimé).
+* Rechercher 35.
+* Les cases ni supprimées, ni occupées sont vides.
+* Expliquer se qui se passe si on utilise?
+$$
+C_1(N,i)=(h(N)+5\cdot i)\mod 10.
+$$
+
diff --git a/slides_2022/cours_14.md b/slides_2022/cours_14.md
new file mode 100644
index 0000000000000000000000000000000000000000..b4057a4e02b56786460cb9fa9782f0b67a8fa1b4
--- /dev/null
+++ b/slides_2022/cours_14.md
@@ -0,0 +1,367 @@
+---
+title: "Tables de hachage"
+date: "2023-02-24"
+patat:
+ eval:
+ tai:
+ command: fish
+ fragment: false
+ replace: true
+ ccc:
+ command: fish
+ fragment: false
+ replace: true
+ images:
+ backend: auto
+---
+
+# Rappel sur les tables de hachage (1/N)
+
+## Définition? Qui se souvient?
+
+. . .
+
+Structure de données abstraite où chaque *valeur* (ou élément) est associée à une *clé* (ou
+argument).
+
+On parle de paires *clé-valeur* (*key-value pairs*).
+
+## Donnez des exemples de telles paires
+
+. . .
+
+* Annuaire (nom-téléphone),
+* Catalogue (objet-prix),
+* Table de valeur fonctions (nombre-nombre),
+* Index (nombre-page)
+* ...
+
+# Rappel sur les tables de hachage (1/N)
+
+## Opérations principales sur les tables
+
+* Insertion d'élément (`insert(clé, valeur)`{.C}), insère la paire `clé-valeur`
+* Consultation (`get(clé)`{.C}), retourne la `valeur` correspondant à `clé`
+* Suppression (`remove(clé)`{.C}), supprime la paire `clé-valeur`
+
+## Transformation de clé (hashing)
+
+* Format: 106.3123.8492.13
+
+ ```
+ Numéro AVS | Nom
+ 0000000000000 | -------
+ ... | ...
+ 1063123849213 | Paul
+ ... | ...
+ 3066713878328 | Orestis
+ ... | ...
+ 9999999999999 | -------
+ ```
+* Nombre de numéros >> nombre d'entrées.
+
+# Fonctions de transformation de clé (hash functions)
+
+* La table est représentée avec un tableau.
+* La taille du tableau est beaucoup plus petit que le nombre de clés.
+* On produit un indice du tableau à partir d'une clé:
+$$
+h(key) = n,\quad n\in\mathbb{N}.
+$$
+En français: on transforme `key` en nombre entier qui sera l'indice dans le
+tableau correspondant à `key`.
+
+## La fonction de hash
+
+* La taille du domaine des clés est beaucoup plus grand que le domaine des
+ indices.
+* Plusieurs indices peuvent correspondre à la **même clé**:
+ * Il faut traiter les **collisions**.
+* L'ensemble des indices doit être plus petit ou égal à la taille de la table.
+
+## Une bonne fonction de hash
+
+* Distribue uniformément les clés sur l'ensemble des indices.
+
+# Fonctions de transformation de clés: exemple
+
+## Méthode par division modulo
+
+Taille de l'index: `N` chiffres.
+
+```
+h(key) = key % N.
+```
+
+## Quelle doit être la taille de la table?
+
+. . .
+
+Oui comme vous le pensiez au moins `N`.
+
+# Traitement des collisions
+
+## La collision
+
+```
+key1 != key2, h(key1) == h(key2)
+```
+
+## Traitement (une idée?)
+
+. . .
+
+* La première clé occupe la place prévue dans le tableau.
+* La deuxième (troisième, etc.) est placée ailleurs de façon **déterministe**.
+
+Dans ce qui suit la taille de la table est `table_size`.
+
+# La méthode séquentielle
+
+\footnotesize
+
+* Quand l'index est déjà occupé on regarde sur la position suivante, jusqu'à en
+ trouver une libre.
+
+```C
+index = h(key);
+while (table[index].state == OCCUPIED && table[index].key != key) {
+ index = (index + 1) % table_size; // attention à pas dépasser
+}
+table[index].key = key;
+table[index].state = OCCUPIED;
+```
+
+# Méthode de chaînage
+
+## Comment ça marche?
+
+* Chaque index de la table contient un pointeur vers une liste chaînée
+ contenant les paires clés-valeurs.
+
+## Un petit dessin
+
+```
+
+
+
+
+
+
+
+
+
+
+
+```
+
+# Méthode de chaînage
+
+## Exemple
+
+On hash avec la fonction `h(key) = key % 11` (`key` est le numéro de la lettre
+de l'alphabet)
+
+```
+ U | N | E | X | E | M | P | L | E | D | E | T | A | B | L | E
+ 10 | 3 | 5 | 2 | 5 | 2 | 5 | 1 | 5 | 4 | 5 | 9 | 1 | 2 | 1 | 5
+```
+
+## Comment on représente ça? (à vous)
+
+. . .
+
+{width=80%}
+
+# Exercice 1
+
+* Construire une table à partir de la liste de clés suivante:
+ ```
+ R, E, C, O, U, P, A, N, T
+ ```
+
+* On suppose que la table est initialement vide, de taille $n = 13$.
+* Utiliser la fonction $h1(k)= k \mod 13$ où k est la $k$-ème lettre de l'alphabet et un traitement séquentiel des collisions.
+
+# Exercice 2
+
+* Reprendre l'exercice 1 et utiliser la technique de double hachage pour traiter
+ les collisions avec
+
+\begin{align*}
+h_1(k)&=k\mod 13,\\
+h_2(k)&=1+(k\mod 11).
+\end{align*}
+
+* La fonction de hachage est donc $h(k)=(h(k)+h_2(k)) \% 13$ en cas de
+ collision.
+
+
+# Exercice 3
+
+* Stocker les numéros de téléphones internes d'une entreprise suivants dans un
+tableau de 10 positions.
+* Les numéros sont compris entre 100 et 299.
+* Soit $N$ le numéro de téléphone, la fonction de hachage est
+$$
+h(N)=N\mod 10.
+$$
+* La fonction de gestion des collisions est
+$$
+C_1(N,i)=(h(N)+3\cdot i)\mod 10.
+$$
+* Placer 145, 167, 110, 175, 210, 215 (mettre son état à occupé).
+* Supprimer 175 (rechercher 175, et mettre son état à supprimé).
+* Rechercher 35.
+* Les cases ni supprimées, ni occupées sont vides.
+* Expliquer se qui se passe si on utilise?
+$$
+C_1(N,i)=(h(N)+5\cdot i)\mod 10.
+$$
+
+# Préambule
+
+\small
+
+* On considère pas le cas du chaînage en cas de collisions.
+* L'insertion est construite avec une forme du type
+
+ ```C
+ index = h(key);
+ while (table[index].state == OCCUPIED
+ && table[index].key != key) {
+ index = (index + k) % table_size; // attention à pas dépasser
+ }
+ table[index].key = key;
+ table[index].state = OCCUPIED;
+ ```
+\normalsize
+
+* Gestion de l'état d'une case *explicite*
+
+ ```C
+ typedef enum {EMPTY, OCCUPIED, DELETED} state;
+ ```
+
+# L'insertion
+
+## Pseudocode?
+
+. . .
+
+```C
+insert(table, key, value) {
+ index = hash de la clé;
+ index =
+ si "index" est déjà "occupé"
+ et la clé correspondante n'est pas "key"
+ alors gérer la collision;
+
+ changer l'état de la case "index" à "occupé";
+ changer la valeur de la case "index" Ã "value";
+}
+```
+
+# La suppression
+
+## Pseudocode?
+
+. . .
+
+```C
+value_t remove(table, key) {
+ index = hash de la clé;
+ tant que l'état de la case n'est pas "vide"
+ si "index" est "occupé" et la clé est "key"
+ changer l'état de la case à "supprimé"
+ sinon
+ index = rehash
+}
+```
+
+# La recherche
+
+## Pseudocode?
+
+. . .
+
+```C
+bool search(table, key, value) {
+ index = hash de la clé;
+ tant que l'état de la case n'est pas "vide"
+ si "index" est "occupé" et la clé est "key"
+ retourner vrai
+ sinon
+ index = rehash
+}
+```
+
+# Écrivons le code!
+
+* Mais avant:
+ * Quelles sont les structures de données dont nous avons besoin?
+ * Y a-t-il des fonctions auxiliaires à écrire?
+ * Écrire les signatures des fonctions.
+
+. . .
+
+## Structures de données
+
+\footnotesize
+
+. . .
+
+```C
+typedef enum {empty, deleted, occupied};
+typedef ... key_t;
+typedef ... value_t;
+typedef struct _cell_t {
+ key_t key;
+ value_t value;
+ state_t state;
+} cell_t;
+typedef struct _hm {
+ cell_t *table;
+ int capacity;
+ int size;
+} hm;
+```
+
+# Écrivons le code!
+
+## Fonctions auxiliaires
+
+. . .
+
+```C
+static int hash(key_t key);
+static int rehash(int index, key_t key);
+static int find_index(hm h, key_t key);
+```
+
+## Signature de l'API
+
+. . .
+
+```C
+void hm_init(hm *h, int capacity);
+void hm_destroy(hm *h);
+bool hm_set(hm *h, key_t key, value_t *value);
+bool hm_get(hm h, key_t key, value_t *value);
+bool hm_remove(hm *h, key_t key, value_t *value);
+bool hm_search(hm h, key_t key);
+void hm_print(hm h);
+```
+
+# Live code session!
+
+0. Offered to you by ProtonVPN[^1]!
+
+. . .
+
+1. Like the video.
+2. Subscribe to the channel.
+3. Use our one time voucher for ProtonVPN: `PAULISAWESOME`.
+4. Consider donating on our patreon.
+
+[^1]: The fastest way to connect to BBB!
diff --git a/slides_2022/cours_15.md b/slides_2022/cours_15.md
new file mode 100644
index 0000000000000000000000000000000000000000..09a8ed06a9c28de6eb60a4ad216f6dbdbddf54a6
--- /dev/null
+++ b/slides_2022/cours_15.md
@@ -0,0 +1,929 @@
+---
+title: "Arbres"
+date: "2023-03-10"
+---
+
+# Les arbres: définition
+
+"Un arbre est un graphe acyclique orienté possédant une unique racine, et tel que tous les nœuds sauf la racine ont un unique parent."
+
+. . .
+
+**Santé!**
+
+## Plus sérieusement
+
+* Ensemble de **nœuds** et d'**arêtes** (graphe),
+* Les arêtes relient les nœuds entre eux, mais pas n'importe comment: chaque
+ nœud a au plus un **parent**,
+* Le seul nœud sans parent est la **racine**,
+* Chaque nœud a un nombre fini de **fils**,
+* La hiérarchie des nœuds rend les arêtes **orientées** (parent -> fils), et empêche les
+ **cycles** (acyclique, orienté).
+* La **feuille** ou **nœud terminal** est un nœud sans enfants,
+* Le **niveau** est 1 Ã la racine et **niveau+1** pour les fils,
+* Le **degré** d'un nœud est le nombre de fils du nœud.
+
+. . .
+
+* Chaque nœud est un arbre en lui même.
+* La **récursivité** sera très utile!
+
+
+# Arbre ou pas arbre?
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ 1-->2;
+ 1-->3;
+ 3-->2;
+ 3-->4;
+ 3-->5;
+```
+::::
+
+. . .
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ 1-->2;
+ 1-->3;
+ 3-->4;
+ 3-->5;
+ 3-->6;
+```
+::::
+
+:::
+
+# Arbre ou pas arbre?
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ 1-->2;
+ 1-->3;
+ 3-->4;
+ 3-->5;
+ 3-->6;
+ 6-->7;
+ 7-->3;
+```
+::::
+
+. . .
+
+:::: column
+```{.mermaid format=pdf width=300 loc=figs/}
+graph TD;
+ 1;
+```
+::::
+
+:::
+
+# Arbre ou pas arbre?
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ 1---2;
+ 1---3;
+ 3---4;
+ 3---5;
+```
+::::
+
+. . .
+
+:::: column
+```{.mermaid format=pdf width=300 loc=figs/}
+graph BT;
+ 1-->2;
+ 1-->3;
+ 3-->4;
+ 3-->5;
+ 3-->6;
+```
+::::
+
+:::
+
+# Degré et niveau
+
+* Illustration du degré (nombre de fils) et du niveau (profondeur)
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ 1[degré 2]-->2[degré 0];
+ 1-->3[degré 3];
+ 3-->4[degré 0];
+ 3-->5[degré 0];
+ 3-->6[degré 0];
+```
+::::
+
+. . .
+
+:::: column
+```{.mermaid format=pdf width=300 loc=figs/}
+graph TD;
+ 1[niveau 1]-->2[niveau 2];
+ 1-->3[niveau 2];
+ 3-->4[niveau 3];
+ 3-->5[niveau 3];
+ 3-->6[niveau 3];
+```
+::::
+
+:::
+
+* Les nœuds de degré 0, sont des feuilles.
+
+# Application: recherche rapide
+
+## Pouvez vous construire un arbre pour résoudre le nombre secret?
+
+ . . .
+
+* Le nombre secret ou la recherche dichotomique (nombre entre 0 et 10).
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 5-->|<|2;
+ 5-->|>|7;
+ 7-->|>|8;
+ 7-->|<|6;
+ 8-->|>|9;
+ 9-->|>|10;
+ 2-->|<|1;
+ 2-->|>|3;
+ 3-->|>|4;
+ 1-->|<|0;
+```
+::::
+
+:::: column
+
+**Question:** Quelle est la complexité pour trouver un nombre?
+
+::::
+
+:::
+
+# Autres représentation
+
+* Botanique
+* **Exercice:** Ajouter les degrés/niveaux et feuilles
+
+```{.mermaid width=250 format=pdf loc=figs/}
+graph TD;
+ A-->B;
+ A-->C;
+ B-->D;
+ B-->E;
+ B-->F;
+ F-->I;
+ F-->J;
+ C-->G;
+ C-->H;
+ H-->K;
+```
+
+# Autres représentation
+
+* Ensembliste
+
+::: columns
+
+:::: column
+```{.mermaid width=300 format=pdf loc=figs/}
+graph TD;
+ A-->B;
+ A-->C;
+ B-->D;
+ B-->E;
+ B-->F;
+ F-->I;
+ F-->J;
+ C-->G;
+ C-->H;
+ H-->K;
+```
+::::
+
+. . .
+
+:::: column
+
+::::
+
+:::
+
+# Autres représentation
+
+* Liste
+
+::: columns
+
+:::: column
+```{.mermaid width=400 format=pdf loc=figs/}
+graph TD;
+ A-->B;
+ A-->C;
+ B-->D;
+ B-->E;
+ B-->F;
+ F-->I;
+ F-->J;
+ C-->G;
+ C-->H;
+ H-->K;
+```
+::::
+
+. . .
+
+:::: column
+```
+(A
+ (B
+ (D)
+ (E)
+ (F
+ (I)
+ (J)
+ )
+ )
+ (C
+ (G)
+ (H
+ (K)
+ )
+ )
+)
+```
+::::
+
+:::
+
+# Autres représentation
+
+* Par niveau
+
+::: columns
+
+:::: column
+```{.mermaid width=400 format=pdf loc=figs/}
+graph TD;
+ A-->B;
+ A-->C;
+ B-->D;
+ B-->E;
+ B-->F;
+ F-->I;
+ F-->J;
+ C-->G;
+ C-->H;
+ H-->K;
+```
+::::
+
+. . .
+
+:::: column
+```
+1 2 3 4
+-------------------------
+A
+ B
+ D
+ E
+ F
+ I
+ J
+ C
+ G
+ H
+ K
+```
+::::
+
+:::
+
+# L'arbre binaire
+
+* Structure de données abstraite,
+* Chaque nœud a au plus deux fils: gauche et droite,
+* Chaque fils est un arbre.
+
+## Comment représenteriez vous une telle structure?
+
+. . .
+
+```C
+<R, G, D>
+ R: racine
+ G: sous-arbre gauche
+ D: sous-arbre droite
+```
+
+## Comment cela s'écrirait en C?
+
+. . .
+
+```C
+typedef struct _node {
+ contenu info;
+ struct _node *left, *right;
+} node;
+typedef node *tree;
+```
+
+# L'arbre binaire
+
+## Que se passerait-il avec
+
+```C
+typedef struct _node {
+ int info;
+ struct _node left, right;
+} node;
+```
+
+* On ne sait pas quelle est la taille de node, on ne peut pas l'allouer!
+
+## Interface minimale
+
+* Qu'y mettriez vous?
+
+. . .
+
+```C
+NULL -> arbre (vide)
+<n, arbre, arbre> -> arbre
+visiter(arbre) -> nœud (la racine de l'arbre)
+gauche(arbre) -> arbre (sous-arbre de gauche)
+droite(arbre) -> arbre (sous-arbre de droite)
+```
+
+* Les autres opérations (insertion, parcours, etc) dépendent de ce qu'on stocke
+ dans l'arbre.
+
+# Exemple d'arbre binaire
+
+* Représentez `(c - a * b) * (d + e / f)` à l'aide d'un arbre binaire (matrix)
+
+. . .
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ A[*]-->B[-];
+ B-->C[c];
+ B-->D[*];
+ D-->E[a];
+ D-->F[b];
+ A-->G[+];
+ G-->H[d];
+ G-->I["/"];
+ I-->J[e];
+ I-->K[f];
+```
+::::
+
+
+:::: column
+
+## Remarques
+
+* L'arbre est **hétérogène**: le genre d'info est pas le même sur chaque nœud
+ (opérateur, opérande).
+ * Les feuilles contiennent les opérandes.
+ * Les nœuds internes contiennent les opérateurs.
+
+::::
+
+:::
+
+# Parcours d'arbres binaires
+
+* Appliquer une opération à tous les nœuds de l'arbre,
+* Nécessité de **parcourir** l'arbre,
+* Utiliser uniquement l'interface: visiter, gauche,
+ droite.
+
+## Une idée de comment parcourir cet arbre?
+
+* 3 parcours (R: Racine, G: sous-arbre gauche, D: sous-arbre droit):
+
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ A[*]-->B[-];
+ B-->C[c];
+ B-->D[*];
+ D-->E[a];
+ D-->F[b];
+ A-->G[+];
+ G-->H[d];
+ G-->I["/"];
+ I-->J[e];
+ I-->K[f];
+```
+::::
+
+:::: column
+
+1. Parcours **préfixe** (R, G D),
+2. Parcours **infixe** (G, R, D),
+3. Parcours **postfixe** (G, D, R).
+
+::::
+
+:::
+
+# Le parcours infixe (G, R, D)
+
+* Gauche, Racine, Droite:
+ 1. On descend dans l'arbre de gauche tant qu'il est pas vide,
+ 2. On visite la racine du sous arbre,
+ 3. On descend dans le sous-arbre de droite (s'il est pas vide),
+ 4. On recommence.
+
+. . .
+
+## Incompréhensible?
+
+* La récursivité c'est la vie.
+
+```
+parcours_infixe(arbre a)
+ si est_pas_vide(gauche(a))
+ parcours_infixe(gauche(a))
+ visiter(A)
+ si est_pas_vide(droite(A))
+ parcours_infixe(droite(A))
+```
+
+# Graphiquement (dessinons)
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ A[*]-->B[-];
+ B-->C[c];
+ B-->D[*];
+ D-->E[a];
+ D-->F[b];
+ A-->G[+];
+ G-->H[d];
+ G-->I["/"];
+ I-->J[e];
+ I-->K[f];
+```
+::::
+
+:::: column
+
+```
+parcours_infixe(arbre a)
+ si est_pas_vide(gauche(a))
+ parcours_infixe(gauche(a))
+ visiter(A)
+ si est_pas_vide(droite(A))
+ parcours_infixe(droite(A))
+```
+
+::::
+
+:::
+
+
+# Graphiquement (`mermaid` c'est super)
+
+::: columns
+
+:::: column
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ A[*]-->B[-];
+ A[*]-.->|1|B[-];
+ B-->C[c];
+ B-.->|2|C[c];
+ C-.->|3|B;
+ B-->D[*];
+ B-.->|4|D;
+ D-->E[a];
+ D-.->|5|E;
+ E-.->|6|D;
+ D-->F[b];
+ D-.->|7|F;
+ F-.->|8|A;
+ A-->G[+];
+ A-.->|9|G;
+ G-->H[d];
+ G-.->|10|H;
+ H-.->|11|G;
+ G-->I["/"];
+ G-.->|12|I;
+ I-->J[e];
+ I-.->|13|J;
+ J-.->|14|I;
+ I-->K[f];
+ I-.->|15|K;
+```
+::::
+
+:::: column
+
+```
+parcours_infixe(arbre a)
+ si est_pas_vide(gauche(a))
+ parcours_infixe(gauche(a))
+ visiter(A)
+ si est_pas_vide(droite(A))
+ parcours_infixe(droite(A))
+```
+
+## Remarque
+
+Le nœud est visité à la **remontée**.
+
+## Résultat
+
+```
+c - a * b * d + e / f
+```
+
+::::
+
+:::
+
+# Et en C?
+
+## Live code
+
+\footnotesize
+
+. . .
+
+```C
+typedef int data;
+typedef struct _node {
+ data info;
+ struct _node* left;
+ struct _node* right;
+} node;
+typedef node* tree_t;
+void tree_print(tree_t tree, int n) {
+ if (NULL != tree) {
+ tree_print(tree->left, n+1);
+ for (int i = 0; i < n; i++) {
+ printf(" ");
+ }
+ printf("%d\n", tree->info);
+ tree_print(tree->right, n+1);
+ }
+}
+```
+
+# Question
+
+## Avez-vous compris le fonctionnement?
+
+. . .
+
+## Vous en êtes sûr·e·s?
+
+. . .
+
+## OK, alors deux exercices:
+
+1. Écrire le pseudo-code pour le parcours R, G, D (matrix).
+2. Écrire le pseudo-code pour la parcours G, D, R (matrix),
+
+## Rappel
+
+```
+parcours_infixe(arbre a)
+ si est_pas_vide(gauche(a))
+ parcours_infixe(gauche(a))
+ visiter(A)
+ si est_pas_vide(droite(A))
+ parcours_infixe(droite(A))
+```
+
+# Correction
+
+\footnotesize
+
+* Les deux parcours sont des modifications **triviales**[^1] de l'algorithme
+ infixe.
+
+## Le parcours postfixe
+
+```python
+parcours_postfixe(arbre a)
+ si est_pas_vide(gauche(a))
+ parcours_postfixe(gauche(a))
+ si est_pas_vide(droite(a))
+ parcours_postfixe(droite(a))
+ visiter(a)
+```
+
+## Le parcours préfixe
+
+```python
+parcours_préfixe(arbre a)
+ visiter(a)
+ si est_pas_vide(gauche(a))
+ parcours_préfixe(gauche(a))
+ si est_pas_vide(droite(a))
+ parcours_préfixe(droite(a))
+```
+
+. . .
+
+**Attention:** L'implémentation de ces fonctions en C sont **à faire** en
+exercice (inspirez vous de ce qu'on a fait avant)!
+
+# Exercice: parcours
+
+## Comment imprimer l'arbre ci-dessous?
+
+```
+ f
+ /
+ e
+ +
+ d
+*
+ c
+ -
+ b
+ *
+ a
+```
+
+. . .
+
+## Bravo vous avez trouvé!
+
+* Il s'agissait du parcours D, R, G.
+
+# Implémentation
+
+## Vous avez 5 min pour implémenter cette fonction et la poster sur matrix!
+
+. . .
+
+```C
+void pretty_print(tree_t tree, int n) {
+ if (NULL != tree) {
+ pretty_print(tree->right, n+1);
+ for (int i = 0; i < n; ++i) {
+ printf(" ");
+ }
+ printf("%d\n", tree->info);
+ pretty_print(tree->left, n+1);
+ }
+}
+```
+
+# Exercice supplémentaire (sans corrigé)
+
+Écrire le code de la fonction
+
+```C
+int depth(tree_t t);
+```
+
+qui retourne la profondeur maximale d'un arbre.
+
+Indice: la profondeur à chaque niveau peut-être calculée à partir du niveau des
+sous-arbres de gauche et de droite.
+
+# La recherche dans un arbre binaire
+
+* Les arbres binaires peuvent retrouver une information très rapidement.
+* À quelle complexité? À quelle condition?
+
+. . .
+
+## Condition
+
+* Le contenu de l'arbre est **ordonné** (il y a une relation d'ordre (`<`, `>`
+ entre les éléments).
+
+## Complexité
+
+* La profondeur de l'arbre (ou le $\mathcal{O}(\log_2(N))$)
+
+. . .
+
+## Exemple: les arbres lexicographiques
+
+* Chaque nœud contient une information de type ordonné, la **clé**,
+* Par construction, pour chaque nœud $N$:
+ * Toutes clé du sous-arbre à gauche de $N$ sont inférieurs à la clé de $N$.
+ * Toutes clé du sous-arbre à droite de $N$ sont inférieurs à la clé de $N$.
+
+# Algorithme de recherche
+
+* Retourner le nœud si la clé est trouvée dans l'arbre.
+
+```python
+arbre recherche(clé, arbre)
+ tante_que est_non_vide(arbre)
+ si clé < clé(arbre)
+ arbre = gauche(arbre)
+ sinon si clé > clé(arbre)
+ arbre = droite(arbre)
+ sinon
+ retourne arbre
+ retourne NULL
+```
+
+# Algorithme de recherche, implémentation (live)
+
+\footnotesize
+
+. . .
+
+```C
+typedef int key_t;
+typedef struct _node {
+ key_t key;
+ struct _node* left;
+ struct _node* right;
+} node;
+typedef node* tree_t;
+tree_t search(key_t key, tree_t tree) {
+ tree_t current = tree;
+ while (NULL != current) {
+ if (current->key > X) {
+ current = current->gauche;
+ } else if (current->key < X){
+ current = current->droite;
+ } else {
+ return current;
+ }
+ }
+ return NULL;
+}
+```
+
+# Exercice (5-10min)
+
+Écrire le code de la fonction
+
+```C
+int tree_size(tree_t tree);
+```
+
+qui retourne le nombre total de nœuds d'un arbre et poster le résultat sur
+matrix.
+
+Indication: la taille, est 1 + le nombre de nœuds du sous-arbre de gauche
+additionné au nombre de nœuds dans le sous-arbre de droite.
+
+. . .
+
+```C
+int arbre_size(tree_t tree) {
+ if (NULL == tree) {
+ return 0;
+ } else {
+ return 1 + tree_size(tree->left)
+ + tree_size(tree->right);
+ }
+}
+```
+
+# L'insertion dans un arbre binaire
+
+* C'est bien joli de pouvoir faire des parcours, recherches, mais si on peut
+ pas construire l'arbre....
+
+## Pour un arbre lexicographique
+
+* Rechercher la position dans l'arbre où insérer.
+* Créer un nœud avec la clé et le rattacher à l'arbre.
+
+# Exemple d'insertions
+
+* Clés uniques pour simplifier.
+* Insertion de 5, 15, 10, 25, 2, -5, 12, 14, 11.
+* Rappel:
+ * Plus petit que la clé courante => gauche,
+ * Plus grand que la clé courante => droite.
+* Faisons le dessins ensemble
+
+```
+
+
+
+
+
+
+
+
+
+```
+
+## Exercice (3min, puis matrix)
+
+* Dessiner l'arbre en insérant 20, 30, 60, 40, 10, 15, 25, -5
+
+
+# Pseudo-code d'insertion (1/2)
+
+* Deux parties:
+ * Recherche le parent où se passe l'insertion.
+ * Ajout du fils dans l'arbre.
+
+## Recherche du parent
+
+```
+arbre position(arbre, clé)
+ si est_non_vide(arbre)
+ si clé < clé(arbre)
+ suivant = gauche(arbre)
+ sinon
+ suivant = droite(arbre)
+ tant que clé(arbre) != clé && est_non_vide(suivant)
+ arbre = suivant
+ si clé < clé(arbre)
+ suivant = gauche(arbre)
+ sinon
+ suivant = droite(arbre)
+
+ retourne arbre
+```
+
+# Pseudo-code d'insertion (2/2)
+
+* Deux parties:
+ * Recherche de la position.
+ * Ajout dans l'arbre.
+
+## Ajout du fils
+
+```
+ajout(arbre, clé)
+ si est_vide(arbre)
+ arbre = nœud(clé)
+ sinon
+ si clé < clé(arbre)
+ gauche(arbre) = nœud(clé)
+ sinon si clé > clé(arbre)
+ droite(arbre) = nœud(clé)
+ sinon
+ retourne
+```
+
+# Code d'insertion en C (1/2)
+
+## Recherche du parent (ensemble)
+
+. . .
+
+```C
+tree_t position(tree_t tree, key_t key) {
+ tree_t current = tree;
+ if (NULL != current) {
+ tree_t subtree = key > current->key ? current->right :
+ current->left;
+ while (key != current->key && NULL != subtree) {
+ current = subtree;
+ subtree = key > current->key ? current->right :
+ current->left;
+ }
+ }
+ return current;
+}
+```
+
+[^1]: Copyright cours de mathématiques pendant trop d'années.
diff --git a/slides_2022/cours_16.md b/slides_2022/cours_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..6e284d61eacb43efcff6d264b12bf025814ddcc6
--- /dev/null
+++ b/slides_2022/cours_16.md
@@ -0,0 +1,1277 @@
+---
+title: "Arbres et tri par tas"
+date: "2023-03-17"
+---
+
+# Un joli site
+
+## Visualisation d'algorithmes
+
+* <https://visualgo.net/>
+* Allons nous rafraîchir la mémoire sur l'insertion / recherche dans un arbre
+ binaire.
+
+
+
+# L'insertion (1/4)
+
+* Deux parties:
+ * Recherche le parent où se passe l'insertion.
+ * Ajout du fils dans l'arbre.
+
+## Recherche du parent (pseudo-code)
+
+```
+arbre position(arbre, clé)
+ si est_non_vide(arbre)
+ si clé < clé(arbre)
+ suivant = gauche(arbre)
+ sinon
+ suivant = droite(arbre)
+ tant que clé(arbre) != clé && est_non_vide(suivant)
+ arbre = suivant
+ si clé < clé(arbre)
+ suivant = gauche(arbre)
+ sinon
+ suivant = droite(arbre)
+
+ retourne arbre
+```
+
+# L'insertion (2/4)
+
+## Recherche du parent (code)
+
+. . .
+
+```C
+tree_t position(tree_t tree, key_t key) {
+ tree_t curr = tree;
+ if (NULL != curr) {
+ tree_t subtree =
+ key > curr->key ? curr->right : curr->left;
+ while (key != curr->key && NULL != subtree) {
+ curr = subtree;
+ subtree = key > curr->key ? curr->right :
+ curr->left;
+ }
+ }
+ return curr;
+}
+```
+
+# L'insertion (3/4)
+
+* Deux parties:
+ * Recherche de la position.
+ * Ajout dans l'arbre.
+
+## Ajout du fils (pseudo-code)
+
+```
+rien ajout(arbre, clé)
+ si est_vide(arbre)
+ arbre = nœud(clé)
+ sinon
+ arbre = position(arbre, clé)
+ si clé < clé(arbre)
+ gauche(arbre) = nœud(clé)
+ sinon si clé > clé(arbre)
+ droite(arbre) = nœud(clé)
+ sinon
+ retourne
+```
+
+
+
+# L'insertion (4/4)
+
+## Ajout du fils (code)
+
+\scriptsize
+
+* 2 cas: arbre vide ou pas.
+* on retourne un pointeur vers le nœud ajouté (ou `NULL`)
+
+. . .
+
+```C
+tree_t add_key(tree_t *tree, key_t key) {
+ node_t *new_node = calloc(1, sizeof(*new_node));
+ new_node->key = key;
+ if (NULL == *tree) {
+ *tree = new_node;
+ } else {
+ tree_t subtree = position(*tree, key);
+ if (key == subtree->key) {
+ return NULL;
+ } else {
+ if (key > subtree->key) {
+ subtree->right = new_node;
+ } else {
+ subtree->left = new_node;
+ }
+ }
+ }
+ return new_node;
+}
+```
+
+# La version PK (1/5)
+
+```C
+typedef struct _node {
+ int info;
+ struct _node *left, *right;
+} node;
+typedef node *tree;
+void parcours_infixe(tree arbre, int n){
+ if(arbre!=NULL){
+ parcours_infixe(arbre->left, n+1);
+ for(int i=0; i<n; i++){
+ printf(" ");
+ }
+ printf("%d\n", arbre->info);
+ parcours_infixe(arbre->right, n+1);
+ }
+}
+```
+
+# La version PK (2/5)
+
+```C
+tree recherche(int cle, tree arbre){
+ while(arbre != NULL){
+ if(arbre->info == cle) return arbre;
+ if(arbre->info > cle){
+ arbre = arbre->left;
+ }else if(arbre->info < cle){
+ arbre = arbre->right;
+ }
+ }
+ return NULL;
+}
+
+```
+
+# La version PK (3/5)
+
+\footnotesize
+
+```C
+node* parent_insertion(int donnee, tree arbre){
+ if(arbre != NULL){
+ node* suivant = NULL;
+ if(arbre->info > donnee){
+ suivant = arbre->left;
+ } else {
+ suivant = arbre->right;
+ }
+ while(suivant != NULL && arbre->info != donnee){
+ arbre = suivant;
+ if(arbre->info > donnee){
+ suivant = arbre->left;
+ } else {
+ suivant = arbre->right;
+ }
+ }
+ }
+ return arbre;
+}
+
+```
+
+# La version PK (4/5)
+
+\footnotesize
+
+```C
+node* nouveau_noeud(int donnee){
+ node* new_node = malloc(sizeof(node));
+ new_node->info = donnee;
+ new_node->left = NULL;
+ new_node->right = NULL;
+ return new_node;
+}
+tree insertion(int donnee, tree arbre){
+ if(arbre == NULL){
+ arbre = nouveau_noeud(donnee);
+ } else {
+ node* parent = parent_insertion(donnee, arbre);
+ if(donnee > parent->info){
+ parent->right = nouveau_noeud(donnee);
+ } else if(donnee < parent->info) {
+ parent->left = nouveau_noeud(donnee);
+ }
+ }
+ return arbre;
+}
+```
+
+# La version PK (5/5)
+
+```C
+int main(){
+ tree arbre = NULL;
+
+ arbre = insertion(2, arbre);
+ arbre = insertion(1, arbre);
+ arbre = insertion(8, arbre);
+ arbre = insertion(10, arbre);
+ arbre = insertion(5, arbre);
+
+ parcours_infixe(arbre, 0);
+}
+```
+
+# La suppression de clé
+
+
+::: columns
+
+:::: column
+
+## Cas simples:
+
+* le nœud est absent,
+* le nœud est une feuille
+* le nœuds a un seul fils.
+
+## Une feuille (le 19 p.ex.).
+
+```{.mermaid format=pdf width=150 loc=figs/}
+flowchart TB;
+ 10-->20;
+ 10-->5
+ 20-->21
+ 20-->19
+```
+
+::::
+
+:::: column
+
+## Un seul fils (le 20 p.ex.).
+
+```{.mermaid format=pdf width=400 loc=figs/}
+flowchart TB;
+ 10-->20;
+ 10-->5
+ 20-->25
+ 20-->18
+ 25-->24
+ 25-->30
+ 5-->4;
+ 5-->8;
+ style 18 fill:#fff,stroke:#fff,color:#fff
+```
+
+## Dans tous les cas
+
+* Chercher le nœud à supprimer: utiliser `position()`.
+
+::::
+
+:::
+
+# La suppression de clé
+
+
+::: columns
+
+:::: column
+
+## Cas compliqué
+
+* Le nœud à supprimer à (au moins) deux descendants (10).
+
+```{.mermaid format=pdf width=400 loc=figs/}
+flowchart TB;
+ 10-->20;
+ 10-->5
+ 20-->25
+ 20-->18
+ 25-->24
+ 25-->30
+ 5-->4;
+ 5-->8;
+```
+
+::::
+
+:::: column
+
+* Si on enlève 10 il se passe quoi?
+
+. . .
+
+* On peut pas juste enlever `10` et recoller...
+* Proposez une solution bon sang!
+
+. . .
+
+## Solution
+
+* Échange de la valeur à droite dans le sous-arbre de gauche ou
+ ...
+* de la valeur de gauche dans le sous-arbre de droite!
+* Puis, on retire le nœud.
+
+::::
+
+:::
+
+# Le pseudo-code de la suppression
+
+## Pour une feuille ou absent (ensemble)
+
+```
+arbre suppression(arbre, clé)
+ sous_arbre = position(arbre, clé)
+ si est_vide(sous_arbre) ou clé(sous_arbre) != clé
+ retourne vide
+ sinon
+ si est_feuille(sous_arbre) et clé(sous_arbre) == clé
+ nouvelle_feuille = parent(arbre, sous_arbre)
+ si est_vide(nouvelle_feuille)
+ arbre = vide
+ sinon
+ si gauche(nouvelle_feuille) == sous_arbre
+ gauche(nouvelle_feuille) = vide
+ sinon
+ droite(nouvelle_feuille) = vide
+ retourne sous_arbre
+```
+
+# Il nous manque le code pour le `parent`
+## Pseudo-code pour trouver le parent (5min -> matrix)
+
+. . .
+
+```
+arbre parent(arbre, sous_arbre)
+ si est_non_vide(arbre)
+ actuel = arbre
+ parent = actuel
+ clé = clé(sous_arbre)
+ faire
+ si (clé != clé(actuel))
+ parent = actuel
+ si clé < clé(actuel)
+ actuel = gauche(actuel)
+ sinon
+ actuel = droite(actuel)
+ sinon
+ retour parent
+ tant_que (actuel != sous_arbre)
+ retourne vide
+```
+
+# Le pseudo-code de la suppression
+
+\footnotesize
+
+## Pour un seul enfant (5min -> matrix)
+
+. . .
+
+```
+arbre suppression(arbre, clé)
+ sous_arbre = position(arbre, clé)
+ si est_vide(gauche(sous_arbre)) ou est_vide(droite(sous_arbre))
+ parent = parent(arbre, sous_arbre)
+ si est_vide(gauche(sous_arbre))
+ si droite(parent) == sous_arbre
+ droite(parent) = droite(sous_arbre)
+ sinon
+ gauche(parent) = droite(sous_arbre)
+ sinon
+ si droite(parent) == sous_arbreou est_
+ droite(parent) = gauche(sous_arbre)
+ sinon
+ gauche(parent) = gauche(sous_arbre)
+ retourne sous_arbre
+```
+
+
+# Le pseudo-code de la suppression
+
+\footnotesize
+
+## Pour au moins deux enfants (ensemble)
+
+```
+arbre suppression(arbre, clé)
+ sous_arbre = position(arbre, clé) # on revérifie pas que c'est bien la clé
+ si est_non_vide(gauche(sous_arbre)) et est_non_vide(droite(sous_arbre))
+ max_gauche = position(gauche(sous_arbre), clé)
+ échange(clé(max_gauche), clé(sous_arbre))
+ suppression(gauche(sous_arbre), clé)
+```
+
+# Exercices (poster sur matrix)
+
+1. Écrire le pseudo-code de l'insertion purement en récursif.
+
+. . .
+
+```
+arbre insertion(arbre, clé)
+ si est_vide(arbre)
+ retourne nœud(clé)
+
+ si (clé < arbre->clé)
+ gauche(arbre) = insert(gauche(arbre), clé)
+ sinon
+ droite(arbre) = insert(droite(arbre), clé)
+ retourne arbre
+```
+
+# Exercices (poster sur matrix)
+
+2. Écrire le pseudo-code de la recherche purement en récursif.
+
+. . .
+
+```
+bool recherche(arbre, clé)
+ si est_vide(arbre)
+ retourne faux // pas trouvée
+ si clé(arbre) == clé
+ retourne vrai // trouvée
+ si clé < clé(arbre)
+ retourne recherche(gauche(arbre), clé)
+ sinon
+ retourne recherche(droite(arbre), clé)
+```
+
+# Exercices (Ã la maison)
+
+3. Écrire une fonction qui insère des mots dans un arbre et ensuite affiche
+ l'arbre.
+
+# Trier un tableau à l'aide d'un arbre binaire
+
+* Tableau représenté comme un arbre binaire.
+* Aide à comprendre "comment" trier, mais on ne construit jamais l'arbre.
+* Complexité $O(N\log_2 N)$ en moyenne et grande stabilité (pas de cas
+ dégénérés).
+
+# Lien entre arbre et tableau
+
+* La racine de l'arbre set le premier élément du tableau.
+* Les deux fils d'un nœud d'indice $i$, ont pour indices $2i+1$ et $2i+2$:
+ * Les fils du nœud $i=0$, sont à $2\cdot 0+1=1$ et $2\cdot 0+2=2$.
+ * Les fils du nœud $i=1$, sont à $2\cdot 1+1=3$ et $2\cdot 1+2=4$.
+ * Les fils du nœud $i=2$, sont à $2\cdot 2+2=5$ et $2\cdot 1+2=6$.
+ * Les fils du nœud $i=3$, sont à $2\cdot 3+1=7$ et $2\cdot 3+2=8$.
+* Un élément d'indice $i$ a pour parent l'élément $(i-1)/2$ (division entière):
+ * Le parent du nœud $i=8$ est $(8-1)/2=3$.
+ * Le parent du nœud $i=7$ est $(7-1)/2=3$.
+
+# Visuellement
+
+::: columns
+
+:::: column
+
+* Où vont les indices correspondant du tableau?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0(( ))-->id1(( ));
+ id0-->id2(( ));
+ id1-->id3(( ));
+ id1-->id4(( ));
+ id2-->id5(( ));
+ id2-->id6(( ));
+ id3-->id7(( ));
+ id3-->id8(( ));
+ id4-->id9(( ));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+::::
+
+:::: column
+
+* Les flèche de gauche à droite, parent -> enfants.
+* Les flèche de droite à gauche, enfants -> parent.
+
+
+
+::::
+
+:::
+
+**Propriétés:**
+
+1. les feuilles sont toutes sur l'avant dernier ou dernier niveau.
+2. les feuilles de profondeur maximale sont "tassée" à gauche.
+
+# Le tas (ou heap)
+
+## Définition
+
+* Un arbre est un tas, si la valeur de chacun de ses descendants est inférieure
+ ou égale à sa propre valeur.
+
+## Exemples (ou pas)
+
+```
+16 8 14 6 2 10 12 4 5 # Tas
+16 14 8 6 2 10 12 4 5 # Non-tas, 10 > 8 et 12 > 8
+```
+
+## Exercices (ou pas)
+
+```
+19 18 12 12 17 1 13 4 5 # Tas ou pas tas?
+19 18 16 12 17 1 12 4 5 # Tas ou pas tas?
+```
+
+. . .
+
+```
+19 18 12 12 17 1 13 4 5 # Pas tas! 13 > 12
+19 18 16 12 17 1 12 4 5 # Tas!
+```
+
+# Exemple de tri par tas (1/N)
+
+```
+ | 1 | 16 | 5 | 12 | 4 | 2 | 8 | 10 | 6 | 7 |
+```
+
+::: columns
+
+:::: column
+
+* Quel est l'arbre que cela représente?
+
+. . .
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((5));
+ id1-->id3((12));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6((8));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((7));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* On commence à l'indice $N/2 = 5$: `4`.
+* `7 > 4` (enfant `>` parent).
+* intervertir `4` et `7`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((5));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((8));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+. . .
+
+```
+ * *
+ | 1 | 16 | 5 | 12 | 7 | 2 | 8 | 10 | 6 | 4 |
+```
+
+# Exemple de tri par tas (2/N)
+
+```
+ | 1 | 16 | 5 | 12 | 7 | 2 | 8 | 10 | 6 | 4 |
+```
+
+::: columns
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* On continue à l'indice $N/2-1 = 4$: `12`.
+* Déjà un tas, rien à faire.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((5));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((8));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* On continue à l'indice $N/2-2 = 3$: `5`.
+* `5 < 8`, échanger `8` et `5` (aka `max(2, 5, 8)`)
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((8));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+. . .
+
+```
+ | 1 | 16 | 8 | 12 | 7 | 2 | 5 | 10 | 6 | 4 |
+```
+
+# Exemple de tri par tas (3/N)
+
+```
+ | 1 | 16 | 5 | 12 | 7 | 2 | 8 | 10 | 6 | 4 |
+```
+
+::: columns
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Indice $N/2-1 = 4$: `12`.
+* Déjà un tas, rien à faire.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((5));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((8));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Indice $N/2-2 = 3$: `5`.
+* `5 < 8`, `5 <=> max(2, 5, 8)`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((8));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ * *
+ | 1 | 16 | 8 | 12 | 7 | 2 | 5 | 10 | 6 | 4 |
+```
+
+# Exemple de tri par tas (4/N)
+
+```
+ | 1 | 16 | 8 | 12 | 7 | 2 | 5 | 10 | 6 | 4 |
+```
+
+::: columns
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Indice $N/2-3 = 1$: `16`.
+* Déjà un tas, rien à faire.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((8));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Indice $N/2-4 = 1$: `1`.
+* `1 < 16 && 1 < 8`, `1 <=> max(1, 16, 8)`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((16))-->id1((1));
+ id0-->id2((8));
+ id1-->id3((12));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ * *
+ | 16 | 1 | 8 | 12 | 7 | 2 | 5 | 10 | 6 | 4 |
+```
+
+
+# Exemple de tri par tas (5/N)
+
+```
+ | 16 | 1 | 8 | 12 | 7 | 2 | 5 | 10 | 6 | 4 |
+```
+
+::: columns
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Recommencer avec `1`.
+* `1 <=> max(1, 12, 7)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((16))-->id1((12));
+ id0-->id2((8));
+ id1-->id3((1));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Transformer l'arbre en tas.
+
+* Recommencer avec `1`.
+* `1 <=> max(1, 10, 6)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((16))-->id1((12));
+ id0-->id2((8));
+ id1-->id3((10));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((1));
+ id3-->id8((6));
+ id4-->id9((4));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ * * *
+ | 16 | 12 | 8 | 10 | 7 | 2 | 5 | 1 | 6 | 4 |
+```
+
+* L'arbre est un tas.
+
+# Exemple de tri par tas (6/N)
+
+```
+ | 16 | 12 | 8 | 10 | 7 | 2 | 5 | 1 | 6 | 4 |
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `16` (`max` de l'arbre) avec `4`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((4))-->id1((12));
+ id0-->id2((8));
+ id1-->id3((10));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((1));
+ id3-->id8((6));
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `4 <=> max(4, 12, 8)`.
+* `4 <=> max(4, 10, 7)`.
+* `4 <=> max(4, 1, 6)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((10));
+ id0-->id2((8));
+ id1-->id3((6));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((1));
+ id3-->id8((4));
+```
+
+::::
+
+:::
+
+```
+ | 12 | 10 | 8 | 6 | 7 | 2 | 5 | 1 | 4 || 16
+```
+
+
+# Exemple de tri par tas (7/N)
+
+```
+ | 12 | 10 | 8 | 6 | 7 | 2 | 5 | 1 | 4 || 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `12` (`max` de l'arbre) avec `4`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((4))-->id1((10));
+ id0-->id2((8));
+ id1-->id3((6));
+ id1-->id4((7));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((1));
+ id3-->id8(( ));
+ style id8 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `4 <=> max(4, 10, 8)`.
+* `4 <=> max(4, 6, 7)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((10))-->id1((7));
+ id0-->id2((8));
+ id1-->id3((6));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6((5));
+ id3-->id7((1));
+ id3-->id8(( ));
+ style id8 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ | 10 | 7 | 8 | 6 | 4 | 2 | 5 | 1 || 12 | 16
+```
+
+# Exemple de tri par tas (8/N)
+
+```
+ | 10 | 7 | 8 | 6 | 4 | 2 | 5 | 1 || 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `10` (`max` de l'arbre) avec `1`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((7));
+ id0-->id2((8));
+ id1-->id3((6));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6((5));
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `1 <=> max(1, 7, 8)`.
+* `5 <=> max(1, 2, 5)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((7));
+ id0-->id2((5));
+ id1-->id3((6));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6((1));
+```
+
+::::
+
+:::
+
+```
+ | 8 | 7 | 5 | 6 | 4 | 2 | 1 || 10 | 12 | 16
+```
+
+# Exemple de tri par tas (9/N)
+
+```
+ | 8 | 7 | 5 | 6 | 4 | 2 | 1 || 10 | 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `8` (`max` de l'arbre) avec `1`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((7));
+ id0-->id2((5));
+ id1-->id3((6));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6(( ));
+ style id6 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `1 <=> max(1, 7, 5)`.
+* `1 <=> max(1, 6, 4)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((7))-->id1((6));
+ id0-->id2((5));
+ id1-->id3((1));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6(( ));
+ style id6 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ | 7 | 6 | 5 | 1 | 4 | 2 || 8 | 10 | 12 | 16
+```
+
+# Exemple de tri par tas (10/N)
+
+```
+ | 7 | 6 | 5 | 1 | 4 | 2 || 8 | 10 | 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `7` (`max` de l'arbre) avec `2`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=150 loc=figs/}
+graph TD;
+ id0((2))-->id1((6));
+ id0-->id2((5));
+ id1-->id3((1));
+ id1-->id4((4));
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `2 <=> max(2, 6, 5)`.
+* `2 <=> max(2, 1, 4)`.
+
+```{.mermaid format=pdf width=150 loc=figs/}
+graph TD;
+ id0((6))-->id1((4));
+ id0-->id2((5));
+ id1-->id3((1));
+ id1-->id4((2));
+```
+
+::::
+
+:::
+
+```
+ | 6 | 4 | 5 | 1 | 2 || 8 | 10 | 12 | 16
+```
+
+# Exemple de tri par tas (11/N)
+
+```
+ | 6 | 4 | 5 | 1 | 2 || 8 | 10 | 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `6` (`max` de l'arbre) avec `2`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=150 loc=figs/}
+graph TD;
+ id0((2))-->id1((4));
+ id0-->id2((5));
+ id1-->id3((1));
+ id1-->id4(( ));
+ style id4 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `2 <=> max(2, 4, 5)`.
+* `2 <=> max(2, 1, 4)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((5))-->id1((4));
+ id0-->id2((2));
+ id1-->id3((1));
+ id1-->id4(( ));
+ style id4 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+```
+ | 5 | 4 | 2 | 1 || 6 | 8 | 10 | 12 | 16
+```
+
+# Exemple de tri par tas (12/N)
+
+```
+ | 5 | 4 | 2 | 1 || 6 | 8 | 10 | 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `5` (`max` de l'arbre) avec `1`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((4));
+ id0-->id2((2));
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas.
+
+* `1 <=> max(1, 4, 2)`.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((4))-->id1((1));
+ id0-->id2((2));
+```
+
+::::
+
+:::
+
+```
+ | 4 | 1 | 2 || 5 | 6 | 8 | 10 | 12 | 16
+```
+
+# Exemple de tri par tas (13/N)
+
+```
+ | 4 | 1 | 2 || 5 | 6 | 8 | 10 | 12 | 16
+```
+
+::: columns
+
+:::: column
+
+**But:** Trier les tas.
+
+* Échanger la racine, `4` (`max` de l'arbre) avec `2`.
+* Traiter la racine.
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((2))-->id1((1));
+ id0-->id2(( ));
+ style id2 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+**But:** Trier les tas. Plus rien à trier
+
+* On fait les 2 dernières étapes en vitesse.
+* Échange `2` avec `1`.
+* Il reste que `1`. GGWP!
+
+
+::::
+
+:::
+
+```
+ | 1 | 2 | 4 | 5 | 6 | 8 | 10 | 12 | 16
+```
+
+# Exercice (10min)
+
+* Trier par tas le tableau
+
+```
+ | 1 | 2 | 4 | 5 | 6 | 8 | 10 | 12 | 16
+```
+
+* Mettez autant de détails que possible.
+* Que constatez-vous?
+* Postez le résultat sur matrix.
+
+
diff --git a/slides_2022/cours_17.md b/slides_2022/cours_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..014b123603906078459f4cccf1270f923e0656be
--- /dev/null
+++ b/slides_2022/cours_17.md
@@ -0,0 +1,695 @@
+---
+title: "Tri par tas et arbres AVL"
+date: "2023-03-24"
+---
+
+# Questions sur les notions du dernier cours (1/2)
+
+* Comment représenter un tableau sous forme d'arbre binaire?
+
+. . .
+
+* Qu'est-ce qu'un tas?
+
+. . .
+
+```
+ | 1 | 16 | 5 | 12 | 4 | 2 | 8 | 10 | 6 | 7 |
+```
+
+* Quel est l'arbre que cela représente?
+
+
+# Questions sur les notions du dernier cours (2/2)
+
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((1))-->id1((16));
+ id0-->id2((5));
+ id1-->id3((12));
+ id1-->id4((4));
+ id2-->id5((2));
+ id2-->id6((8));
+ id3-->id7((10));
+ id3-->id8((6));
+ id4-->id9((7));
+ id4-->id10(( ));
+ style id10 fill:#fff,stroke:#fff
+```
+
+# Exercice
+
+* Trier par tas le tableau
+
+```
+ | 1 | 2 | 4 | 5 | 6 | 8 | 10 | 12 | 16
+```
+
+* Mettez autant de détails que possible.
+* Que constatez-vous?
+* Postez le résultat sur matrix.
+
+# L'algorithme du tri par tas (1/2)
+
+\footnotesize
+
+## Deux étapes
+
+1. Entassement: transformer l'arbre en tas.
+2. Échanger la racine avec le dernier élément et entasser la racine.
+
+## Pseudo-code d'entassement de l'arbre (15 min, matrix)
+
+. . .
+
+```python
+rien tri_par_tas(tab)
+ entassement(tab)
+ échanger(tab[0], tab[size(tab)-1])
+ pour i de size(tab)-1 Ã 2
+ tamisage(tab, 0)
+ échanger(tab[0], tab[i-1])
+
+rien entassement(tab)
+ pour i de size(tab)/2-1 Ã 0
+ tamisage(tab, i)
+
+rien tamisage(tab, i)
+ ind_max = ind_max(tab, i, gauche(i), droite(i))
+ si i != ind_max
+ échanger(tab[i], tab[ind_max])
+ tamisage(tab, ind_max)
+```
+
+# L'algorithme du tri par tas (2/2)
+
+* Fonctions utilitaires
+
+ ```python
+ entier ind_max(tab, i, g, d)
+ ind_max = i
+ si tab[ind_max] < tab[g] && size(tab) > g
+ ind_max = g
+ si tab[ind_mx] < tab[d] && size(tab) > d
+ ind_max = d
+ retourne ind_max
+
+ entier gauche(i)
+ retourne 2 * i + 1
+
+ entier droite(i)
+ retourne 2 * i + 2
+ ```
+
+
+<!-- # L'algorithme du tri par tas (3/4)
+
+\footnotesize
+
+## Implémenter en C l'algorithme du tri par tas (matrix, 20min)
+
+. . .
+
+```C
+void heapsort(int size, int tab[size]) {
+ heapify(size, tab);
+ swap(tab, tab + size - 1);
+ for (int s = size - 1; s > 1; s--) {
+ sift_up(s, tab, 0);
+ swap(tab, tab + s - 1);
+ }
+}
+void heapify(int size, int tab[size]) {
+ for (int i = size / 2 - 1; i >= 0; i--) {
+ sift_up(size, tab, i);
+ }
+}
+void sift_up(int size, int tab[size], int i) {
+ int ind_max = ind_max3(size, tab, i, left(i), right(i));
+ if (i != ind_max) {
+ swap(tab + i, tab + ind_max);
+ sift_up(size, tab, ind_max);
+ }
+}
+```
+
+# L'algorithme du tri par tas (4/4)
+
+\footnotesize
+
+## Fonctions utilitaires
+
+. . .
+
+```C
+int ind_max3(int size, int tab[size], int i, int l, int r) {
+ int ind_max = i;
+ if (l < size && tab[ind_max] < tab[l]) {
+ ind_max = l;
+ }
+ if (r < size && tab[ind_max] < tab[r]) {
+ ind_max = r;
+ }
+ return ind_max;
+}
+void swap(int *a, int *b) {
+ int tmp = *a;
+ *a = *b;
+ *b = tmp;
+}
+int left(int i) {
+ return 2 * i + 1;
+}
+int right(int i) {
+ return 2 * i + 2;
+}
+``` -->
+
+
+# Complexités
+
+::: columns
+
+:::: column
+
+## Complexité de la recherche
+
+1. En moyenne?
+2. Dans le pire des cas?
+
+. . .
+
+1. $O(\log_2(N))$
+2. $O(N)$
+
+::::
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((10))-->id1((9));
+ id0-->id8(( ));
+ id1-->id2((7));
+ id1-->id9(( ));
+ id2-->id3((6));
+ id2-->id10(( ));
+ id3-->id4((5));
+ id3-->id11(( ));
+ style id8 fill:#fff,stroke:#fff
+ style id9 fill:#fff,stroke:#fff
+ style id10 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Un meilleur arbre
+
+* Quelle propriété doit satisfaire un arbre pour être $O(\log_2(N))$?
+
+. . .
+
+* Si on a environ le même nombre de nœuds dans les sous-arbres.
+
+## Définition
+
+Un arbre binaire est parfaitement équilibré si, pour chaque
+nœud, la différence entre les nombres de nœuds des sous-
+arbres gauche et droit vaut au plus 1.
+
+* Si l'arbre est **parfaitement équilibré**, alors tout ira bien.
+* Quelle est la hauteur (profondeur) d'un arbre parfaitement équilibré?
+
+. . .
+
+* $O(\log_2(N))$.
+
+
+# Équilibre parfait ou pas?
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((W))-->id1((B));
+ id0-->id8((Y));
+ id1-->id2((A));
+ id1-->id9(( ));
+ id8-->id10((X));
+ id8-->id11(( ));
+ style id9 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+```
+É
+Q
+U
+I
+L
+I
+B
+R
+É
+```
+
+::::
+
+:::
+
+# Équilibre parfait ou pas?
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((16))-->id1((10));
+ id0-->id2((19));
+ id1-->id3((8));
+ id1-->id4((12));
+ id4-->id5((11));
+ id4-->id6(( ));
+ id2-->id7((17));
+ id2-->id8(( ));
+ id7-->id9(( ));
+ id7-->id10((18));
+ style id6 fill:#fff,stroke:#fff
+ style id8 fill:#fff,stroke:#fff
+ style id9 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+```
+P
+A
+S
+
+É
+Q
+U
+I
+L
+I
+B
+R
+É
+```
+
+::::
+
+:::
+
+# Arbres AVL
+
+\footnotesize
+
+* Quand est-ce qu'on équilibre un arbre?
+
+. . .
+
+* A l'insertion/suppression.
+* Maintenir un arbre en état d'équilibre parfait: cher (insertion, suppression).
+* On peut "relaxer" la condition d'équilibre: profondeur (hauteur) au lieu du
+ nombre de nœuds:
+ * La hauteur $\sim\log_2(N)$.
+
+## Définition
+
+Un arbre AVL (Adelson-Velskii et Landis) est un arbre binaire de recherche dans
+lequel, pour chaque nœud, la différence de hauteur entre le sous-arbre gauche et droit vaut au plus 1.
+
+* Relation entre nœuds et hauteur:
+$$
+\log_2(1+N)\leq 1+H\leq 1.44\cdot\log_2(2+N),\quad N=10^5,\ 17\leq H \leq 25.
+$$
+* Permet l'équilibrage en temps constant: insertion/suppression $O(\log_2(N))$.
+
+# Notation
+
+* `hg`: hauteur du sous-arbre gauche.
+* `hd`: hauteur du sous-arbre droit.
+* `facteur d'équilibre = fe = hd - hg`
+* Que vaut `fe` si l'arbre est AVL?
+
+. . .
+
+* `fe = {-1, 0, 1}`
+
+
+# AVL ou pas?
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((10));
+ id0-->id2((19));
+ id2-->id3((17));
+ id2-->id4((97));
+```
+
+::::
+
+:::: column
+
+. . .
+
+```
+A
+V
+L
+```
+
+::::
+
+:::
+
+# AVL ou pas?
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id2-->id3((17));
+ id2-->id4((97));
+ id4-->id5((37));
+ id4-->id6(( ));
+ style id6 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+```
+P
+A
+S
+
+A
+V
+L
+```
+
+::::
+
+:::
+
+# Insertion dans un arbre AVL (1/N)
+
+1. On part d'un arbre AVL.
+2. On insère un nouvel élément.
+
+::: columns
+
+:::: column
+
+* `hd ? hg`.
+* Insertion de `4`?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+```
+
+::::
+
+:::: column
+
+. . .
+
+* `hd = hg`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id1-->id4(( ));
+ id1-->id5((4));
+ style id4 fill:#fff,stroke:#fff
+```
+
+* `fe = -1`
+
+::::
+
+:::
+
+# Insertion dans un arbre AVL (2/N)
+
+1. On part d'un arbre AVL.
+2. On insère un nouvel élément.
+
+::: columns
+
+:::: column
+
+* `hd ? hg`.
+* Insertion de `4`?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id2-->id3((18));
+ id2-->id4(( ));
+ style id4 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+* `hd < hg`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id2-->id3((18));
+ id2-->id4(( ));
+ id1-->id5(( ));
+ id1-->id6((4));
+ style id4 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+```
+
+* `fe = 0`
+
+::::
+
+:::
+
+# Insertion dans un arbre AVL (3/N)
+
+\footnotesize
+
+1. On part d'un arbre AVL.
+2. On insère un nouvel élément.
+
+::: columns
+
+:::: column
+
+* `hd ? hg`.
+* Insertion de `4`?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id1-->id3(( ));
+ id1-->id4((6));
+ id2-->id5(( ));
+ id2-->id6(( ));
+ style id3 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+ style id6 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+* `hd < hg`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id1-->id3(( ));
+ id1-->id4((6));
+ id4-->id5((4));
+ id4-->id6(( ));
+ id2-->id7(( ));
+ id2-->id8(( ));
+ style id3 fill:#fff,stroke:#fff
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id8 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+**Déséquilibre!** Que vaut `fe`?
+
+. . .
+
+* `fe = 2`
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 1a
+
+* `u`, `v`, `w` même hauteur.
+* déséquilibre en `B` après insertion dans `u`
+
+
+
+::::
+
+:::: column
+
+## Cas 1a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v` `w` à la même hauteur.
+* `v` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 1b (symétrique 1a)
+
+
+
+::::
+
+:::: column
+
+## Cas 1b (symétrique 1a)
+
+* Comment rééquilibrer?
+
+. . .
+
+
+
+::::
+
+:::
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 2a
+
+* `h(v1)=h(v2), h(u)=h(w)`.
+* déséquilibre en `C` après insertion dans `v2`
+
+
+
+::::
+
+:::: column
+
+## Cas 2a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v2`, `w` à la même hauteur (`v1` pas tout à fait).
+* `v2` Ã droite de `B` (gauche de `C`)
+* `B` Ã droite de `A` (gauche de `C`)
+* `v1` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 2b (symétrique 2a)
+
+
+
+::::
+
+:::: column
+
+## Cas 2b (symétrique 2a)
+
+* Comment rééquilibrer?
+
+. . .
+
+
+
+::::
+
+:::
diff --git a/slides_2022/cours_18.md b/slides_2022/cours_18.md
new file mode 100644
index 0000000000000000000000000000000000000000..15f609fb6e379e873a28ac473e768c4e444bbd24
--- /dev/null
+++ b/slides_2022/cours_18.md
@@ -0,0 +1,947 @@
+---
+title: "Arbres AVL"
+date: "2023-03-31"
+---
+
+# Questions sur les notions du dernier cours
+
+* Qu'est-ce qu'un arbre AVL?
+
+. . .
+
+* Un arbre binaire qui a la propriété suivante:
+ * La différence de hauteur de chaque noeud est d'au plus 1.
+ * Tous les noeuds ont `fe = hd - hg = {-1, 0, 1}`.
+
+* Pourquoi utiliser un arbre AVL plutôt qu'un arbre binaire de recherche?
+
+. . .
+
+* Insertion/recherche/... toujours en $O(\log_2(N))$.
+
+# AVL ou pas?
+
+\footnotesize
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((21))-->id1((9));
+ id0-->id2((40));
+ id1-->id3((5));
+ id1-->id4((10));
+ id3-->id5((3));
+ id3-->id6((7))
+ id6-->id7((6))
+ id6-->id8(( ))
+ id2-->id9((33))
+ id2-->id10((61))
+ id9-->id11((22))
+ id9-->id12((39))
+ id10-->id13(( ))
+ id10-->id14((81))
+ style id8 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+. . .
+
+* Ajouter un noeud pour qu'il le soit plus.
+
+# Insertion dans un arbre AVL
+
+\footnotesize
+
+1. On part d'un arbre AVL.
+2. On insère un nouvel élément.
+
+::: columns
+
+:::: column
+
+* `hd ? hg`.
+* Insertion de `4`?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id1-->id3(( ));
+ id1-->id4((6));
+ id2-->id5(( ));
+ id2-->id6(( ));
+ style id3 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+ style id6 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+* `hd > hg`
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((12))-->id1((1));
+ id0-->id2((19));
+ id1-->id3(( ));
+ id1-->id4((6));
+ id4-->id5((4));
+ id4-->id6(( ));
+ id2-->id7(( ));
+ id2-->id8(( ));
+ style id3 fill:#fff,stroke:#fff
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id8 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+**Déséquilibre!** Que vaut `fe`?
+
+. . .
+
+* `fe = 2`
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 1a
+
+* `u`, `v`, `w` même hauteur.
+* déséquilibre en `B` après insertion dans `u`
+
+
+
+::::
+
+:::: column
+
+## Cas 1a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v` `w` à la même hauteur.
+* `v` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 1b (symétrique 1a)
+
+
+
+::::
+
+:::: column
+
+## Cas 1b (symétrique 1a)
+
+* Comment rééquilibrer?
+
+. . .
+
+
+
+::::
+
+:::
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 2a
+
+* `h(v1)=h(v2), h(u)=h(w)`.
+* déséquilibre en `C` après insertion dans `v2`
+
+
+
+::::
+
+:::: column
+
+## Cas 2a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v2`, `w` à la même hauteur (`v1` pas tout à fait).
+* `v2` Ã droite de `B` (gauche de `C`)
+* `B` Ã droite de `A` (gauche de `C`)
+* `v1` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+
+# Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 2b (symétrique 2a)
+
+
+
+::::
+
+:::: column
+
+## Cas 2b (symétrique 2a)
+
+* Comment rééquilibrer?
+
+. . .
+
+
+
+::::
+
+:::
+
+# Le facteur d'équilibre (balance factor)
+
+## Définition
+
+```
+fe(arbre) = hauteur(droite(arbre)) - hauteur(gauche(arbre))
+```
+
+## Valeurs possibles?
+
+. . .
+
+```
+fe = {-1, 0, 1} // arbre AVL
+fe = {-2, 2} // arbre déséquilibré
+```
+
+{width=40%}
+
+# Algorithme d'insertion
+
+* Insérer le noeud comme d'habitude.
+* Mettre à jour les facteurs d'équilibre jusqu'à la racine (ou au premier
+ noeud déséquilibré).
+* Rééquilibrer le noeud si nécessaire.
+
+## Cas possibles
+
+::: columns
+
+:::: column
+
+## Sous-arbre gauche (avant)
+
+```
+fe(P) = 1
+fe(P) = 0
+fe(P) = -1
+```
+
+::::
+
+:::: column
+
+## Sous-arbre gauche (après)
+
+. . .
+
+```
+=> fe(P) = 0
+=> fe(P) = -1
+=> fe(P) = -2 // Rééquilibrer P
+```
+
+::::
+
+:::
+
+# Algorithme d'insertion
+
+* Insérer le noeud comme d'habitude.
+* Mettre à jour les facteurs d'équilibre jusqu'à la racine (ou au premier
+ noeud déséquilibré).
+* Rééquilibrer le noeud si nécessaire.
+
+## Cas possibles
+
+::: columns
+
+:::: column
+
+## Sous-arbre droit (avant)
+
+```
+fe(P) = 1
+fe(P) = 0
+fe(P) = -1
+```
+
+::::
+
+:::: column
+
+## Sous-arbre droit (après)
+
+. . .
+
+```
+=> fe(P) = 0
+=> fe(P) = +1
+=> fe(P) = +2 // Rééquilibrer P
+```
+
+::::
+
+:::
+
+# Rééquilibrage
+
+## Lien avec les cas vus plus tôt
+
+```
+fe(P) = -2 && fe(gauche(P)) = -1 => cas 1a
+fe(P) = -2 && fe(gauche(P)) = +1 => cas 2a
+
+fe(P) = +2 && fe(droite(P)) = -1 => cas 2b
+fe(P) = +2 && fe(droite(P)) = +1 => cas 1b
+```
+
+## Dessiner les différents cas, sur le dessin ci-dessous
+
+
+
+# La rotation
+
+## La rotation gauche (5min, matrix)
+
+
+
+. . .
+
+\footnotesize
+```
+arbre rotation_gauche(arbre P)
+ si est_non_vide(P)
+ Q = droite(P)
+ droite(P) = gauche(Q)
+ gauche(Q) = P
+ retourne Q
+ retourne P
+```
+
+# La rotation en C (1/2)
+
+## La rotation gauche
+
+```
+arbre rotation_gauche(arbre P)
+ si est_non_vide(P)
+ Q = droite(P)
+ droite(P) = gauche(Q)
+ gauche(Q) = P
+ retourne Q
+ retourne P
+```
+
+## Écrire le code C correspondant (5min, matrix)
+
+1. Structure de données
+2. Fonction `tree_t rotation_left(tree_t tree)`
+
+. . .
+
+\footnotesize
+```C
+typedef struct _node {
+ int key;
+ struct _node *left, *right;
+ int bf; // balance factor
+} node;
+typedef node *tree_t;
+```
+
+# La rotation en C (2/2)
+
+\footnotesize
+
+```C
+tree_t rotation_left(tree_t tree) {
+ tree_t subtree = NULL;
+ if (NULL != tree) {
+ subtree = tree->right;
+ tree->right = subtree->left;
+ subtree->left = tree;
+ }
+ return subtree;
+}
+```
+
+. . .
+
+* Et la rotation à droite, pseudo-code (5min)?
+
+. . .
+
+```
+arbre rotation_droite(arbre P)
+ si est_non_vide(P)
+ Q = gauche(P)
+ gauche(P) = droite(Q)
+ droite(Q) = P
+ retourne Q
+ retourne P
+```
+
+<!-- ```C
+tree_t rotation_right(tree_t tree) {
+ tree_t subtree = NULL;
+ if (NULL != tree) {
+ subtree = tree->left;
+ tree->left = subtree->right;
+ subtree->right = tree;
+ }
+ return subtree;
+}
+``` -->
+
+# Exemple de rotation (1/2)
+
+## Insertion de 9?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((5))-->id1((1));
+ id0-->id2((6));
+ id2-->id3(( ));
+ id2-->id4((8));
+ style id3 fill:#fff,stroke:#fff
+```
+
+# Exemple de rotation (2/2)
+
+::: columns
+
+:::: column
+
+## Quelle rotation et sur quel noeud (5 ou 6)?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((5))-->id1((1));
+ id0-->id2((6));
+ id2-->id3(( ));
+ id2-->id4((8));
+ id4-->id5(( ));
+ id4-->id6((9));
+ style id3 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Sur le plus jeune évidemment!
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((5))-->id1((1));
+ id0-->id2((8));
+ id2-->id3((6));
+ id2-->id4((9));
+```
+
+::::
+
+:::
+
+* Cas `1a/b` *check*!
+
+
+# La rotation gauche-droite
+
+## LÃ c'est plus difficile (cas 2a/b)
+
+
+
+# Exercices
+
+## Faire l'implémentation de la double rotation (pas corrigé, 5min)
+
+# Exercices
+
+::: columns
+
+:::: column
+
+## Insérer 50, ex 10min (matrix)
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((89))-->id1((71));
+ id0-->id2((90));
+ id1-->id3((44));
+ id3-->id4((37));
+ id3-->id5((61));
+ id1-->id6((81))
+ id2-->id7(( ))
+ id2-->id8((100))
+ style id7 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Où se fait la rotation?
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((89))-->id1((71));
+ id0-->id2((90));
+ id1-->id3((44));
+ id3-->id4((37));
+ id3-->id5((61));
+ id1-->id6((81))
+ id2-->id7(( ))
+ id2-->id8((100))
+ id5-->id9((50))
+ id5-->id10(( ))
+ style id7 fill:#fff,stroke:#fff
+ style id10 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Exercices
+
+::: columns
+
+:::: column
+
+## Rotation gauche en 44
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((89))-->id1((71));
+ id0-->id2((90));
+ id1-->id3((61));
+ id1-->id10((81));
+ id3-->id4((44));
+ id3-->id5(( ));
+ id4-->id6((37))
+ id4-->id7((50))
+ id2-->id8(( ))
+ id2-->id9((100))
+ style id5 fill:#fff,stroke:#fff
+ style id8 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Rotation à droite en 71
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((89))-->id1((61));
+ id0-->id2((90));
+ id1-->id3((44));
+ id1-->id10((71));
+ id3-->id4((37));
+ id3-->id5((50));
+ id2-->id8(( ));
+ id2-->id9((100));
+ id10-->id11(( ))
+ id10-->id12((81))
+ style id8 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Exercice de la mort
+
+Soit l’arbre AVL suivant:
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((60))-->id1((40));
+ id0-->id2((120));
+ id1-->id3((20));
+ id1-->id4((50));
+ id3-->id5((10));
+ id3-->id6((30));
+ id2-->id7((100));
+ id2-->id8((140));
+ id7-->id9((80))
+ id7-->id10((110))
+ id9-->id11((70))
+ id9-->id12((90))
+ id8-->id13((130))
+ id8-->id14((160))
+ id14-->id15((150))
+ id14-->id16((170))
+```
+
+::::
+
+:::: column
+
+1. Montrer les positions des insertions de feuille qui conduiront à un arbre
+ désequilibré.
+2. Donner les facteurs d’equilgaucheibre.
+3. Dessiner et expliquer les modifications de l’arbre lors de l’insertion de la
+ valeur `65`. On mentionnera les modifications des facteurs
+ d’équilibre.
+
+::::
+
+:::
+
+# Encore un petit exercice
+
+* Insérer les nœuds suivants dans un arbre AVL
+
+```
+25 | 60 | 35 | 10 | 5 | 20 | 65 | 45 | 70 | 40
+ | 50 | 55 | 30 | 15
+```
+
+## Un à un et le/la premier/ère qui poste la bonne réponse sur matrix a un point
+
+# Suppression dans un arbre AVL
+
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: supprimer 10
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((10));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11((12))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: supprimer 10
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL
+
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: supprimer 8
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation de 12
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((9))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9(( ))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id9 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: rotation de 12
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((9))-->id1((4));
+ id0-->id2((14));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((12))
+ id2-->id10((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+1. On supprime comme d'habitude.
+2. On rééquilibre si besoin à l'endroit de la suppression.
+
+* Facile non?
+
+. . .
+
+* Plus dur....
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL 2.0
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: suppression de 30
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((30));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4((40));
+ id3-->id5(( ));
+ id3-->id6((20))
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id5 fill:#fff,stroke:#fff
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation GD autour de 40
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((40));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4(( ));
+ id3-->id5(( ));
+ id3-->id6((20))
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id4 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL 2.0
+
+::: columns
+
+:::: column
+
+## Argl! 50 est déséquilibré!
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((20));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4((40));
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation DG autour de 50
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((80))-->id1((50));
+ id0-->id2((100));
+ id1-->id3((20));
+ id1-->id4((70));
+ id3-->id5((10));
+ id3-->id6((40));
+ id4-->id9((60))
+ id4-->id10(( ))
+ id2-->id7((90))
+ id2-->id8((200))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id10 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Résumé de la suppression
+
+1. On supprime comme pour un arbre binaire de recherche.
+2. Si un nœud est déséquilibré, on le rééquilibre.
+ * Cette opération peut déséquilibrer un autre nœud.
+3. On continue à rééquilibrer tant qu'il y a des nœuds à équilibrer.
diff --git a/slides_2022/cours_19.md b/slides_2022/cours_19.md
new file mode 100644
index 0000000000000000000000000000000000000000..4394b42c615b1a53fc3240807123112286508aeb
--- /dev/null
+++ b/slides_2022/cours_19.md
@@ -0,0 +1,488 @@
+---
+title: "Arbres quaternaires"
+date: "2023-04-28"
+---
+
+
+# Les arbres quaternaires
+
+## Définition
+
+Arbre dont chaque nœud a 4 enfants ou aucun.
+
+
+
+# Les arbres quaternaires
+
+## Cas d'utilisation
+
+Typiquement utilisés pour représenter des données bidimensionnelles.
+
+Son équivalent tri-dimensionnel est l'octree (chaque nœud a 8 enfants ou aucun).
+
+## Cas d'utilisation: images
+
+* Stockage: compression.
+* Transformations: symétries, rotations, etc.
+
+## Cas d'utilisation: simulation
+
+* Indexation spatiale.
+* Détection de collisions.
+* Simulation de galaxies, Barnes-Hut.
+
+# Exemple de compression
+
+::: columns
+
+:::: {.column width=30%}
+
+## Comment représenter l'image
+
+
+
+::::
+
+:::: {.column width=70%}
+
+## Sous la forme d'un arbre quaternaire?
+
+. . .
+
+
+
+**Économie?**
+
+. . .
+
+Image 64 pixels, arbre 25 nœuds.
+
+::::
+
+:::
+
+
+# Structure de données
+
+::: columns
+
+:::: {.column width=50%}
+
+## Pseudo-code?
+
+. . .
+
+```python
+struct node
+ info
+ node sup_gauche, sup_droit,
+ inf_gauche, inf_droit
+```
+
+
+
+::::
+
+:::: {.column width=50%}
+
+## En C?
+
+. . .
+
+```C
+struct _node {
+ int info;
+ struct _node *sup_left;
+ struct _node *sup_right;
+ struct _node *inf_left;
+ struct _node *inf_right;
+};
+```
+
+* Pourquoi le `*` est important?
+
+. . .
+
+* Type récursif => taille inconnue à la compilation.
+
+::::
+
+:::
+
+# Une fonctionnalité simple
+
+\footnotesize
+
+## La fonction `est_feuille(noeud)`
+
+* Problème avec cette implémentation?
+
+```python
+bool est_feuille(noeud)
+ retourne
+ est_vide(sup_gauche(noeud)) &&
+ est_vide(sup_droit(noeud)) &&
+ est_vide(inf_gauche(noeud)) &&
+ est_vide(inf_droit(noeud))
+```
+
+. . .
+
+* Inutile d'avoir 4 conditions (soit 4 enfants soit aucun!)
+* Facile d'en oublier un!
+* Comment changer la structure pour que ça soit moins terrible?
+
+. . .
+
+```python
+struct node
+ info
+ node enfant[4]
+```
+
+# Structure de données
+
+## En C?
+
+. . .
+
+```C
+typedef struct _node {
+ int info;
+ struct _node *child[4];
+} node;
+```
+
+## Fonction `is_leaf(node *tree)`?
+
+. . .
+
+```C
+bool is_leaf(node *tree) {
+ return (NULL == tree->child[0]); // only first matters
+}
+```
+
+# Problème à résoudre
+
+* Construire un arbre quaternaire à partir d'une image:
+ * Créer l'arbre (allouer la mémoire pour tous les nœuds),
+ * Le remplir avec les valeurs des pixels.
+* Compression de l'image:
+ * Si les pixels sont les mêmes dans le quadrant on supprime le sous-arbre (sans perte)
+ * Si les pixels dévient pas trop on supprime le quadrant (avec perte)
+
+# Création de l'arbre
+
+## Comment créer un arbre de profondeur `prof` (3min)?
+
+. . .
+
+```python
+arbre creer_arbre(prof)
+ n = nouveau_noeud() # alloue la mémoire
+ si prof > 0
+ pour i = 0 Ã 3
+ n.enfant[i] = creer_arbre(prof-1)
+ retourne n
+```
+
+## En `C` (3 min, matrix)?
+
+. . .
+
+```C
+node *qt_create(int depth) {
+ node *n = calloc(1, sizeof(node));
+ if (depth > 0) {
+ for (int i = 0; i < 4; ++i) {
+ n->child[i] = qt_create(depth-1);
+ }
+ }
+ return n;
+}
+```
+
+# Le nombre de nœuds?
+
+## Comment implémenter la fonction (pseudo-code, 5min, matrix)?
+
+. . .
+
+```C
+entier nombre_nœuds(arbre)
+ si est_feuille(arbre)
+ retourne 1
+ sinon
+ somme = 1
+ pour i de 0 Ã 3
+ somme += nombre_nœuds(arbre.enfant[i])
+ retourne somme
+```
+
+# Le nombre de nœuds?
+
+## Comment implémenter la fonction en C (3min, matrix)?
+
+. . .
+
+```C
+int size(node *qt) {
+ if (is_leaf(qt)) {
+ return 1;
+ } else {
+ int sum = 1;
+ for (int i = 0; i < 4; ++i) {
+ sum += size(qt->child[i]);
+ }
+ return sum;
+ }
+}
+```
+
+# La profondeur en C?
+
+## Implémentation (5min, matrix)
+
+. . .
+
+\footnotesize
+
+```C
+int max(int x, int y) {
+ return (x >= y ? x : y);
+}
+int max_depth(int depths[4]) {
+ int m = depths[0];
+ for (int i = 1; i < 4; ++i) {
+ m = max(m, depths[i]);
+ }
+ return m;
+}
+int depth(node *qt) {
+ int depths[] = {0, 0, 0, 0};
+ if (is_leaf(qt)) {
+ return 0;
+ } else {
+ for (int i = 0; i < 4; ++i) {
+ depths[i] = depth(qt->child[i]);
+ }
+ return 1 + max_depth(depths);
+ }
+}
+```
+
+# Fonctions utiles (1/4)
+
+## Comment remplir un arbre depuis une matrice?
+
+```
+ SG=0 | SD=1
+ 21 | 12 | 4 | 4
+ 9 | 7 | 4 | 4
+-----------------
+ 1 | 1 | 0 | 31
+ 1 | 1 | 3 | 27
+ IG=2 | ID=3
+```
+
+## Quel arbre cela représente?
+
+. . .
+
+
+
+# Fonctions utiles (2/4)
+
+* On veut transformer une ligne/colonne en feuille.
+* Comment?
+
+::: columns
+
+:::: {.column width=40%}
+
+## Soit `ligne=2`, `colonne=3`
+
+```
+ SG=0 | SD=1
+ 21 | 12 | 4 | 4
+ 9 | 7 | 4 | 4
+-----------------
+ 1 | 1 | 0 | 31
+ 1 | 1 | 3 | 27
+ IG=2 | ID=3
+```
+
+::::
+
+:::: {.column width=70%}
+
+## Trouver un algorithme
+
+
+
+* Quelle feuille pour 31 (`li=2`, `co=3`)?
+* Plus important: quel chemin?
+
+. . .
+
+* `co -> G/D`, `li -> S/I`,
+* `2 * (li / 2) + co / 2 -> 2 * 1 + 1 = 3`
+* `2 * ((li % 2) / 1) + (co % 2) / 1 -> 2 * 0 + 1 = 1`
+* Comment généraliser?
+
+::::
+
+:::
+
+# Fonctions utiles (3/4)
+
+::: columns
+
+:::: {.column width=40%}
+
+## Soit `ligne=2`, `colonne=3`
+
+```
+ SG=0 | SD=1
+ 21 | 12 | 4 | 4
+ 9 | 7 | 4 | 4
+-----------------
+ 1 | 1 | 0 | 31
+ 1 | 1 | 3 | 27
+ IG=2 | ID=3
+```
+
+::::
+
+:::: {.column width=70%}
+
+## Trouver un algorithme (prendre plusieurs exemples, 15min, matrix)
+
+
+
+* Comment généraliser?
+
+. . .
+
+```C
+noeud position(li, co, arbre)
+ d = profondeur(arbre);
+ tant_que (d >= 1)
+ index = 2 * ((li % 2^d) / 2^(d-1)) +
+ (col % 2^d) / 2^(d-1)
+ arbre = arbre.enfant[index]
+ d -= 1
+ retourne arbre
+```
+
+
+::::
+
+:::
+
+# Fonctions utiles (4/4)
+
+\footnotesize
+
+## Pseudo-code
+
+```C
+noeud position(li, co, arbre)
+ d = profondeur(arbre);
+ tant_que (d >= 1)
+ index = 2 * ((li % 2^d) / 2^(d-1)) +
+ (col % 2^d) / 2^(d-1)
+ arbre = arbre.enfant[index]
+ d -= 1
+ retourne arbre
+```
+
+## Écrire le code `C` correspondant (5min, matrix)
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+```
+
+# Remplir l'arbre
+
+## A partir d'une matrice (pseudo-code, 5min, matrix)?
+
+. . .
+
+```C
+arbre matrice_Ã _arbre(matrice)
+ arbre = creer_arbre(profondeur)
+ pour li de 0 Ã nb_lignes(matrice)
+ pour co de 0 Ã nb_colonnes(matrice)
+ noeud = position(li, co, arbre)
+ noeud.info = matrice[co][li]
+ retourne arbre
+```
+
+. . .
+
+## A partir d'une matrice (C, 5min, matrix)?
+
+. . .
+
+\footnotesize
+
+```C
+node *matrix_to_qt(int nb_li, int nb_co, int matrix[nb_li][nb_co], int depth)
+{
+ node *qt = qt_create(depth);
+ for (int li = 0; li < nd_li; ++li) {
+ for (int co = 0; co < nd_co; ++co) {
+ node *current = position(li, co, qt);
+ current->info = matrix[li][co];
+ }
+ }
+ return qt;
+}
+```
+
+
+# Remplir la matrice
+
+## A partir de l'arbre (pseudo-code, 3min, matrix)?
+
+. . .
+
+```C
+matrice arbre_Ã _matrice(arbre)
+ matrice = creer_matrice(nb_lignes(arbre), nb_colonnes(arbre))
+ pour li de 0 Ã nb_lignes(matrice)
+ pour co de 0 Ã nb_colonnes(matrice)
+ noeud = position(li, co, arbre)
+ matrice[co][li] = noeud.info
+ retourne matrice
+```
+
+. . .
+
+## A partir de l'arbre (C, 3min, matrix)?
+
+. . .
+
+\footnotesize
+
+```C
+void qt_to_matrix(node *qt, int nb_li, int nb_co, int matrix[nb_li][nb_co])
+ for (int li = 0; li < nd_li; ++li) {
+ for (int co = 0; co < nd_co; ++co) {
+ node *current = position(li, co, qt);
+ matrix[li][co] = current->info;
+ }
+ }
+```
diff --git a/slides_2022/cours_2.md b/slides_2022/cours_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..660bf8184e15e88a3c1bca0a25bc600d7c0ce712
--- /dev/null
+++ b/slides_2022/cours_2.md
@@ -0,0 +1,317 @@
+---
+title: "Introduction aux algorithmes"
+date: "2022-09-28"
+---
+
+# Rappel (1/2)
+
+* Quel est l'algorithme pour le calcul des nombres 1ers?
+
+. . .
+
+```C
+bool est_premier(int nombre) {
+ int i = 2; // bonne pratique!
+ while (i < nombre) { // penser à bien indenter!
+ if (0 == nombre % i) { // ça rend le code LISIBLE!
+ return false;
+ }
+ i += 1;
+ }
+ return true;
+}
+```
+
+# Rappel (2/2)
+
+* Quelles structures de contrôles avons nous vues?
+
+. . .
+
+* La boucle `while`,
+* La boucle `do ... while`,
+* La boucle `for`,
+* La condition `if ... else if ... else`,
+
+# Exercice: la factorielle
+
+Écrire un programme qui calcule la factorielle d'un nombre
+$$
+N! = 1\cdot 2\cdot ... \cdot (N-1)\cdot N.
+$$
+
+## Par groupe de 3: écrire un pseudo-code
+
+. . .
+
+```python
+entier factorielle(entier n)
+ i = 1
+ fact = 1
+ tant que i <= n
+ fact *= i
+ i += 1
+ retourne fact
+```
+
+# Exercice: la factorielle
+
+\footnotesize
+
+Écrire un programme qui calcule la factorielle d'un nombre
+$$
+N! = 1\cdot 2\cdot ... \cdot (N-1)\cdot N.
+$$
+
+## Par groupe de 3: écrire un code en C
+
+Quand vous avez fini postez le code sur le salon matrix.
+
+. . .
+
+```C
+#include <stdio.h>
+int main() {
+ int nb = 10;
+ int fact = 1;
+ int iter = 2;
+ while (iter <= nb) {
+ fact *= iter;
+ iter++;
+ }
+ printf("La factorielle de %d est %d\n", nb, fact);
+}
+```
+
+. . .
+
+## Comment améliorer ce code? (notez ça sur une feuille)
+
+
+# Exercice: la factorielle en mieux
+
+## Individuellement
+
+1. Écrivez l'algorithme de calcul de deux façon différentes.
+2. Que se passe-t-il si $n>=15$?
+3. Pour celles et ceux qui ont fini pendant que les autres essaient: faites-le
+ en récursif (sans aide).
+
+. . .
+
+## Postez vos solutions sur **matrix**!
+
+# Exercice: test si un nombre est premier
+
+## Avec tout ce que vous avez appris jusqu'ici:
+
+* Écrivez le code testant si un nombre est premier.
+* Quels sont les ajouts possibles par rapport au code de la semaine passée?
+* Rencontrez-vous des problèmes éventuels de compilation?
+* Voyez-vous une façon de générer des nombres premiers avec votre programme?
+
+. . .
+
+## Implémentez-la et postez votre code sur le salon matrix (10 min).
+
+# Corrigé: enfin pas vraiment mais juste un possible
+
+\footnotesize
+
+```C
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <stdbool.h>
+int main() {
+ int nb = 1;
+ printf("Entrez un nombre: ");
+ scanf("%d", &nb);
+ bool premier = true;
+ for (int div = 2; div <= sqrt(nb); div++) {
+ if (nb % div == 0) {
+ premier = false;
+ break;
+ }
+ }
+ if (premier) {
+ printf("Le nombre %d est premier\n", nb);
+ } else {
+ printf("Le nombre %d n'est pas premier\n", nb);
+ }
+ return 0;
+}
+```
+
+# Quelques algorithmes simples
+
+## Voyons quelques algorithmes supplémentaires
+
+- Plus petit commun multiple (PPCM) de deux nombres
+- Autre algorithme de calcul du PPCM de deux nombres
+- Plus grand commun diviseur (PGCD) de deux nombres
+
+# Le calcul du PPCM (1/5)
+
+## Définition
+
+Le plus petit commun multiple (PPCM), `p`, de deux entiers non nuls, `a` et `b`,
+est le plus petit entier strictement positif qui soit multiple de ces deux
+nombres.
+
+Exemples:
+
+```C
+PPCM(3, 4) = 12,
+PPCM(4, 6) = 12,
+PPCM(5, 15) = 15.
+```
+
+. . .
+
+## Mathématiquement
+
+Décomposition en nombres premiers:
+
+$$
+36 = 2^2\cdot 3^2,\quad 90=2\cdot 5\cdot 3^2,
+$$
+On garde tous les premiers à la puissance la plus élevée
+$$
+PPCM(36, 90)=2^2\cdot 3^2\cdot 5=180.
+$$
+
+# Le calcul du PPCM (2/5)
+
+## Exemple d'algorithme
+
+```C
+PPCM(36, 90):
+36 < 90 // 36 + 36
+72 < 90 // 72 + 36
+108 > 90 // 90 + 90
+108 < 180 // 108 + 36
+144 < 180 // 144 + 36
+180 = 180 // The End!
+```
+
+* 5 additions, 5 assignations, et 6 comparaisons.
+
+. . .
+
+## Transcrivez cet exemple en algorithme (groupe de 3), 5min
+
+. . .
+
+## et codez-le!
+
+
+# Le calcul du PPCM (3/5)
+
+## Tentative de correction
+
+```C
+int main() {
+ int m = 15, n = 12;
+ int mult_m = m, mult_n = n;
+ while (mult_m != mult_n) {
+ if (mult_m > mult_n) {
+ mult_n += n;
+ } else {
+ mult_m += m;
+ }
+ }
+ printf("Le ppcm de %d et %d est %d\n", n, m, mult_m);
+}
+```
+
+. . .
+
+* Combien d'additions / comparaisons au pire?
+
+# Le calcul du PPCM (4/5)
+
+## Réusinage: Comment décrire une fonction qui ferait ce calcul (arguments, sorties)?
+
+. . .
+
+En `C` on pourrait la décrire comme
+
+```C
+int ppcm(int a, int b); // La **signature** de cette fonction
+```
+
+. . .
+
+## Algorithme
+
+Par groupe de 3 (5-10min):
+
+* réfléchissez à un algorithme alternatif donnant le PPCM de deux nombres;
+* écrivez l'algorithme en pseudo-code.
+
+# Le calcul du PPCM (5/5)
+
+## Indication
+
+Si un nombre, `p`, est multiple de `a` et de `b` alors il peut s'écrire `p = a
+* i = b * j` ou encore `p / a = i` et `p / b = j`.
+
+. . .
+
+## Pseudo-code
+
+```C
+int ppcm(int a, int b) {
+ for (i in [1, b]) {
+ if a * i est divisible par b {
+ return a * i
+ }
+ }
+}
+```
+
+# Le code du PPCM de 2 nombres (1/2)
+
+## Implémentez le pseudo-code et postez le code sur matrix (5min).
+
+. . .
+
+## Un corrigé possible
+
+
+```C
+#include <stdio.h>
+#include <stdlib.h>
+int main() {
+ int n = 15, m = 12;
+ int i = 1;
+ while (n * i % m != 0) {
+ i++;
+ }
+ printf("Le ppcm de %d et %d est %d\n", n, m, n*i);
+}
+```
+
+# Le code du PPCM de 2 nombres (2/2)
+
+## Corrigé alternatif
+
+```C
+#include <stdio.h>
+#include <stdlib.h>
+int main() {
+ int res = n*m;
+ for (int i = 2; i <= m; i++) {
+ if (n * i % m == 0) {
+ res = n * i;
+ break;
+ }
+ }
+ printf("Le ppcm de %d et %d est %d\n", n, m, res);
+}
+```
+
+
+
+
diff --git a/slides_2022/cours_20.md b/slides_2022/cours_20.md
new file mode 100644
index 0000000000000000000000000000000000000000..5fbbc844e15c8fe6607c6442350e374f6d462e47
--- /dev/null
+++ b/slides_2022/cours_20.md
@@ -0,0 +1,795 @@
+---
+title: "Arbres quarternaires, compression et Barnes-Hut"
+date: "2023-05-05"
+---
+
+# Le cours précédent (1/2)
+
+## Questions
+
+* Qu'est-ce qu'un arbre quaternaire?
+
+. . .
+
+* Un arbre où chaque nœud a soit **4 enfants** soit **aucun**.
+* A quoi peut servir un arbre quaternaire?
+
+. . .
+
+* Compression
+
+# Le cours précédent (2/2)
+
+## Questions
+
+
+* Structure de données d'un arbre quaternaire?
+
+. . .
+
+```C
+typedef struct _node {
+ int info;
+ struct _node *child[4];
+} node;
+```
+
+. . .
+
+* Dessin d'un nœud d'arbre quaternaire (avec correspondance `node`)?
+
+. . .
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((" "))-->|"child[0]"| id1(("info"));
+ id0-->|"child[1]"| id2(("info"));
+ id0-->|"child[2]"| id3(("info"));
+ id0-->|"child[3]"| id4(("info"));
+```
+
+# Transformations avec un arbre quaternaire
+
+## A faire
+
+* Symétrie axiale (horizontale/verticale).
+* Rotation quart de cercle (gauche/droite).
+* Compression.
+
+# La symétrie verticale
+
+## Que donne la symétrie verticale de
+
+```
+ SG=0 | SD=1
+ 21 | 12 | 4 | 4
+ 9 | 7 | 4 | 4
+-----------------
+ 1 | 1 | 0 | 31
+ 1 | 1 | 3 | 27
+ IG=2 | ID=3
+```
+
+. . .
+
+```
+ SG=0 | SD=1
+ 4 | 4 | 12 | 21
+ 4 | 4 | 7 | 9
+------------------
+ 31 | 0 | 1 | 1
+ 27 | 3 | 1 | 1
+ IG=2 | ID=3
+```
+
+# La symétrie d'axe vertical
+
+## Comment faire sur une matrice (3min, matrix)?
+
+. . .
+
+\footnotesize
+
+```C
+matrice symétrie(matrice)
+ pour i de 0 Ã nb_colonnes(matrice) / 2
+ pour j de 0 Ã nb_lignes(matrice)
+ échanger(matrice[i][j], matrice[nb_colonnes(matrice)-1-i][j])
+ retourne matrice
+```
+
+# La symétrie d'axe vertical
+
+## Comment faire sur un arbre?
+
+* Faire un dessin de l'arbre avant/après (5min, matrix)
+
+ ```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 12 | 21
+ 9 | 7 | 4 | 4 4 | 4 | 7 | 9
+ ----------------- => ----------------
+ 1 | 1 | 0 | 31 31 | 0 | 1 | 1
+ 1 | 1 | 3 | 27 27 | 3 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+ ```
+
+* Écrire le pseudo-code (3min, matrix)
+
+. . .
+
+\footnotesize
+
+```C
+arbre symétrie(arbre)
+ si !est_feuille(arbre)
+ échanger(arbre.enfant[0], arbre.enfant[1])
+ échanger(arbre.enfant[2], arbre.enfant[3])
+ pour i de 0 Ã 3
+ symétrie(arbre.enfant[i])
+ retourne arbre
+```
+
+# La symétrie d'axe horizontal
+
+* Trivial de faire l'axe horizontal (exercice à la maison)
+
+# Rotation d'un quart de cercle
+
+## Comment faire sur un arbre?
+
+* Faire un dessin de l'arbre avant/après (5min, matrix)
+
+ ```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 31 | 27
+ 9 | 7 | 4 | 4 4 | 4 | 0 | 3
+ ----------------- => -----------------
+ 1 | 1 | 0 | 31 12 | 7 | 1 | 1
+ 1 | 1 | 3 | 27 21 | 9 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+
+ ```
+
+* Écrire le pseudo-code (3min, matrix)
+
+. . .
+
+```C
+rien rotation_gauche(arbre)
+ si !est_feuille(arbre)
+ échange_cyclique_gauche(arbre.enfant)
+ pour i de 0 Ã 3
+ rotation_gauche(arbre.enfant[i])
+```
+
+# Rotation d'un quart de cercle
+
+\footnotesize
+
+## Comment faire sur un arbre?
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 31 | 27
+ 9 | 7 | 4 | 4 4 | 4 | 0 | 3
+----------------- => -----------------
+ 1 | 1 | 0 | 31 12 | 7 | 1 | 1
+ 1 | 1 | 3 | 27 21 | 9 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* Écrire le vrai (5min, matrix)
+
+. . .
+
+```C
+void rotate(node *qt) {
+ if (!is_leaf(qt)) {
+ node *tmp = qt->child[2];
+ qt->child[2] = qt->child[0];
+ qt->child[0] = qt->child[1];
+ qt->child[1] = qt->child[3];
+ qt->child[3] = tmp;
+ for (int i=0;i < 4; i++) {
+ rotate(qt->child[i]);
+ }
+ }
+}
+```
+
+# Compression sans perte (1/5)
+
+## Idée générale
+
+* Regrouper les pixels par valeur
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 21 | 12 | 4
+ 9 | 7 | 4 | 4 9 | 7 |
+----------------- => -----------------
+ 1 | 1 | 0 | 31 1 | 0 | 31
+ 1 | 1 | 3 | 27 | 3 | 27
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* Comment faire?
+
+# Compression sans perte (2/5)
+
+## Que devient l'arbre suivant?
+
+
+
+. . .
+
+## Arbre compressé
+
+
+
+# Compression sans perte (3/5)
+
+* Si un nœud a tous ses enfants égaux:
+ * Donner la valeur au nœud,
+ * Supprimer les enfants.
+* Remonter jusqu'Ã la racine.
+
+## Écrire le pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien compression_sans_pertes(arbre)
+ si !est_feuille(arbre)
+ pour i de 0 Ã 3
+ compression_sans_pertes(arbre.enfant[i])
+ si derniere_branche(arbre)
+ valeur, toutes_égales = valeur_enfants(arbre)
+ si toutes_egales
+ arbre.info = valeur
+ detruire_enfants(arbre)
+```
+
+# Compression sans perte (4/5)
+
+\footnotesize
+
+## Écrire le code C (5min, matrix)
+
+. . .
+
+```C
+void lossless_compression(node *qt) {
+ if (!is_leaf(qt)) {
+ for (int i = 0; i < CHILDREN; i++) {
+ lossless_compression(qt->child[i]);
+ }
+ if (is_last_branch(qt)) {
+ int val = -1;
+ if (last_value(qt, &val)) {
+ qt->info = val;
+ for (int i = 0; i < 4; ++i) {
+ free(qt->child[i]);
+ qt->child[i] = NULL;
+ }
+ }
+ }
+ }
+}
+```
+
+# Compression sans perte (5/5)
+
+\footnotesize
+
+```C
+bool is_last_branch(node *qt) {
+ for (int i = 0; i < 4; ++i) {
+ if (!is_leaf(qt)) {
+ return false;
+ }
+ }
+ return true;
+}
+bool last_value(node *qt, int *val) {
+ int info = qt->child[0];
+ for (int i = 1; i < 4; ++i) {
+ if (info != qt->child[i]) {
+ return false;
+ }
+ }
+ *val = info;
+ return true;
+}
+```
+
+
+# Compression avec perte (1/5)
+
+## Idée générale
+
+* Regrouper les pixels par valeur sous certaines conditions
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 3 21 | 12 | 4
+ 9 | 7 | 4 | 4 9 | 7 |
+----------------- => ------------------
+ 1 | 1 | 0 | 31 1 | 0 | 31
+ 2 | 1 | 3 | 27 | 3 | 27
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* On enlève si l'écart à la moyenne est "petit"?
+
+# Compression avec perte (2/5)
+
+## Que devient l'arbre suivant si l'écart est petit?
+
+
+
+. . .
+
+## Arbre compressé
+
+
+
+# Compression avec perte (3/5)
+
+## Comment mesurer l'écart à la moyenne?
+
+. . .
+
+* Avec l'écart-type
+
+\begin{equation*}
+\mu = \frac{1}{4}\sum_{i=0}^{3} p[i],\quad \sigma = \sqrt{\frac{1}{4}\sum_{i=0}^3 (\mu-p[i])
+^2} = \sqrt{\frac{1}{4}\left(\sum_{i=0}^3p[i]^2\right)-\mu^2}
+\end{equation*}
+
+## Que devient l'algorithme?
+
+. . .
+
+* Si $\sigma<\theta$, $\theta$ est la **tolérance**:
+ * Remplacer la valeur du pixel par la moyenne des enfants.
+ * Remonter les valeurs dans l'arbre.
+
+## Quelle influence de la valeur de $\theta$ sur la compression?
+
+. . .
+
+* Plus $\theta$ est grand, plus l'image sera compressée.
+
+# Compression avec perte (4/5)
+
+## Que devient l'arbre avec $\theta=0.5$?
+
+
+
+. . .
+
+
+
+# Compression avec perte (5/5)
+
+## Modifications sur la structure de données?
+
+. . .
+
+* On stocke la moyenne, et la moyenne des carrés.
+
+```C
+struct noeud
+ flottant moyenne, moyenne_carre
+ node enfants[4]
+```
+
+* Comment on calcule `moyenne` et `moyenne_carre` sur chaque nœud (pseudo-code)?
+
+# Calcul de la moyenne
+
+## Pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien moyenne(arbre) {
+ si !est_feuille(arbre)
+ pour enfant dans arbre.enfants
+ moyenne(enfant)
+ pour enfant dans arbre.enfants
+ arbre.moyenne += enfant.moyenne
+ arbre.moyenne_carre += enfant.moyenne_carre
+ arbre.moyenne /= 4
+ arbre.moyenne_carre /= 4
+```
+
+# La compression avec pertes
+
+\footnotesize
+
+## Pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien compression_avec_pertes(arbre, theta)
+ si !est_feuille(arbre)
+ pour i de 0 Ã 3
+ compression_avec_pertes(arbre.enfant[i])
+ si derniere_branche(arbre)
+ si racine(arbre.moyenne_carre - arbre.moyenne^2) < theta
+ detruire_enfants(arbre)
+```
+
+## Le code en entier
+
+```C
+arbre = matrice_Ã _arbre(matrice)
+moyenne(arbre)
+compression_sans_pertes(arbre)
+```
+
+# La dynamique des corps célestes
+
+## Slides très fortement inspirés du cours de J. Latt, Unige
+
+## Simulation du problème à $N$-corps
+
+* Prédiction du mouvement d'un grand nombre de corps célestes.
+* Modélisation:
+ * On se limite aux étoiles;
+ * Chaque étoile est caractérisée par un point (coordonnées) et une masse;
+ * On simule en deux dimensions.
+ * Interactions uniquement par les lois de la gravitation Newtonienne (oui-oui c'est de la **physique**!).
+
+
+# Les équations du mouvement
+
+## Mouvement de la $i$-ème étoile
+
+* Algorithme de Verlet ($t_{n+1}=t_n+\delta t$)
+
+ $$
+ \vec x_i(t_{n+1})= 2\vec x_i(t_n)-\vec x_i(t_{n-1})+\vec a_i(t_n)\delta t^2.
+ $$
+
+## Force de gravitation
+
+* $\vec a_i(t_n)=\vec F_i/m_i$.
+* Sur l'étoile $i$, la force résultante est donnée par
+
+ $$
+ \vec F_i=\sum_{j=1,j\neq i}^N \vec F_{ij}.
+ $$
+ avec
+ $$
+ \vec F_{ij}=\frac{G m_i m_j(\vec x_j-\vec x_i)}{||\vec x_j-\vec x_i||^3}.
+ $$
+
+# Algorithme du problème à $n$-corps
+
+## Pseudo-code: structure de données
+
+```C
+struct étoile
+ flottant m
+ vec x, x_precedent, f
+```
+
+## Pseudo-code: itération temporelle
+
+```C
+rien iteration_temporelle(étoiles, dt)
+ pour étoile_une dans étoiles
+ étoile_une.f = 0
+ pour étoile_deux dans étoiles
+ si (étoile_un != étoile_deux)
+ étoile_une.f +=
+ force(étoile_une, étoile_deux)
+ pour étoile dans étoiles
+ étoile.x, étoile.x_precedent =
+ verlet(étoile.x, étoile.x_precedent,
+ étoile.f / étoile.m, dt)
+```
+
+# Algorithme du problème à $n$-corps
+
+## Complexité
+
+* Complexité de chacune des parties?
+
+. . .
+
+* $\mathcal{O}(N^2)$, $\mathcal{O}(N)$.
+
+## En temps CPU pour **une itération**
+
+\footnotesize
+
+* Si temps pour $N=1$ on calcule en $1\mu s$:
+
++--------+-------+-------+-----------+
+| N | N^2 | t [s] | t [réel] |
++--------+-------+-------+-----------+
+| 10 | 10^2 | 1e-4 | |
++--------+-------+-------+-----------+
+| 10^4 | 10^8 | 1e+2 | ~1min |
++--------+-------+-------+-----------+
+| 10^6 | 10^12 | 1e+6 | ~11j |
++--------+-------+-------+-----------+
+| 10^9 | 10^18 | 1e+12 | ~30k ans |
++--------+-------+-------+-----------+
+| 10^11 | 10^22 | 1e+16 | ~300M ans |
++--------+-------+-------+-----------+
+
+* Typiquement il y a des milliers-millions d'itérations.
+* Il y a $10^{11}$ étoiles dans la galaxie.
+* Houston we have a problem.
+
+# Question
+
+## Comment faire mieux, des idées?
+
+. . .
+
+* Si un groupe d'étoiles est suffisamment loin, on le modélise comme un corps unique situé en son centre de masse.
+* Exemple: Si on simule plusieurs galaxies, on considère chaque galaxie comme un corps unique!
+* Un arbre quaternaire est une structure parfaite pour regrouper les étoiles.
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+::::
+
+:::: {.column width=50%}
+
+## Problématique
+
+* On veut calculer la force sur $1$.
+
+::::
+
+:::
+
+. . .
+
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+
+::::
+
+:::: {.column width=50%}
+
+## Résultat
+
+* Calcul et somme des forces venant des $9$ autre corps.
+
+::::
+
+:::
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Réduction d'un groupe à un seul corps
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Idée
+
+* On accélère le calcul en traitant un groupe comme un seul corps.
+* Fonctionne uniquement si le groupe est assez loin.
+* Autrement l'approximation est trop grossière.
+
+::::
+
+:::
+
+# Solution: l'arbre quaternaire
+
+## Corps célestes - arbre
+
+
+
+* On omet les nœuds vides pour éviter la surcharge.
+* La numérotation est:
+ * 0: ID
+ * 1: SD
+ * 2: IG
+ * 3: SG
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 1
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant ID.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 2
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (1/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est prise par 2.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (2/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 2
+
+{width=100%}
+
+* On crée un nouveau nœud.
+* Deux corps dans le nœud ID.
+* On crée un nouveau nœud.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (3/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 3
+
+{width=100%}
+
+* 2 va dans ID.
+* 3 va dans SG.
+* C'est des feuilles vides, tout va bien.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Que fait-on avec les nœuds intérieurs?
+
+* On les utilise pour:
+ * stocker la masse totale;
+ * stocker le centre de masse.
+
+\begin{align}
+m&=m_2+m_3,\\
+\vec x &= \frac{m_2\vec x_2+m_3\vec x_3}{m}.
+\end{align}
+
+## Chaque feuille contient **une étoile**
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre
+
+{width=100%}
+
+::::
+
+:::
+
+# Résumé
+
+* Insertion du corps `c` dans le nœud `n` en partant de la racine.
+* Si le nœud `n`
+ * ne contient pas de corps, on y dépose `c`,
+ * est interne, on met à jour masse et centre de masse. `c` est inséré récursivement dans le bon quadrant.
+ * est externe, on subdivise `n`, on met à jour la masse et centre de masse, on insère récursivement les deux nœuds dans les quadrants appropriés.
+
+## Remarque
+
+* Il faut stocker les coordonnées des quadrants.
+* Un nœud a un comportement différent s'il est interne ou externe.
+
diff --git a/slides_2022/cours_21.md b/slides_2022/cours_21.md
new file mode 100644
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@@ -0,0 +1,731 @@
+---
+title: "Barnes-Hut et B-arbres"
+date: "2023-05-12"
+---
+
+# Le cours précédent
+
+## A quoi sert l'algorithme de Barnes-Hut?
+
+. . .
+
+* A accélérer la résolution du problème à $n$-corps avec la gravitation,
+* si on peut vivre avec une réduction de précision.
+
+## Sur quelle structure de données se base l'algorithme?
+
+* L'arbre quaternaire.
+
+. . .
+
+## Quelle est l'idée générale?
+
+. . .
+
+* Si un groupe d'étoiles est suffisamment loin, on le modélise comme un corps unique situé en son centre de masse.
+* Exemple: Si on simule plusieurs galaxies, on considère chaque galaxie comme un corps unique!
+* Un arbre quaternaire est une structure parfaite pour regrouper les étoiles.
+
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+::::
+
+:::: {.column width=50%}
+
+## Problématique
+
+* On veut calculer la force sur $1$.
+
+::::
+
+:::
+
+. . .
+
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+
+::::
+
+:::: {.column width=50%}
+
+## Résultat
+
+* Calcul et somme des forces venant des $9$ autre corps.
+
+::::
+
+:::
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Réduction d'un groupe à un seul corps
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Idée
+
+* On accélère le calcul en traitant un groupe comme un seul corps.
+* Fonctionne uniquement si le groupe est assez loin.
+* Autrement l'approximation est trop grossière.
+
+::::
+
+:::
+
+# Solution: l'arbre quaternaire
+
+## Corps célestes - arbre
+
+
+
+* On omet les nœuds vides pour éviter la surcharge.
+* La numérotation est:
+ * 0: ID
+ * 1: SD
+ * 2: IG
+ * 3: SG
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 1
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant ID.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 2
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (1/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est prise par 2.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (2/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 2
+
+{width=100%}
+
+* On crée un nouveau nœud.
+* Deux corps dans le nœud ID.
+* On crée un nouveau nœud.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (3/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 3
+
+{width=100%}
+
+* 2 va dans ID.
+* 3 va dans SG.
+* C'est des feuilles vides, tout va bien.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Que fait-on avec les nœuds intérieurs?
+
+* On les utilise pour:
+ * stocker la masse totale;
+ * stocker le centre de masse.
+
+\begin{align}
+m&=m_2+m_3,\\
+\vec x &= \frac{m_2\vec x_2+m_3\vec x_3}{m}.
+\end{align}
+
+## Chaque feuille contient **une étoile**
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre
+
+{width=100%}
+
+::::
+
+:::
+
+# Résumé
+
+* Insertion du corps `c` dans le nœud `n` en partant de la racine.
+* Si le nœud `n`
+ * ne contient pas de corps, on y dépose `c`,
+ * est interne, on met à jour masse et centre de masse. `c` est inséré récursivement dans le bon quadrant.
+ * est externe, on subdivise `n`, on met à jour la masse et centre de masse, on insère récursivement les deux nœuds dans les quadrants appropriés.
+
+## Remarque
+
+* Il faut stocker les coordonnées des quadrants.
+* Un nœud a un comportement différent s'il est interne ou externe.
+
+# Algorithme d'insertion
+
+## Structure de données
+
+```C
+struct node
+ etoile e // externe: pour stocker
+ etoile sup_etoile // interne: pour stocker m, x
+ quadrant q // coordonnées du quadrant
+ node enfants[4]
+```
+
+## Remarque:
+
+* On fait une simplification "moche": `sup_etoile` pourrait juste avoir une masse et une position.
+
+# Algorithme d'insertion
+
+\footnotesize
+
+## Algorithme d'insertion, pseudo-code (15min, matrix)
+
+. . .
+
+```C
+rien insertion_etoile(arbre, e)
+ si (!est_vide(arbre) && dans_le_quadrant(arbre.q, e.x)) {
+ si (est_feuille(arbre))
+ si (!contient_etoile(arbre))
+ arbre.e = e
+ sinon
+ // on crée enfants et arbre.sup_etoile est initialisée
+ subdivision_arbre(arbre, e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, arbre.e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, e)
+ destruction(arbre.e)
+ sinon
+ maj_masse_cdm(arbre.sup_etoile, e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, e)
+```
+
+# Utilisation de l'arbre
+
+* L'arbre est rempli: comment on calcule la force sur le corps 1?
+* Parcours de l'arbre:
+ * si la distance entre 1 et le centre de masse est suffisante, on utilise la masse totale et centre de masse pour calculer la force.
+ * sinon, on continue le parcours
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Le cadrant ID ne contient que `1`, rien à faire.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Le cadrant SG ne contient `5` corps.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* La distance entre `1` et le centre de masse de SG est `d`.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* La distance entre `1` et le centre de masse de SG est `d`.
+* Est-ce que `d` est assez grand?
+* On va comparer avec la distance `d` avec la taille du quadrant `s`.
+
+# Critère $\theta$
+
+* On compare $d=||\vec x_1-\vec x_{cm}||$ avec $s$ la taille du quadrant.
+* Le domain est assez éloigné si
+
+ $$
+ \frac{s}{d}<\theta,
+ $$
+* $\theta$ est la valeur de seuil.
+* Une valeur typique est $\theta=0.5$, donc la condition devient
+
+ $$
+ d>2s.
+ $$
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Ici $d<2s$, domaine rejeté.
+* ON descend dans l'arbre.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* `s` est plus petit, mais....
+* Cela ne suffit pas $d<2s$, domaine rejeté.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Les nœuds sont des feuilles, on calcule la force.
+* On ajoute la force qu'ils exercent sur `1`.
+
+# Algorithme pour le calcul de la force
+
+Pour calculer la force sur un corps `c`, on parcourt l'arbre en commençant par la racine:
+
+* Si le nœud `n` est une feuille et n'est pas `c`, on ajoute la force dûe à `n` sur `c`;
+* Sinon si $s/d<\theta$, on traite `n` comme une feuille et on ajoute la force dûe à `n` sur `c`;
+* Sinon on continue sur les enfants récursivement.
+
+
+## Continuons notre exemple précédent!
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Il y a deux corps dans le quadrant vert.
+* Quel est le critère pour remplacer les étoiles par leur centre de masse?
+
+. . .
+
+* Et oui! $d>2s$ on peut remplacer les étoiles par leur centre de masse!
+
+# Algorithme du calcul de force
+
+## Écrire le pseudo-code-code du calcul de la force
+
+\footnotesize
+
+```C
+rien maj_force_sur_etoile(arbre, e, theta)
+ si est_vide(arbre)
+ retourne
+
+ si est_feuille(arbre) && contient_etoile(arbre) && dans_le_quadrant(arbre.q, e.x)
+ maj_force(e, arbre.e)
+ sinon si noeud_assez_loin(arbre, e, theta)
+ maj_force(e, arbre.sup_etoile)
+ sinon
+ pour enfant dans enfants
+ maj_force_sur_etoile(enfant, e, theta)
+```
+
+# Les B-arbres
+
+## Problématique
+
+* Grands jeux de données (en 1970).
+* Stockage dans un arbre, mais l'arbre tiens pas en mémoire.
+* Regrouper les sous-arbres en **pages** qui tiennent en mémoire.
+
+## Exemple
+
+* 100 nœuds par page et l'arbre comporte $10^6$ nœuds:
+ * Recherche B-arbre: $\log_{100}(10^6)=3$;
+ * Recherche ABR: $\log_2(10^6)=20$.
+* Si on doit lire depuis le disque: $10\mathrm{ms}$ par recherche+lecture:
+ * $30\mathrm{ms}$ (lecture beaucoup plus rapide que recherche) vs $200\mathrm{ms}=0.2\mathrm{s}$.
+
+## Remarques
+
+* On sait pas ce que veut dire `B`: Bayer, Boeing, Balanced?
+* Variante plus récente B+-arbres.
+
+# Les B-arbres
+
+## Illustration, arbre divisé en pages de 3 nœuds
+
+
+
+. . .
+
+## Utilisation
+
+* Bases de données (souvent très grandes donc sur le disque);
+* Système de fichier.
+
+# Les B-arbres
+
+## Avantages
+
+* Arbres moins profonds;
+* Diminue les opération de rééquilibrage;
+* Complexité toujours en $\log(N)$;
+
+. . .
+
+## Définition: B-arbre d'ordre $n$
+
+* Chaque page d'un arbre contient au plus $2\cdot n$ *clés*;
+* Chaque page (excepté la racine) contient au moins $n$ clés;
+* Chaque page qui contient $m$ clés contient soit:
+ * $0$ descendants;
+ * $m+1$ descendants.
+* Toutes les pages terminales apparaissent au même niveau.
+
+# Les B-arbres
+
+## Est-ce un B-arbre?
+
+
+
+. . .
+
+## Oui!
+
+* Dans chaque nœud les clés sont **triées**.
+* Chaque page contient au plus $n$ nœuds: check;
+* Chaque nœud avec $m$ clés a $m+1$ descendants;
+* Toutes les feuilles apparaissent au même niveau.
+
+# Les B-arbres
+
+## Exemple de recherche: trouver `32`
+
+
+
+. . .
+
+* Si `n` plus petit que la 1e clé ou plus grand que la dernière descendre.
+* Sinon parcourir (par bissection ou séquentiellement) jusqu'à trouver ou descendre entre 2 éléments.
+
+# Les B-arbres
+
+## La recherche de la clé `C` algorithme
+
+0. En partant de la racine.
+1. Si on est dans une feuille:
+ * Si la `C` est dans une page, retourner la page;
+ * Sinon c'est perdu.
+2. Sinon:
+ * Tant que `C > page` passer à la page suivante
+ * Descendre
+
+# Les B-arbres
+
+## Disclaimer
+
+* Inspiration de <https://en.wikipedia.org/wiki/B-tree>
+
+## Exemples d'insertion: `1`
+
+
+
+. . .
+
+* L'arbre est vide, on insère juste dans la première page.
+
+# Les B-arbres
+
+## Exemples d'insertion: `2`
+
+
+
+. . .
+
+* La première page est pas pleine, on insère dans l'ordre (après 1).
+
+# Les B-arbres
+
+## Exemples d'insertion: `3`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `3`
+
+{width=50%}
+
+. . .
+
+* La page est pleine, on crée deux enfants.
+* On choisit, `2`, la médiane de `1, 2, 3` et il est inséré à la racine.
+* `1` descend à gauche, `3` descend à droite.
+
+# Les B-arbres
+
+## Exemples d'insertion: `4`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `4`
+
+{width=50%}
+
+. . .
+
+* On pourrait insérer à droite de `2`, mais... ça ferait 2 parents pour 2 enfants (mais `m` parents => `m+1` enfants ou `0`);
+* On descend à droite (`4 > 2`);
+* On insère à droite de `3`.
+
+# Les B-arbres
+
+## Exemples d'insertion: `5`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `5`
+
+
+
+. . .
+
+* On descend à droite (on peut pas insérer à la racine comme pour `4`);
+* On dépasse la capacité de l'enfant droite;
+* `4`, médiane de `3, 4, 5`, remonte à la racine;
+* On crée un nouveau nœud à droite de `4`;
+* La règle `m => m+1` est ok.
+
+# Les B-arbres
+
+## Exemples d'insertion: `6`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `6`
+
+
+
+. . .
+
+* `6 > 4` on descend à droite;
+* `6 > 5` et on a à la place à droite, on insère.
+
+# Les B-arbres
+
+## Exemples d'insertion: `7`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `7`
+
+{width=50%}
+
+. . .
+
+* `7 > 4` on descend à droite;
+* `7 > 6` mais on a dépassé la capacité;
+* `6` est la médiane de `5, 6, 7`, remonte à la racine;
+* `5` reste à gauche, `7` à droite, mais `6` fait dépasser la capacité de la racine;
+* `4` est la médiane de `2, 4, 6`, `4` remonte, `2` reste à gauche, `6` à droite.
+
+# Les B-arbres
+
+## L'algorithme d'insertion
+
+0. Rechercher la feuille (la page a aucun enfant) où insérer;
+1. Si la page n'est pas pleine insérer dans l'ordre croissant.
+2. Si la page est pleine, on sépare la page en son milieu :
+ 1. On trouve la médiane, `M`, de la page;
+ 2. On met les éléments `< M` dans la page de gauche de `M` et les `> M` dans la page de droite de `M`;
+ 3. `M` est insérée récursivement dans la page parent.
+
+# Les B-arbres
+
+## Exercice: insérer `22, 45, 50` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `5` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `32, 55, 60` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `41` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## Exercice (matrix, 15min)
+
+* Insérer 20, 40, 10, 30, 15, 35, 7, 26, 18, 22, 5, 42, 13, 46, 27, 8, 32, 38, 24, 45, 25, 2, 14, 28, 32, 41,
+* Dans un B-arbre d'ordre 2.
+
diff --git a/slides_2022/cours_22.md b/slides_2022/cours_22.md
new file mode 100644
index 0000000000000000000000000000000000000000..3940c6d559336882800283d028a1be66cc483cce
--- /dev/null
+++ b/slides_2022/cours_22.md
@@ -0,0 +1,351 @@
+---
+title: "B-arbres"
+date: "2023-05-19"
+---
+
+# Rappel: Les B-arbres
+
+## Pourquoi utiliser un B-arbre?
+
+. . .
+
+## À quoi ressemble un B-arbre?
+
+. . .
+
+## Qu'est-ce qu'un B-arbre d'ordre $n$
+
+* Chaque page d'un arbre contient au plus $2\cdot n$ *clés*;
+* Chaque page (excepté la racine) contient au moins $n$ clés;
+* Chaque page qui contient $m$ clés contient soit:
+ * $0$ descendants;
+ * $m+1$ descendants.
+* Toutes les pages terminales apparaissent au même niveau.
+
+
+# Rappel: Les B-arbres
+
+## Quelques propriétés
+
+* Dans chaque nœud les clés sont **triées**.
+* Chaque page contient au plus $n$ nœuds;
+* Chaque nœud avec $m$ clés a $m+1$ descendants;
+* Toutes les feuilles apparaissent au même niveau.
+
+# Les B-arbres
+
+\footnotesize
+
+## Structure de données
+
+* Chaque page a une contrainte de remplissage, par rapport à l'ordre de l'arbre;
+* Un nœud (page) est composé d'un tableau de clés/pointeurs vers les enfants;
+
+```
+P_0 | K_1 | P_1 | K_2 | .. | P_i | K_{i+1} | .. | P_{m-1} | K_m | P_m
+```
+
+* `P_0`, ..., `P_m` pointeurs vers enfants;
+* `K_1`, ..., `K_m` les clés.
+* Il y a `m+1` pointeurs mais `m` clés.
+* Comment faire pour gérer l'insertion?
+
+# Les B-arbres
+
+## Faire un dessin de la structure de données (3min matrix)?
+
+. . .
+
+
+
+1. On veut un tableau de `P_i, K_i => struct`;
+2. `K_0` va être en "trop";
+3. Pour simplifier l'insertion dans une page, on ajoute un élément de plus.
+
+# Les B-arbres
+
+## L'insertion cas nœud pas plein, insertion `4`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+{width=50%}
+
+# Les B-arbres
+
+## L'insertion cas nœud pas plein, insertion `N`
+
+* On décale les éléments plus grand que `N`;
+* On insère `N` dans la place "vide";
+* Si la page n'est pas pleine, on a terminé.
+
+# Les B-arbres
+
+## L'insertion cas nœud plein, insertion `2`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+{width=50%}
+
+# Les B-arbres
+
+## L'insertion cas nœud plein, promotion `3`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+
+
+# Les B-arbres
+
+## L'insertion cas nœud plein, insertion `N`
+
+* On décale les éléments plus grand que `N`;
+* On insère `N` dans la place "vide";
+* Si la page est pleine:
+ * On trouve la valeur médiane `M` de la page (quel indice?);
+ * On crée une nouvelle page de droite;
+ * On copie les valeur à droite de `M` dans la nouvelle page;
+ * On promeut `M` dans la page du dessus;
+ * On connecte le pointeur de gauche de `M` et de droite de `M` avec l'ancienne et la nouvelle page respectivement.
+
+# Les B-arbres
+
+## Pseudo-code structure de données (3min, matrix)?
+
+. . .
+
+```C
+struct page
+ entier ordre, nb
+ element tab[2*ordre + 2]
+```
+
+```C
+struct element
+ entier clé
+ page pg
+```
+
+# Les B-arbres
+
+\footnotesize
+
+## Les fonctions utilitaires (5min matrix)
+
+```C
+booléen est_feuille(page) // la page est elle une feuille?
+entier position(page, valeur) // à quelle indice on insère?
+booléen est_dans_page(page, valeur) // la valeur est dans la page
+```
+
+. . .
+
+```C
+booléen est_feuille(page)
+ retourne (page.tab[0].pg == vide)
+
+entier position(page, valeur)
+ i = 0
+ tant que i < page.nb && val >= page.tab[i+1].clef
+ i += 1
+ retourne i
+
+booléen est_dans_page(page, valeur)
+ i = position(page, valeur)
+ retourne (page.nb > 0 && page.tab[i].val == valeur)
+```
+
+# Les B-arbres
+
+\footnotesize
+
+## Les fonctions utilitaires (5min matrix)
+
+```C
+page nouvelle_page(ordre) // créer une page
+rien liberer_memoire(page) // libérer tout un arbre!
+```
+. . .
+
+```C
+page nouvelle_page(ordre)
+ page = allouer(page)
+ page.ordre = ordre
+ page.nb = 0
+ page.tab = allouer(2*ordre+2)
+ retourner page
+
+rien liberer_memoire(page)
+ si est_feuille(page)
+ liberer(page.tab)
+ liberer(page)
+ sinon
+ pour fille dans page.tab
+ liberer_memoire(fille)
+ liberer(page.tab)
+ liberer(page)
+```
+
+# Les B-arbres
+
+## Les fonctions (5min matrix)
+
+```C
+page recherche(page, valeur) // retourner la page contenant
+ // la valeur ou vide
+```
+
+. . .
+
+```C
+page recherche(page, valeur)
+ si est_dans_page(page, valeur)
+ retourne page
+ sinon si est_feuille(page)
+ retourne vide
+ sinon
+ recherche(page.tab[position(page, valeur) - 1], valeur)
+```
+
+# Les B-arbres
+
+## Les fonctions
+
+```C
+page inserer_valeur(page, valeur) // insérer une valeur
+```
+
+. . .
+
+```C
+page inserer_valeur(page, valeur)
+ element = nouvel_element(valeur)
+ // ici élément est modifié pour savoir
+ // s'il faut le remonter
+ inserer_element(page, element)
+ si element.page != vide && page.nb > 2*page.ordre
+ // si on atteint le sommet!
+ page = ajouter_niveau(page, element)
+ retourne page
+```
+
+# Les B-arbres
+
+## Les fonctions
+
+```C
+rien inserer_element(page, element) // insérer un element
+ // et voir s'il remonte
+```
+
+. . .
+
+```C
+rien inserer_element(page, element)
+ si est_feuille(page)
+ placer(page, element)
+ sinon
+ sous_page = page.tab[position(page, element.clé) - 1].page
+ inserer_element(sous_page, element)
+ // un element a été promu
+ si element.page != vide
+ placer(page, element)
+```
+
+# Les B-arbres
+
+## Les fonctions (5min matrix)
+
+```C
+rien placer(page, element) // inserer un élément
+```
+
+. . .
+
+```C
+rien placer(page, element)
+ pos = position(page, element.clé)
+ pour i de 2*page.ordre à pos+1
+ page.tab[i+1] = page.tab[i]
+ page.tab[pos+1] = element
+ page.nb += 1
+ si page.nb > 2*page.ordre
+ scinder(page, element)
+```
+
+# Les B-arbres
+
+## Les fonctions (5min matrix)
+
+```C
+rien scinder(page, element) // casser une page et remonter
+```
+
+. . .
+
+```C
+rien scinder(page, element)
+ nouvelle_page = nouvelle_page(page.ordre)
+ nouvelle_page.nb = page.ordre
+ pour i de 0 Ã ordre inclu
+ nouvelle_page.tab[i] = page.tab[i+ordre+1]
+ element.clé = page.tab[ordre+1].clé
+ element.page = nouvelle_page
+```
+
+# Les B-arbres
+
+## Les fonctions (5min matrix)
+
+```C
+page ajouter_niveau(page, element) // si on remonte à la
+ // racine, on doit créer
+ // une nouvelle racine
+```
+
+. . .
+
+```C
+page ajouter_niveau(page, element)
+ tmp = nouvelle_page(page.ordre)
+ tmp.tab[0].page = page
+ tmp.tab[1].clé = element.clé
+ tmp.tab[1].page = element.page
+ retourne tmp
+```
+
+
+
+
+
+<!-- # Les B-arbres -->
+
+<!-- ## Structure de données en C (3min, matrix) -->
+
+<!-- . . . -->
+
+<!-- ```C -->
+<!-- typedef struct _page { -->
+<!-- int order, nb; -->
+<!-- struct _element *tab; -->
+<!-- } page; -->
+<!-- ``` -->
+
+<!-- ```C -->
+<!-- typedef struct element { -->
+<!-- int key; -->
+<!-- struct _page *pg; -->
+<!-- } element; -->
+<!-- ``` -->
+
diff --git a/slides_2022/cours_23.md b/slides_2022/cours_23.md
new file mode 100644
index 0000000000000000000000000000000000000000..60f01e4b14c71331b3b09872c34ee768a6f134df
--- /dev/null
+++ b/slides_2022/cours_23.md
@@ -0,0 +1,1013 @@
+---
+title: "B-arbres et Graphes"
+date: "2023-05-24"
+patat:
+ eval:
+ tai:
+ command: fish
+ fragment: false
+ replace: true
+ ccc:
+ command: fish
+ fragment: false
+ replace: true
+ images:
+ backend: auto
+---
+
+
+# Les B-arbres: suppression
+
+## Cas simplissime
+
+{width=80%}
+
+. . .
+
+{width=80%}
+
+# Les B-arbres: suppression
+
+\footnotesize
+
+## Cas simple
+
+
+{width=60%}
+
+. . .
+
+* On retire 27, mais....
+ * Chaque page doit avoir au moins 2 éléments.
+ * On doit déplacer des éléments dans une autre feuille! Mais comment?
+
+. . .
+
+{width=60%}
+
+# Les B-arbres: suppression
+
+## Cas moins simple
+
+{width=60%}
+
+. . .
+
+* Un élément à droite, comment on fait?
+ * Remonter `7`, serait ok si racine, mais... c'est pas forcément.
+ * On redistribue les feuilles.
+
+. . .
+
+{width=60%}
+
+# Les B-arbres: suppression
+
+\footnotesize
+
+## Cas ultra moins simple
+
+{width=60%}
+
+. . .
+
+* `7` seul:
+ * Fusionner les feuilles et redistribuer, comment?
+
+. . .
+
+{width=60%}
+
+# Les B-arbres: suppression
+
+## Cas ultra moins simple
+
+{width=60%}
+
+. . .
+
+* `8` est seul, c'est plus un B-arbre :
+ * Fusionner le niveau 2 et redistribuer, comment?
+
+. . .
+
+{width=40%}
+
+. . .
+
+* La profondeur a diminué de 1.
+
+# Les B-arbres: suppression
+
+## Algorithme pour les feuilles!
+
+* Si la clé est supprimée d'une feuille:
+ * Si on a toujours `n` (ordre de l'arbre) clés dans la feuille on décale simplement les clés.
+ * Sinon on combine (récursivement) avec le nœud voisin et on descend la clé médiane.
+
+# Les B-arbres: suppression
+
+## Cas non-feuille!
+
+{width=60%}
+
+. . .
+
+* On sait comment effacer une valeur d'une feuille, donc?
+
+. . .
+
+{width=60%}
+
+* Ensuite?
+
+# Les B-arbres: suppression
+
+## Cas non-feuille!
+
+{width=60%}
+
+. . .
+
+* On sait comment effacer une valeur d'une feuille!
+
+. . .
+
+{width=60%}
+
+# Les B-arbres: suppression
+
+## Algorithme pour les non-feuilles!
+
+* Si la clé est supprimée d'une page qui n'est pas une feuille:
+ * On échange la valeur avec la valeur de droite de la page de gauche
+ * On supprime comme pour une feuille!
+
+## Et maintenant des exercices par millions!
+
+# Les graphes! Historique
+
+**Un mini-peu d'histoire...**
+
+## L. Euler et les 7 ponts de Koenigsberg:
+
+* Existe-t-il une promenade sympa, passant **une seule fois** par les 7 ponts et revenant au point de départ?
+
+{width=50%}
+
+. . .
+
+* Réponse: ben non!
+
+# Utilisation quotidienne
+
+## Réseau social
+
+{width=40%}
+
+* Chaque sommet est un individu.
+* Chaque trait une relation d'amitié.
+* Miam, Miam, Facebook.
+
+# Utilisation quotidienne
+
+## Moteurs de recherche
+
+{width=40%}
+
+* Sommet est un site.
+* Liens sortants;
+* Liens entrants;
+* Notion d'importance d'un site: combien de liens entrants, pondérés par l'importance du site.
+* Miam, Miam, Google (PageRank).
+
+# Introduction
+
+## Définition, plus ou moins
+
+* Un graphe est un ensemble de sommets, reliés par des lignes ou des flèches.
+
+
+
+* Des sommets (numérotés 1 à 6);
+* Connectés ou pas par des traits ou des flèches!
+
+# Généralités
+
+## Définitions
+
+* Un **graphe** $G(V, E)$ est constitué
+ * $V$: un ensemble de sommets;
+ * $E$: un ensemble d'arêtes.
+* Une **arête** relie une **paire** de sommets de $V$.
+
+## Remarques
+
+* Il y a **au plus** une arête $E$ par paire de sommets de $V$.
+* La **complexité** d'un algorithme dans un graphe se mesure en terme de $|E|$ et $|V|$, le nombre d'éléments de $E$ et $V$ respectivement.
+
+# Généralités
+
+## Notations
+
+* Une arête d'un graphe **non-orienté** est représentée par une paire **non-ordonnée** $(u,v)=(v,u)$, avec $u,v\in V$.
+* Les arêtes ne sont pas orientées dans un graphe non-orienté (elles sont bi-directionnelles, peuvent être parcourues dans n'importe quel ordre).
+
+## Exemple
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+## Que valent $V$, $|V|$, $E$, et $|E|$?
+
+. . .
+
+\begin{align*}
+V&=\{1, 2, 3, 4\},\\
+|V|&=4,\\
+E&=\{(1,2),(2,3),(2,4),(4,1)\},\\
+|E|&=4.
+\end{align*}
+
+::::
+
+:::
+
+# Généralités
+
+## Notations
+
+* Une arête d'un graphe **orienté** est représentée par une paire **ordonnée** $(u,v)\neq(v,u)$, avec $u,v\in V$.
+* Les arêtes sont orientées dans un graphe orienté (étonnant non?).
+
+## Exemple
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+## Que valent $V$, $|V|$, $E$, et $|E|$?
+
+. . .
+
+\begin{align*}
+V&=\{1, 2, 3, 4\},\\
+|V|&=4,\\
+E&=\{(1,2),(2,3),(2,4),(4,1),(4,2)\},\\
+|E|&=5.
+\end{align*}
+
+::::
+
+:::
+
+# Généralités
+
+## Définition
+
+* Le somme $v$ est **adjacent** au sommet $u$, si et seulement si $(u,v)\in E$;
+* Si un graphe non-orienté contient une arête $(u,v)$, $v$ est adjacent à $u$ et $u$ et adjacent à $v$.
+
+## Exemple
+
+::: columns
+
+:::: column
+
+{width=80%}
+
+::::
+
+:::: column
+
+{width=80%}
+
+::::
+
+:::
+
+# Généralités
+
+## Définition
+
+* Un **graphe pondéré** ou **valué** est un graphe dont chaque arête a un
+ poids associé, habituellement donné par une fonction de pondération $w:E\rightarrow\mathbb{R}$.
+
+## Exemples
+
+{width=80%}
+
+
+# Généralités
+
+## Définition
+
+* Dans un graphe $G(V,E)$, une **chaîne** reliant un sommet $u$ à un sommet $v$ est une suite d'arêtes entre les sommets, $w_0$, $w_1$, ..., $w_k$, telles que
+$$
+(w_i, w_{i+1})\in E,\quad u=w_0,\quad v=w_k,\quad \mbox{pour }0\leq i< k,
+$$
+avec $k$ la longueur de la chaîne (le nombre d'arêtes du chemin).
+
+## Exemples
+
+{width=80%}
+
+# Généralités
+
+## Définition
+
+* Une **chaîne élémentaire** est une chaîne dont tous les sommets sont distincts, sauf les extrémités qui peuvent être égales
+
+## Exemples
+
+{width=80%}
+
+# Généralités
+
+## Définition
+
+* Une **boucle** est une arête $(v,v)$ d'un sommet vers lui-même.
+
+## Exemples
+
+{width=40%}
+
+# Généralités
+
+## Définition
+
+* Un graphe non-orienté est dit **connexe**, s'il existe un chemin reliant n'importe quelle paire de sommets distincts.
+
+## Exemples
+
+\
+
+::: columns
+
+:::: column
+
+{width=80%}
+
+::::
+
+:::: column
+{width=60%}
+
+::::
+
+:::
+
+# Généralités
+
+## Définition
+
+* Un graphe orienté est dit **fortement connexe**, s'il existe un chemin reliant n'importe quelle paire de sommets distincts.
+
+## Exemples
+
+\
+
+::: columns
+
+:::: column
+
+{width=60%}
+
+::::
+
+:::: column
+
+{width=100%}
+
+::::
+
+:::
+
+# Généralités
+
+## Définition
+
+* Un **cycle** dans un graphe *non-orienté* est une chaîne de longueur $\geq 3$ telle que le 1er sommet de la chaîne est le même que le dernier, et dont les arêtes sont distinctes.
+* Pour un graphe *orienté* on parle de **circuit**.
+* Un graphe sans cycles est dit **acyclique**.
+
+## Exemples
+
+{width=100%}
+
+# Question de la mort
+
+* Qu'est-ce qu'un graphe connexe acyclique?
+
+. . .
+
+* Un arbre!
+
+# Représentations
+
+* La complexité des algorithmes sur les graphes s'expriment en fonction du nombre de sommets $V$, et du nombre d'arêtes $E$:
+ * Si $|E|\sim |V|^2$, on dit que le graphe est **dense**.
+ * Si $|E|\sim |V|$, on dit que le graphe est **peu dense**.
+* Selon qu'on considère des graphes denses ou peu denses, différentes structure de données peuvent être envisagées.
+
+## Question
+
+* Comment peut-on représenter un graphe informatiquement? Des idées (réflexion de quelques minutes)?
+
+. . .
+
+* Matrice/liste d'adjacence.
+
+# Matrice d'adjacence
+
+* Soit le graphe $G(V,E)$, avec $V=\{1, 2, 3, ..., n\}$;
+* On peut représenter un graphe par une **matrice d'adjacence**, $A$, de dimension $n\times n$ définie par
+$$
+A_{ij}=\left\{ \begin{array}{ll}
+ 1 & \mbox{si } i,j\in E,\\
+ 0 & \mbox{sinon}.
+ \end{array} \right.
+$$
+
+
+::: columns
+
+:::: column
+
+## Exemple
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---4;
+ 2---5;
+ 4---5;
+ 5---3;
+```
+
+::::
+
+:::: column
+
+\footnotesize
+
+## Quelle matrice d'adjacence?
+
+. . .
+
+```
+ || 1 | 2 | 3 | 4 | 5
+===||===|===|===|===|===
+ 1 || 0 | 1 | 0 | 1 | 0
+---||---|---|---|---|---
+ 2 || 1 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 3 || 0 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 4 || 1 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 5 || 0 | 1 | 1 | 1 | 0
+```
+
+::::
+
+:::
+
+# Matrice d'adjacence
+
+## Remarques
+
+* Zéro sur la diagonale.
+* La matrice d'un graphe non-orienté est symétrique
+
+$$
+A_{ij}=A_{ji}, \forall i,j\in[1,n]
+$$.
+
+::: columns
+
+:::: column
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---4;
+ 2---5;
+ 4---5;
+ 5---3;
+```
+
+::::
+
+:::: column
+
+\footnotesize
+
+```
+ || 1 | 2 | 3 | 4 | 5
+===||===|===|===|===|===
+ 1 || 0 | 1 | 0 | 1 | 0
+---||---|---|---|---|---
+ 2 || 1 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 3 || 0 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 4 || 1 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 5 || 0 | 1 | 1 | 1 | 0
+```
+
+::::
+
+:::
+
+# Matrice d'adjacence
+
+* Pour un graphe orienté (digraphe)
+
+::: columns
+
+:::: column
+
+## Exemple
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 2-->1;
+ 1-->4;
+ 2-->5;
+ 5-->2;
+ 4-->5;
+ 5-->3;
+```
+
+::::
+
+:::: column
+
+\footnotesize
+
+## Quelle matrice d'adjacence?
+
+. . .
+
+```
+ || 1 | 2 | 3 | 4 | 5
+===||===|===|===|===|===
+ 1 || 0 | 0 | 0 | 1 | 0
+---||---|---|---|---|---
+ 2 || 1 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 3 || 0 | 0 | 0 | 0 | 0
+---||---|---|---|---|---
+ 4 || 0 | 0 | 0 | 0 | 1
+---||---|---|---|---|---
+ 5 || 0 | 1 | 1 | 0 | 0
+```
+
+::::
+
+:::
+
+* La matrice d'adjacence n'est plus forcément symétrique
+$$
+A_{ij}\neq A_{ji}.
+$$
+
+# Stockage
+
+* Quel est l'espace nécessaire pour stocker une matrice d'adjacence pour un graphe orienté?
+
+. . .
+
+* $\mathcal{O}(|V|^2)$.
+* Quel est l'espace nécessaire pour stocker une matrice d'adjacence pour un graphe non-orienté?
+
+. . .
+
+* $\mathcal{O}(|V|-1)|V|/2$.
+
+# Considérations d'efficacité
+
+* Dans quel type de graphes la matrice d'adjacence est utile?
+
+. . .
+
+* Dans les graphes denses.
+* Pourquoi?
+
+. . .
+
+* Dans les graphes peu denses, la matrice d'adjacence est essentiellement composée de `0`.
+
+## Remarque
+
+* Dans la majorité des cas, les grands graphes sont peu denses.
+* Comment représenter un graphe autrement?
+
+# La liste d'adjacence (non-orienté)
+
+* Pour chaque sommet $v\in V$, stocker les sommets adjacents à $v$-
+* Quelle structure de données pour la liste d'adjacence?
+
+. . .
+
+* Tableau de liste chaînée, vecteur (tableau dynamique), etc.
+
+
+::: columns
+
+:::: column
+
+## Exemple
+
+{width=80%}
+
+::::
+
+:::: column
+
+
+## Quelle liste d'adjacence?
+
+. . .
+
+
+
+
+::::
+
+:::
+
+# La liste d'adjacence (orienté)
+
+
+::: columns
+
+:::: column
+
+## Quelle liste d'adjacence pour...
+
+* Matrix (2min)
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 0-->1;
+ 0-->2;
+ 1-->2;
+ 3-->0;
+ 3-->1;
+ 3-->2;
+```
+
+::::
+
+:::: column
+
+```
+
+
+
+
+
+
+
+
+
+
+
+```
+
+
+::::
+
+:::
+
+# Complexité
+
+## Stockage
+
+* Quelle espace est nécessaire pour stocker une liste d'adjacence (en fonction de $|E|$ et $|V|$)?
+
+. . .
+
+$$
+\mathcal{O}(|E|)
+$$
+
+* Pour les graphes *non-orientés*: $\mathcal{O}(2|E|)$.
+* Pour les graphes *orientés*: $\mathcal{O}(|E|)$.
+
+## Définition
+
+* Le **degré** d'un sommet $v$, est le nombre d'arêtes incidentes du sommet (pour les graphes orientés on a un degré entrant ou sortant).
+* Comment on retrouve le degré de chaque sommet avec la liste d'adjacence?
+
+. . .
+
+* C'est la longueur de la liste chaînée.
+
+
+# Parcours
+
+* Beaucoup d'applications nécessitent de parcourir des graphes:
+ * Trouver un chemin d'un sommet à un autre;
+ * Trouver si le graphe est connexe;
+* Il existe *deux* parcours principaux:
+ * en largeur (Breadth-First Search);
+ * en profondeur (Depth-First Search).
+* Ces parcours créent *un arbre* au fil de l'exploration (si le graphe est non-connexe cela crée une *forêt*, un ensemble d'arbres).
+
+# Illustration: parcours en largeur
+
+{width=80%}
+
+# Exemple
+
+## Étape par étape (blanc non-visité)
+
+{width=50%}
+
+## Étape par étape (gris visité)
+
+{width=50%}
+
+# Exemple
+
+## Étape par étape
+
+{width=50%}
+
+## Étape par étape (vert à visiter)
+
+{width=50%}
+
+# Exemple
+
+## Étape par étape
+
+{width=50%}
+
+## Étape par étape
+
+{width=50%}
+
+# Exemple
+
+## Étape par étape
+
+{width=50%}
+
+## Étape par étape
+
+{width=50%}
+
+# Exemple
+
+## Étape par étape
+
+{width=50%}
+
+## Étape par étape
+
+{width=50%}
+
+# Exemple
+
+## Étape par étape
+
+{width=50%}
+
+## Étape par étape
+
+{width=50%}
+
+# En faisant ce parcours...
+
+
+::: columns
+
+:::: column
+
+## Du parcours de l'arbre
+
+{width=100%}
+
+::::
+
+:::: column
+
+## Quel arbre est créé par le parcours (2min)?
+
+. . .
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 0[x]-->1[w];
+ 0-->2[t];
+ 0-->3[y];
+ 2-->9[u];
+ 1-->4[s];
+ 4-->5[r];
+ 5-->6[v];
+```
+
+::::
+
+:::
+
+## Remarques
+
+* Le parcours dépend du point de départ dans le graphe.
+* L'arbre sera différent en fonction du noeud de départ, et de l'ordre de parcours des voisins d'un noeud.
+
+# Le parcours en largeur
+
+## L'algorithme, idée générale (3min, matrix)?
+
+. . .
+
+```C
+v = un sommet du graphe
+i = 1
+pour sommet dans graphe et sommet non-visité
+ visiter(v, sommet, i) // marquer sommet à distance i visité
+ i += 1
+```
+
+## Remarque
+
+* `i` est la distance de plus cours chemin entre `v` et les sommets en cours de visite.
+
+
+# Le parcours en largeur
+
+## L'algorithme, pseudo-code (3min, matrix)?
+
+* Comment garder la trace de la distance?
+
+. . .
+
+* Utilisation d'une **file**
+
+. . .
+
+```C
+initialiser(graphe) // tous sommets sont non-visités
+file = visiter(sommet, vide) // sommet est un sommet du graphe au hasard
+tant que !est_vide(file)
+ v = défiler(file)
+ file = visiter(v, file)
+```
+
+## Que fait visiter?
+
+```
+file visiter(sommet, file)
+ sommet = visité
+ pour w = chaque arête de sommet
+ si w != visité
+ file = enfiler(file, w)
+ retourne file
+```
+
+# Exercice (5min)
+
+## Appliquer l'algorithme sur le graphe
+
+{width=50%}
+
+* En partant de `v`, `s`, ou `u` (par colonne de classe).
+* Bien mettre à chaque étape l'état de la file.
+
+# Complexité du parcours en largeur
+
+## Étape 1
+
+* Extraire un sommet de la file;
+
+## Étape 2
+
+* Traîter tous les sommets adjacents.
+
+## Quelle est la complexité?
+
+. . .
+
+* Étape 1: $\mathcal{O}(|V|)$,
+* Étape 2: $\mathcal{O}(2|E|)$,
+* Total: $\mathcal{O}(|V| + |2|E|)$.
+
+# Exercice
+
+* Établir la liste d'adjacence et appliquer l'algorithme de parcours en largeur au graphe
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---3;
+ 1---4;
+ 2---3;
+ 2---6;
+ 3---6;
+ 3---4;
+ 3---5;
+ 4---5;
+```
+
+
+# Illustration: parcours en profondeur
+
+{width=80%}
+
+# Parcours en profondeur
+
+## Idée générale
+
+* Initialiser les sommets comme non-lus
+* Visiter un sommet
+* Pour chaque sommet visité, on visite un sommet adjacent s'il est pas encore visité récursivement.
+
+## Remarque
+
+* La récursivité est équivalent à l'utilisation d'une **pile**.
+
+# Parcours en profondeur
+
+## Pseudo-code (5min)
+
+. . .
+
+```C
+initialiser(graphe) // tous sommets sont non-visités
+pile = visiter(sommet, vide) // sommet est un sommet du graphe au hasard
+tant que !est_vide(pile)
+ v = dépiler(pile)
+ pile = visiter(v, pile)
+```
+
+## Que fait visiter?
+
+. . .
+
+```C
+pile visiter(sommet, pile)
+ sommet = visité
+ pour w = chaque arête de sommet
+ si w != visité
+ pile = empiler(pile, w)
+ retourne pile
+```
+
+
+# Exercice
+
+* Établir la liste d'adjacence et appliquer l'algorithme de parcours en profondeur au graphe
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---3;
+ 1---4;
+ 2---3;
+ 2---6;
+ 3---6;
+ 3---4;
+ 3---5;
+ 4---5;
+```
+
+# Interprétation des parcours
+
+* Un graphe vu comme espace d'états (sommet: état, arête: action);
+ * Labyrinthe;
+ * Arbre des coups d'un jeu.
+
+. . .
+
+* BFS (Breadth-First) ou DFS (Depth-First) parcourent l'espace des états à la recherche du meilleur mouvement.
+ * Les deux parcourent *tout* l'espace;
+ * Mais si l'arbre est grand, l'espace est gigantesque!
+
+. . .
+
+* Quand on a un temps limité
+ * BFS explore beaucoup de coups dans un futur proche;
+ * DFS explore peu de coups dans un futur lointain.
diff --git a/slides_2022/cours_24.md b/slides_2022/cours_24.md
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+---
+title: "Graphes - Plus court chemin"
+date: "2023-06-02"
+---
+
+# Rappel du dernier cours
+
+* Qu'est-ce qu'un graphe? Un graphe orienté? Un graphe pondéré?
+
+. . .
+
+* Ensemble de sommets et arêtes, avec une direction, possédant une pondération.
+* Comment représenter un graphe informatiquement?
+
+. . .
+
+* Liste ou matrice d'adjacence.
+* Quel est le parcours que nous avons vu?
+
+. . .
+
+* Le parcours en largeur.
+
+# Le parcours en largeur
+
+## Le pseudo-code
+
+* Utilisation d'une **file**
+
+```C
+initialiser(graphe) // tous sommets sont non-visités
+file = visiter(sommet, vide) // sommet est un sommet
+ // du graphe
+tant que !est_vide(file)
+ v = defiler(file)
+ file = visiter(v, file)
+
+file visiter(sommet, file)
+ sommet = visité
+ pour w = chaque arête de sommet
+ si w != visité
+ file = enfiler(file, w)
+ retourne file
+```
+
+# Complexité du parcours en largeur
+
+## Étape 1
+
+* Extraire un sommet de la file;
+
+## Étape 2
+
+* Traîter tous les sommets adjacents.
+
+## Quelle est la complexité?
+
+. . .
+
+* Étape 1: $\mathcal{O}(|V|)$,
+* Étape 2: $\mathcal{O}(2|E|)$,
+* Total: $\mathcal{O}(|V| + 2|E|)$.
+
+# Exercice
+
+* Établir la liste d'adjacence et appliquer l'algorithme de parcours en largeur au graphe
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---3;
+ 1---4;
+ 2---3;
+ 2---6;
+ 3---6;
+ 3---4;
+ 3---5;
+ 4---5;
+```
+
+
+# Illustration: parcours en profondeur
+
+{width=80%}
+
+# Parcours en profondeur
+
+## Idée générale
+
+* Initialiser les sommets comme non-lus
+* Visiter un sommet
+* Pour chaque sommet visité, on visite un sommet adjacent s'il est pas encore visité récursivement.
+
+## Remarque
+
+* La récursivité est équivalent à l'utilisation d'une **pile**.
+
+# Parcours en profondeur
+
+## Pseudo-code (5min)
+
+. . .
+
+```C
+initialiser(graphe) // tous sommets sont non-visités
+pile = visiter(sommet, vide) // sommet est un
+ // sommet du graphe
+tant que !est_vide(pile)
+ v = dépiler(pile)
+ pile = visiter(v, pile)
+```
+
+## Que fait visiter?
+
+. . .
+
+```C
+pile visiter(sommet, pile)
+ sommet = visité
+ pour w = chaque arête de sommet
+ si w != visité
+ pile = empiler(pile, w)
+ retourne pile
+```
+
+
+# Exercice
+
+* Établir la liste d'adjacence et appliquer l'algorithme de parcours en profondeur au graphe
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph LR;
+ 1---2;
+ 1---3;
+ 1---4;
+ 2---3;
+ 2---6;
+ 3---6;
+ 3---4;
+ 3---5;
+ 4---5;
+```
+
+# Interprétation des parcours
+
+* Un graphe vu comme espace d'états (sommet: état, arête: action);
+ * Labyrinthe;
+ * Arbre des coups d'un jeu.
+
+. . .
+
+* BFS (Breadth-First) ou DFS (Depth-First) parcourent l'espace des états à la recherche du meilleur mouvement.
+ * Les deux parcourent *tout* l'espace;
+ * Mais si l'arbre est grand, l'espace est gigantesque!
+
+. . .
+
+* Quand on a un temps limité
+ * BFS explore beaucoup de coups dans un futur proche;
+ * DFS explore peu de coups dans un futur lointain.
+
+
+## Contexte: les réseaux (informatique, transport, etc.)
+
+* Graphe orienté;
+* Source: sommet `s`;
+* Destination: sommet `t`;
+* Les arêtes ont des poids (coût d'utilisation, distance, etc.);
+* Le coût d'un chemin est la somme des poids des arêtes d'un chemin.
+
+## Problème à résoudre
+
+* Quel est le plus court chemin entre `s` et `t`.
+
+# Exemples d'application de plus court chemin
+
+## Devenir riches!
+
+* On part d'un tableau de taux de change entre devises.
+* Quelle est la meilleure façon de convertir l'or en dollar?
+
+{width=80%}
+
+. . .
+
+* 1kg d'or => 327.25 dollars
+* 1kg d'or => 208.1 livres => 327 dollars
+* 1kg d'or => 455.2 francs => 304.39 euros => 327.28 dollars
+
+# Exemples d'application de plus court chemin
+
+## Formulation sous forme d'un graphe: Comment (3min)?
+
+{width=80%}
+
+
+# Exemples d'application de plus court chemin
+
+## Formulation sous forme d'un graphe: Comment (3min)?
+
+{width=60%}
+
+* Un sommet par devise;
+* Une arête orientée par transaction possible avec le poids égal au taux de change;
+* Trouver le chemin qui maximise le produit des poids.
+
+. . .
+
+## Problème
+
+* On aimerait plutôt avoir une somme...
+
+
+# Exemples d'application de plus court chemin
+
+## Conversion du problème en plus court chemin
+
+* Soit `taux(u, v)` le taux de change entre la devise `u` et `v`.
+* On pose `w(u,w)=-log(taux(u,v))`
+* Trouver le chemin poids minimal pour les poids `w`.
+
+{width=60%}
+
+* Cette conversion se base sur l'idée que
+
+$$
+\log(u\cdot v)=\log(u)+\log(v).
+$$
+
+# Applications de plus courts chemins
+
+## Quelles applications voyez-vous?
+
+. . .
+
+* Déplacement d'un robot;
+* Planificaiton de trajet / trafic urbain;
+* Routage de télécommunications;
+* Réseau électrique optimal;
+* ...
+
+# Plus courts chemins à source unique
+
+* Soit un graphe, $G=(V, E)$, une fonction de pondération $w:E\rightarrow\mathbb{R}$, et un sommet $s\in V$
+ * Trouver pour tout sommet $v\in V$, le chemin de poids minimal reliant $s$ Ã $v$.
+* Algorithmes standards:
+ * Dijkstra (arêtes de poids positif seulement);
+ * Bellman-Ford (arêtes de poids positifs ou négatifs, mais sans cycles).
+* Comment résoudre le problèmes si tous les poids sont les mêmes?
+
+. . .
+
+* Un parcours en largeur!
+
+# Algorithme de Dijkstra
+
+## Comment chercher pour un plus court chemin?
+
+. . .
+
+```
+si distance(u,v) > distance(u,w) + distance(w,v)
+ on passe par w plutôt qu'aller directement
+```
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+* $D$ est le tableau des distances au sommet $1$: $D[7]$ est la distance de 1 Ã 7.
+* Le chemin est pas forcément direct.
+* $S$ est le tableau des sommets visités.
+
+::: columns
+
+:::: column
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![1 visité, `D[2]=1`, `D[4]=3`.](figs/dijkstra_1.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![2 visité, `D[3]=2`, `D[7]=3`.](figs/dijkstra_2.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![3 visité, `D[7]=3` inchangé, `D[6]=6`.](figs/dijkstra_3.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![4 visité, `D[7]=3` inchangé, `D[5]=9`.](figs/dijkstra_4.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![7 visité, `D[5]=7`, `D[6]=6` inchangé.](figs/dijkstra_5.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+. . .
+
+![`6` visité, `D[5]=7` inchangé.](figs/dijkstra_6.png)
+
+::::
+
+:::
+
+# Algorithme de Dijkstra (1 Ã 5)
+
+::: columns
+
+:::: column
+
+
+
+::::
+
+:::: column
+
+. . .
+
+
+
+::::
+
+:::
+
+# Algorithme de Dijkstra
+
+## Idée générale
+
+* On assigne à chaque noeud une distance $0$ pour $s$, $\infty$ pour les autres.
+* Tous les noeuds sont marqués non-visités.
+* Depuis du noeud courant, on suit chaque arête du noeud vers un sommet non visité et on calcule le poids du chemin à chaque voisin et on met à jour sa distance si elle est plus petite que la distance du noeud.
+* Quand tous les voisins du noeud courant ont été visités, le noeud est mis à visité (il ne sera plus jamais visité).
+* Continuer avec le noeud à la distance la plus faible.
+* L'algorithme est terminé losrque le noeud de destination est marqué comme visité, ou qu'on a plus de noeuds qu'on peut visiter et que leur distance est infinie.
+
+# Algorithme de Dijkstra
+
+## Pseudo-code (5min, matrix)
+
+\footnotesize
+
+. . .
+
+```C
+tab dijkstra(graph, s, t)
+ pour chaque v dans graphe
+ distance[v] = infini
+ q = ajouter(q, v)
+ distance[s] = 0
+ tant que non_vide(q)
+ // sélection de u t.q. la distance dans q est min
+ u = min(q, distance)
+ si u == t // on a atteint la cible
+ retourne distance
+ q = remove(q, u)
+ // voisin de u encore dans q
+ pour chaque v dans voisinage(u, q)
+ // on met à jour la distance du voisin en passant par u
+ n_distance = distance[u] + w(u, v)
+ si n_distance < distance[v]
+ distance[v] = n_distance
+ retourne distance
+```
+
+# Algorithme de Dijkstra
+
+* Cet algorithme, nous donne le plus court chemin mais...
+* ne nous donne pas le chemin!
+
+## Comment modifier l'algorithme pour avoir le chemin?
+
+. . .
+
+* Pour chaque nouveau noeud à visiter, il suffit d'enregistrer d'où on est venu!
+* On a besoin d'un tableau `precedent`.
+
+## Modifier le pseudo-code ci-dessus pour ce faire (3min matrix)
+
+# Algorithme de Dijkstra
+
+\footnotesize
+
+```C
+tab, tab dijkstra(graph, s, t)
+ pour chaque v dans graphe
+ distance[v] = infini
+ precedent[v] = indéfini
+ q = ajouter(q, v)
+ distance[s] = 0
+ tant que non_vide(q)
+ // sélection de u t.q. la distance dans q est min
+ u = min(q, distance)
+ si u == t
+ retourne distance
+ q = remove(q, u)
+ // voisin de u encore dans q
+ pour chaque v dans voisinage(u, q)
+ n_distance = distance[u] + w(u, v)
+ si n_distance < distance[v]
+ distance[v] = n_distance
+ precedent[v] = u
+ retourne distance, precedent
+```
+
+# Algorithme de Dijkstra
+
+## Comment reconstruire un chemin ?
+
+. . .
+
+```C
+pile parcours(precedent, s, t)
+ sommets = vide
+ u = t
+ // on a atteint t ou on ne connait pas de chemin
+ si u != s && precedent[u] != indéfini
+ tant que vrai
+ sommets = empiler(sommets, u)
+ u = precedent[u]
+ si u == s // la source est atteinte
+ retourne sommets
+ retourne sommets
+```
+
+# Algorithme de Dijkstra amélioré
+
+## On peut améliorer l'algorithme
+
+* Avec une file de priorité!
+
+## Une file de priorité est
+
+* Une file dont chaque élément possède une priorité,
+* Elle existe en deux saveurs: `min` ou `max`:
+ * File `min`: les éléments les plus petits sont retirés en premier.
+ * File `max`: les éléments les plus grands sont retirés en premier.
+* On regarde l'implémentation de la `max`.
+
+## Comment on fait ça?
+
+. . .
+
+* On insère les éléments à haute priorité tout devant dans la file!
+
+# Les files de priorité
+
+## Trois fonction principales
+
+```C
+booléen est_vide(element) // triviale
+element enfiler(element, data, priorite)
+data defiler(element)
+rien changer_priorite(element, data, priorite)
+nombre priorite(element) // utilitaire
+```
+
+## Pseudo-implémentation: structure (1min)
+
+. . .
+
+```C
+struct element
+ data
+ priorite
+ element suivant
+```
+
+# Les files de priorité
+
+## Pseudo-implémentation: enfiler (2min)
+
+. . .
+
+```C
+element enfiler(element, data, priorite)
+ n_element = creer_element(data, priorite)
+ si est_vide(element)
+ retourne n_element
+ si priorite(n_element) > priorite(element)
+ n_element.suivant = element
+ retourne n_element
+ sinon
+ tmp = element
+ prec = element
+ tant que !est_vide(tmp) && priorite < priorite(tmp)
+ prec = tmp
+ tmp = tmp.suivant
+ prev.suivant = n_element
+ n_element.suivant = tmp
+ retourne element
+```
+
+# Les files de priorité
+
+## Pseudo-implémentation: defiler (2min)
+
+. . .
+
+```C
+data, element defiler(element)
+ si est_vide(element)
+ retourne AARGL!
+ sinon
+ tmp = element.data
+ n_element = element.suivant
+ liberer(element)
+ retourne tmp, n_element
+```
+
+# Algorithme de Dijkstra avec file de priorité min
+
+```C
+distance, precedent dijkstra(graphe, s, t):
+ distance[source] = 0
+ fp = file_p_vide()
+ pour v dans sommets(graphe)
+ si v != s
+ distance[v] = infini
+ precedent[v] = indéfini
+ fp = enfiler(fp, v, distance[v])
+ tant que !est_vide(fp)
+ u, fp = defiler(fp)
+ pour v dans voisinage de u
+ n_distance = distance[u] + w(u, v)
+ si n_distance < distance[v]
+ distance[v] = n_distance
+ precedent[v] = u
+ fp = changer_priorite(fp, v, n_distance)
+ retourne distance, precedent
+```
+
+# Algorithme de Dijkstra avec file
+
+\footnotesize
+
+```C
+distance dijkstra(graphe, s, t)
+---------------------------------------------------------
+ pour v dans sommets(graphe)
+O(V) si v != s
+ distance[v] = infini
+O(V) fp = enfiler(fp, v, distance[v]) // notre impl est nulle
+------------------O(V * V)-------------------------------
+ tant que !est_vide(fp)
+O(1) u, fp = defiler(fp)
+---------------------------------------------------------
+O(E) pour v dans voisinage de u
+ n_distance = distance[u] + w(u, v)
+ si n_distance < distance[v]
+ distance[v] = n_distance
+O(V) fp = changer_priorite(fp, v, n_distance)
+---------------------------------------------------------
+ retourne distance
+```
+
+* Total: $\mathcal{O}(|V|^2+|E|\cdot |V|)$:
+ * Graphe dense: $\mathcal{O}(|V|^3)$
+ * Graphe peu dense: $\mathcal{O}(|V|^2)$
+
+# Algorithme de Dijkstra avec file
+
+## On peut faire mieux
+
+* Avec une meilleure implémentation de la file de priorité:
+ * Tas binaire: $\mathcal{O}(|V|\log|V|+|E|\log|V|)$.
+ * Tas de Fibonnacci: $\mathcal{O}(|V|+|E|\log|V|)$
+* Graphe dense: $\mathcal{O}(|V|^2\log|V|)$.
+* Graphe peu dense: $\mathcal{O}(|V|\log|V|)$.
+
+# Algorithme de Dijkstra (exercice, 5min)
+
+{width=60%}
+
+* Donner la liste de priorité, puis...
+
+## A chaque étape donner:
+
+* Le tableau des distances à `a`;
+* Le tableau des prédécesseurs;
+* L'état de la file de priorité.
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Algorithme de Dijkstra (corrigé)
+
+
+
+# Limitation de l'algorithme de Dijkstra
+
+## Que se passe-t-il pour?
+
+{width=50%}
+
+## Quel est le problème?
+
+. . .
+
+* L'algorithme n'essaiera jamais le chemin `s->x->y->v` et prendra direct `s->v`.
+* Ce problème n'apparaît que s'il y a des poids négatifs.
+
diff --git a/slides_2022/cours_25.md b/slides_2022/cours_25.md
new file mode 100644
index 0000000000000000000000000000000000000000..8deb43bd2a1232c605492fc8ca0e4debc2149af7
--- /dev/null
+++ b/slides_2022/cours_25.md
@@ -0,0 +1,1077 @@
+---
+title: "Graphes - Plus court chemin suite"
+date: "2023-06-23"
+---
+
+# Questions
+
+* A quoi sert l'algorithme de Dijkstra?
+
+. . .
+
+* A trouver le plus court chemin entre un sommet, $s$, d'un graphe pondéré et tous les autres sommets.
+* Quelle est la limitation de l'algorithme de Dijkstra?
+
+. . .
+
+* Les poids doivent être positifs.
+* Résumer les étapes de l'algorithme de Dijkstra.
+
+. . .
+
+* `distance[source] = 0`, `distance[reste]=inf`;
+* enfiler tous les sommets, `distance <=> priorité`;
+* tant qu'il y a des sommets dans la file:
+ * u = défiler;
+ * pour tous les sommets `v` dans le voisinage de `u`;
+ * mettre à jour `distance[v]` (priorité et précédence) si `distance[v] > distance[u] + w(u,v)`.
+
+
+# Plus cours chemin pour toute paire de sommets
+
+## Comment faire pour avoir toutes les paires?
+
+. . .
+
+* Appliquer Dijkstra sur tous les sommets d'origine.
+* Complexité:
+ * Graphe dense: $\mathcal{O}(|V|)\mathcal{O}(|V|^2\log|V|)=\mathcal{O}(|V|^3\log|V|)$.
+ * Graphe peu dense: $\mathcal{O}(|V|)\mathcal{O}(|V|\log|V|)=\mathcal{O}(|V|^2\log|V|)$.
+
+. . .
+
+## Solution alternative: Floyd--Warshall
+
+* Pour toutes paires de sommets $u,v\in V$, trouver le chemin de poids minimal reliant $u$ Ã $v$.
+* Complexité $\mathcal{O}(|V|^3)$, indiqué pour graphes denses.
+* Fonctionne avec la matrice d'adjacence.
+
+# Algorithme de Floyd--Warshall
+
+## Idée générale
+
+* Soit l'ensemble de sommets $V=\{1, 2, 3, 4, ..., n\}$.
+* Pour toute paire de sommets, $i,j$, on considère tous les chemins passant par les sommets intermédiaires $\in\{1, 2, ..., k\}$ avec $k\leq n$.
+* On garde pour chaque $k$ la plus petite valeur.
+
+## Principe
+
+* A chaque étape, $k$, on vérifie s'il est plus court d'aller de $i$ à $j$ en passant par le sommet $k$.
+* Si à l'étape $k-1$, le coût du parcours est $p$, on vérifie si $p$ est plus petit que $p_1+p_2$, le chemin de $i$ à $k$, et $k$ à $j$ respectivement.
+
+# Algorithme de Floyd--Warshall
+
+## The algorithme
+
+Soit $d_{ij}(k)$ le plus court chemin de $i$ Ã $j$ passant par les sommets $\in\{1,2,...,k\}$
+
+$$
+d_{ij}(k)=\left\{
+\begin{array}{ll}
+ w(i,j), & \mbox{si } k=0,\\
+ \min(d_{ij}(k-1),d_{ik}(k-1)+d_{kj}(k-1)), & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(0)}$ (3min)?
+
+. . .
+
+$$
+D^{(0)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 8 & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & \infty & \infty & 0 & 5 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+## On part de $D^{(0)}$?
+
+$$
+D^{(0)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 8 & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & \infty & \infty & 0 & 5 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(1)}$ (3min)?
+
+. . .
+
+$$
+D^{(0)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & \mathbf{6} & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & \mathbf{3} & \mathbf{5} & 0 & \mathbf{4} \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+## On part de $D^{(0)}$
+
+$$
+D^{(0)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 8 & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & \infty & \infty & 0 & 5 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(1)}$ (3min)?
+
+. . .
+
+$$
+D^{(1)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & \mathbf{6} & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & \mathbf{3} & \mathbf{5} & 0 & \mathbf{4} \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+## Exemple
+
+$$
+D_{42}^{(1)}=D_{41}^{(0)}+D_{12}^{(0)}=1+2<\infty.
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(1)}$
+
+$$
+D^{(1)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 6 & \infty & 1 \\
+6 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(2)}$ (3min)?
+
+. . .
+
+$$
+D^{(2)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 6 & \infty & 1 \\
+\mathbf{4} & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(2)}$
+
+$$
+D^{(2)}=\begin{bmatrix}
+0 & 2 & 4 & \infty & 3 \\
+2 & 0 & 6 & \infty & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(3)}$ (3min)?
+
+. . .
+
+$$
+D^{(3)}=\begin{bmatrix}
+0 & 2 & 4 & \mathbf{8} & 3 \\
+2 & 0 & 6 & \mathbf{10} & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(3)}$
+
+$$
+D^{(3)}=\begin{bmatrix}
+0 & 2 & 4 & 8 & 3 \\
+2 & 0 & 6 & 10 & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\infty & \infty & \infty & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::: column
+
+## Que vaut $D^{(4)}$ (3min)?
+
+. . .
+
+$$
+D^{(4)}=\begin{bmatrix}
+0 & 2 & 4 & 8 & 3 \\
+2 & 0 & 6 & 10 & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+\mathbf{2} & \mathbf{4} & \mathbf{6} & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(4)}$
+
+$$
+D^{(4)}=\begin{bmatrix}
+0 & 2 & 4 & 8 & 3 \\
+2 & 0 & 6 & 10 & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+2 & 4 & 6 & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::: column
+
+## Que vaut $D^{(5)}$ (3min)?
+
+. . .
+
+$$
+D^{(5)}=\begin{bmatrix}
+0 & 2 & 4 & \mathbf{4} & 3 \\
+2 & 0 & 6 & \mathbf{2} & 1 \\
+4 & 2 & 0 & 4 & 3 \\
+1 & 3 & 5 & 0 & 4 \\
+2 & 4 & 6 & 1 & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code (10min)
+
+* Quelle structure de données?
+* Quelle initialisation?
+* Quel est le code pour le calcul de la matrice $D$?
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code
+
+* Quelle structure de données?
+
+```C
+int distance[n][n];
+```
+
+. . .
+
+* Quelle initialisation?
+
+```C
+matrice ini_floyd_warshall(distance, n, w)
+ pour i de 1 Ã n
+ pour j de 1 Ã n
+ distance[i][j] = w(i,j)
+ retourne distance
+```
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code
+
+* Quel est le code pour le calcul de la matrice $D$?
+
+```C
+matrice floyd_warshall(distance, n, w)
+ pour k de 1 Ã n
+ pour i de 1 Ã n
+ pour j de 1 Ã n
+ distance[i][j] = min(distance[i][j],
+ distance[i][k] + distance[k][j])
+ retourne distance
+```
+
+# Algorithme de Floyd--Warshall
+
+## La matrice de précédence
+
+* On a pas encore vu comment reconstruire le plus court chemin!
+* On définit, $P_{ij}^{(k)}$, qui est le prédécesseur du sommet $j$ depuis $i$ avec les sommets intermédiaires $\in\{1, 2, ..., k\}$.
+$$
+P^{(0)}_{ij}=\left\{
+\begin{array}{ll}
+ \mbox{vide}, & \mbox{si } i=j\mbox{, ou }w(i,j)=\infty\\
+ i, & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+* Mise à jour
+$$
+P^{(k)}_{ij}=\left\{
+\begin{array}{ll}
+ P^{(k-1)}_{\mathbf{i}j}, & \mbox{si } d_{ij}^{(k)}\leq d_{ik}^{(k-1)}+d_{kj}^{(k-1)}\\
+ P^{(k-1)}_{\mathbf{k}j}, & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+. . .
+
+* Moralité: si le chemin est plus court en passant par $k$, alors il faut utiliser son prédécesseur!
+
+# Algorithme de Floyd--Warshall
+
+## La matrice de précédence (pseudo-code, 3min)
+
+. . .
+
+```C
+matrice, matrice floyd_warshall(distance, n, w)
+ pour k de 1 Ã n
+ pour i de 1 Ã n
+ pour j de 1 Ã n
+ n_distance = distance[i][k] + distance[k][j]
+ if n_distance < distance[i][j]
+ distance[i][j] = n_distance
+ précédence[i][j] = précédence[k][j]
+ retourne distance, précédence
+```
+
+# Algorithme de Floyd--Warshall (exercice)
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+## Que vaut $P^{(0)}$ (3min)?
+
+. . .
+
+$$
+P^{(0)}=\begin{bmatrix}
+- & 1 & 1 & - & 1 \\
+2 & - & 2 & - & 2 \\
+3 & 3 & - & 3 & 3 \\
+4 & - & - & - & 4 \\
+- & - & - & 5 & - \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exercice)
+
+
+::: columns
+
+:::: column
+
+
+
+
+::::
+
+:::: column
+
+## Que vaut $P^{(5)}$ (10min)?
+
+. . .
+
+$$
+P^{(5)}=\begin{bmatrix}
+- & 1 & 1 & 5 & 1 \\
+2 & - & 1 & 5 & 2 \\
+2 & 3 & - & 3 & 3 \\
+4 & 1 & 1 & - & 1 \\
+4 & 1 & 1 & 5 & - \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Exercice: retrouver le chemin entre 1 et 4 (5min)
+
+$$
+P=\begin{bmatrix}
+- & 1 & 1 & 5 & 1 \\
+2 & - & 1 & 5 & 2 \\
+2 & 3 & - & 3 & 3 \\
+4 & 1 & 1 & - & 4 \\
+4 & 1 & 1 & 5 & - \\
+\end{bmatrix}
+$$
+
+. . .
+
+## Solution
+
+* Le sommet $5=P_{14}$, on a donc, $5\rightarrow 4$, on veut connaître le prédécesseur de 5.
+* Le sommet $1=P_{15}$, on a donc, $1\rightarrow 5\rightarrow 4$. The end.
+
+# Exercice complet
+
+## Appliquer l'algorithme de Floyd--Warshall au graphe suivant
+
+{width=50%}
+
+* Bien indiquer l'état de $D$ et $P$ à chaque étape!
+* Ne pas oublier de faire la matrice d'adjacence évidemment...
+
+# La suite
+
+* Sans transition.... la suite!
+
+# Trouver un réseau électrique pour
+
+
+
+# Solution: pas optimale
+
+
+
+* La longueur totale des câbles est super longue!
+
+# Solution: optimale
+
+
+
+# Formalisation: Les arbres couvrants
+
+## Application: minimisation des coûts
+
+* Équipement d'un lotissement avec des lignes électriques/téléphoniques, des canalisations, ...
+
+. . .
+
+* Pour réduire les coûts, on cherche à minimiser la longueur totale des câbles/tuyaux.
+
+. . .
+
+* Les lignes/tuyaux forment un *arbre couvrant*.
+
+. . .
+
+* La meilleure option est un *arbre couvrant minimal*.
+
+
+# Formalisation: Les arbres couvrants
+
+* Qu'est-ce qu'un arbre couvrant? Des idées? De quel objet on part? Où va-t-on?
+
+. . .
+
+* Un arbre couvrant d'un graphe non-orienté et connexe est:
+ * un arbre inclus dans le graphe qui connecte tous les sommets du graphe.
+
+. . .
+
+
+
+# Arbres couvrants
+
+* Quels algorithmes que nous avons déjà vus permettent de construire des arbres couvrants?
+
+. . .
+
+* Les parcours en largeur et en profondeur!
+
+. . .
+
+
+
+# Arbres couvrants minimaux
+
+* Un *arbre couvrant minimal* est un sous-graphe d'un graphe non-orienté pondéré $G(V,E)$, tel quel:
+ * C'est un arbre (graphe acyclique);
+ * Il couvre tous les sommets de $G$ et contient $|V|-1$ arêtes;
+ * Le coût total associé aux arêtes de l'arbre est minimum parmi tous les arbres couvrants possibles.
+
+. . .
+
+* Est-il unique?
+
+. . .
+
+* Pas forcément.
+
+# Arbres couvrants minimaux
+
+* Comment générer un arbre couvrant minimal?
+
+
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+## On part de `e` (au hasard)
+
+
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment?
+
+
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `e->d`
+
+
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment?
+
+
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `d->a`
+
+
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment?
+
+
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `d->c`
+
+
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment?
+
+
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `e->b`
+
+
+
+* Game over!
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+## Structures de données
+
+* Dans quoi allons nous stocker les sommets?
+
+. . .
+
+* File de priorité min.
+* Autre chose?
+
+. . .
+
+* Tableau des distances (comme pour Dijkstra).
+* Autre chose?
+
+. . .
+
+* Tableau des parents (presque comme pour Dijkstra).
+* Autre chose?
+
+. . .
+
+* Non.
+
+# Algorithme de Prim
+
+## Initialisation: Pseudo-code (2min)
+
+. . .
+
+```C
+file_priorité, distance, parent initialisation(graphe)
+ r = aléatoire(graphe)
+ distance[r] = 0
+ parent[r] = indéfini
+ fp = file_p_vide()
+ pour v dans sommets(graphe)
+ si v != r
+ distance[v] = infini
+ parent[v] = indéfini
+ fp = enfiler(fp, v, distance[v])
+ retourne fp, distance, parent
+```
+
+# Algorithme de Prim
+
+## Algorithme: Pseudo-code (5min)
+
+. . .
+
+```C
+sommets, parent prim(file_priorité, distance, parent)
+ sommets = vide
+ tant que !est_vide(file_priorité)
+ u, fp = défiler(file_priorité)
+ sommets = insérer(sommets, u)
+ pour v dans voisinage de u et pas dans sommets
+ // ou dans file_priorité
+ si w(u, v) < distance[v]
+ parent[w] = u
+ distance[w] = w(u, v)
+ fp = changer_priorité(fp, w, w(u, v))
+ retourne sommets, parent
+```
+
+# Exemple d'algorithme de Prim
+
+::: columns
+
+:::: {.column width="40%"}
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+```
+FP | e | d | b | c | a |
+----------------------------------
+D | 0 | inf | inf | inf | inf |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | - | - | - | - |
+```
+
+## Devient?
+
+. . .
+
+```
+FP | d | b | c | a |
+----------------------------
+D | 4 | 5 | 5 | inf |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | e | - |
+```
+
+::::
+
+:::
+
+# Exemple d'algorithme de Prim
+
+::: columns
+
+:::: {.column width="40%"}
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+```
+FP | d | b | c | a |
+----------------------------
+D | 4 | 5 | 5 | inf |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | e | - |
+```
+
+## Devient?
+
+. . .
+
+```
+FP | a | c | b |
+----------------------
+D | 2 | 4 | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+::::
+
+:::
+
+# Exemple d'algorithme de Prim
+
+::: columns
+
+:::: {.column width="40%"}
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+```
+FP | a | c | b |
+----------------------
+D | 2 | 4 | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+## Devient?
+
+. . .
+
+```
+FP | c | b |
+----------------
+D | 4 | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+::::
+
+:::
+
+# Exemple d'algorithme de Prim
+
+::: columns
+
+:::: {.column width="40%"}
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+```
+FP | c | b |
+----------------
+D | 4 | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+## Devient?
+
+. . .
+
+```
+FP | b |
+----------
+D | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+::::
+
+:::
+
+# Exemple d'algorithme de Prim
+
+::: columns
+
+:::: {.column width="40%"}
+
+## Un exemple
+
+
+
+::::
+
+:::: column
+
+```
+FP | b |
+----------
+D | 5 |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+## Devient?
+
+. . .
+
+```
+FP |
+----
+D |
+
+ | e | d | b | c | a |
+----------------------------------
+P | - | e | e | d | d |
+```
+
+::::
+
+:::
+
+# Exercice: algorithme de Prim
+
+## Appliquer l'algorithme de Prim à (15min):
+
+
+
+# Exercice: algorithme de Prim
+
+## Solution
+
+
+
+# Complexité de l'algorithme de Prim
+
+\footnotesize
+
+```C
+file_priorité, distance, parent initialisation(graphe)
+ // choix r et initialisation
+ pour v dans sommets(graphe)
+O(|V|) // initialisation distance et parent
+ fp = enfiler(fp, v, distance[v])
+ retourne fp, distance, parent
+sommets, parent prim(file_priorité, distance, parent)
+ sommets = vide
+ tant que !est_vide(file_priorité)
+O(|V|) u, fp = défiler(file_priorité)
+ sommets = insérer(sommets, u)
+ pour v dans voisinage de u et pas dans sommets
+ O(|E|) si w(u, v) < distance[v]
+ // mà j dista + parent
+ O(|V|) fp = changer_priorité(fp, w, w(u, v))
+ retourne sommets, parent
+```
+
+* $O(|V|)+O(|E|)+O(|V|^2)=O(|E|+|V|^2)$
+* Remarque: $O(|E|)$ n'est pas mutliplié par $O(|V|)$, car les arêtes parcourues qu'une fois en **tout**.
diff --git a/slides_2022/cours_3.md b/slides_2022/cours_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..f2f3788e69a0a66f7e5c83042a010644f15b0cf1
--- /dev/null
+++ b/slides_2022/cours_3.md
@@ -0,0 +1,339 @@
+---
+title: "Introduction aux algorithmes"
+date: "2022-10-05"
+patat:
+ eval:
+ tai:
+ command: fish
+ fragment: false
+ replace: true
+ ccc:
+ command: fish
+ fragment: false
+ replace: true
+ images:
+ backend: auto
+---
+
+# Rappel (1/2)
+
+## Quels algos avons-nous vu la semaine passée?
+
+. . .
+
+* L'algorithme de la factorielle.
+* L'algorithme du PPCM.
+* Le début de l'algorithme du PGCD.
+
+# Rappel (2/2)
+
+## Algorithme du PPCM?
+
+. . .
+
+```C
+int main() {
+ int m = 15, n = 12;
+ int mult_m = m, mult_n = n;
+ while (mult_m != mult_n) {
+ if (mult_m > mult_n) {
+ mult_n += n;
+ } else {
+ mult_m += m;
+ }
+ }
+ printf("Le ppcm de %d et %d est %d\n", n, m, mult_m);
+}
+```
+
+# Le calcul du PGCD (1/5)
+
+## Définition
+
+Le plus grand commun diviseur (PGCD) de deux nombres entiers non nuls est le
+plus grand entier qui les divise en même temps.
+
+## Exemples:
+
+```C
+PGCD(3, 4) = 1,
+PGCD(4, 6) = 2,
+PGCD(5, 15) = 5.
+```
+
+. . .
+
+## Mathématiquement
+
+Décomposition en nombres premiers:
+
+$$
+36 = 2^2\cdot 3^2,\quad 90=2\cdot 5\cdot 3^2,
+$$
+On garde tous les premiers à la puissance la plus basse
+$$
+PGCD(36, 90)=2^{\min{1,2}}\cdot 3^{\min{2,2}}\cdot 5^{\min{0,1}}=18.
+$$
+
+# Le calcul du PGCD (2/5)
+
+## Algorithme
+
+Par groupe de 3 (5-10min):
+
+* réfléchissez à un algorithme alternatif donnant le PGCD de deux nombres;
+* écrivez l'algorithme en pseudo-code.
+
+. . .
+
+## Exemple d'algorithme
+
+```C
+PGCD(36, 90):
+90 % 36 != 0 // otherwise 36 would be PGCD
+90 % 35 != 0 // otherwise 35 would be PGCD
+90 % 34 != 0 // otherwise 34 would be PGCD
+...
+90 % 19 != 0 // otherwise 19 would be PGCD
+90 % 18 == 0 // The end!
+```
+
+* 18 modulos, 18 assignations, et 18 comparaisons.
+
+# Le calcul du PGCD (3/5)
+
+## Transcrivez cet exemple en algorithme (groupe de 3) et codez-le (5-10min)!
+
+. . .
+
+## Optimisation
+
+* Combien d'additions / comparaisons au pire?
+* Un moyen de le rendre plus efficace?
+
+. . .
+
+## Tentative de correction
+
+```C
+void main() {
+ int n = 90, m = 78;
+ int gcd = 1;
+ for (int div = n; div >= 2; div--) { // div = m, sqrt(n)
+ if (n % div == 0 && m % div == 0) {
+ gcd = div;
+ break;
+ }
+ }
+ printf("Le pgcd de %d et %d est %d\n", n, m, gcd);
+}
+```
+
+# Le calcul du PGCD (4/5)
+
+## Réusinage: l'algorithme d'Euclide
+
+`Dividende = Diviseur * Quotient + Reste`
+
+```C
+PGCD(35, 60):
+35 = 60 * 0 + 35 // 60 -> 35, 35 -> 60
+60 = 35 * 1 + 25 // 35 -> 60, 25 -> 35
+35 = 25 * 1 + 10 // 25 -> 35, 20 -> 25
+25 = 10 * 2 + 5 // 10 -> 25, 5 -> 10
+10 = 5 * 2 + 0 // PGCD = 5!
+```
+
+. . .
+
+## Algorithme
+
+Par groupe de 3 (5-10min):
+
+* analysez l'exemple ci-dessus;
+* transcrivez le en pseudo-code.
+
+# Le calcul du PGCD (5/5)
+
+## Pseudo-code
+
+```C
+entier pgcd(m, n) {
+ tmp_n = n
+ tmp_m = m
+ tant que (tmp_m ne divise pas tmp_n) {
+ tmp = tmp_n
+ tmp_n = tmp_m
+ tmp_m = tmp modulo tmp_m
+ }
+ retourne tmp_m
+}
+```
+
+# Le code du PGCD de 2 nombres
+
+## Implémentez le pseudo-code et postez le code sur matrix (5min).
+
+. . .
+
+## Un corrigé possible
+
+```C
+#include <stdio.h>
+void main() {
+ int n = 90;
+ int m = 78;
+ printf("n = %d et m = %d\n", n, m);
+ int tmp_n = n;
+ int tmp_m = m;
+ while (tmp_n%tmp_m > 0) {
+ int tmp = tmp_n;
+ tmp_n = tmp_m;
+ tmp_m = tmp % tmp_m;
+ }
+ printf("Le pgcd de %d et %d est %d\n", n, m, tmp_m);
+}
+```
+
+# Quelques algorithmes simples
+
+* Remplissage d'un tableau et recherche de la valeur minimal
+* Anagrammes
+* Palindromes
+* Crible d'ératosthène
+
+. . .
+
+* Ces algorithme nécessitent d'utiliser des **tableaux**.
+
+# Collections: tableaux statiques
+
+\footnotesize
+
+* Objets de même type: leur nombre est **connu à la compilation**;
+* Stockés contigüement en mémoire (très efficace);
+
+ ```C
+ #define SIZE 10
+ int entiers[] = {2, 1, 4, 5, 7}; // taille 5, initialisé
+ int tab[3]; // taille 3, non initialisé
+ float many_floats[SIZE]; // taille 10, non initialisé
+ ```
+* Les indices sont numérotés de `0` à `taille-1`;
+
+ ```C
+ int premier = entier[0]; // premier = 2
+ int dernier = entier[4]; // dernier = 7
+ ```
+* Les tableaux sont **non-initialisés** par défaut;
+* Les bornes ne sont **jamais** vérifiées.
+
+ ```C
+ int indetermine_1 = tab[1]; // undefined behavior
+ int indetermine_2 = entiers[5]; // UB
+ ```
+
+# Remarques
+
+* Depuis `C99` possibilité d'avoir des tableaux dont la taille est *inconnue Ã
+ la compilation*;
+
+ ```C
+ int size;
+ scanf("%d", &size);
+ char string[size];
+ ```
+
+ . . .
+
+* Considéré comme une mauvaise pratique: que se passe-t-il si `size == 1e9`?
+* On préfère utiliser l'allocation **dynamique** de mémoire pour ce genre de
+ cas-là (spoiler du futur du cours).
+
+# Initialisation
+
+* Les variables ne sont **jamais** initialisées en `C` par défaut.
+* Question: Que contient le tableau suivant?
+
+ ```C
+ double tab[4];
+ ```
+
+. . .
+
+* Réponse: On en sait absolument rien!
+* Comment initialiser un tableau?
+
+. . .
+
+```C
+#define SIZE 10
+double tab[SIZE];
+for (int i = 0; i < SIZE; ++i) {
+ tab[i] = rand() / (double)RAND_MAX * 10.0 - 5.0;
+ // tab[i] contient un double dans [-5;5]
+}
+```
+
+# Recherche du minimum dans un tableau (1/2)
+
+## Problématique
+
+Trouver la valeur minimale contenue dans un tableau et l'indice de l'élément le plus petit.
+
+## Écrire un pseudo-code résolvant ce problème (groupe de 3), 2min
+
+. . .
+
+```C
+index = 0
+min = tab[0]
+pour i de 1 Ã SIZE - 1
+ si min > tab[i]
+ min = tab[i]
+ index = i
+```
+
+# Recherche du minimum dans un tableau (2/2)
+
+## Implémenter ce bout de code en C (groupe de 3), 4min
+
+. . .
+
+```C
+int index = 0;
+float min = tab[0];
+for (int i = 1; i < SIZE; ++i) {
+ if min > tab[i] {
+ min = tab[i];
+ index = i;
+ }
+}
+```
+
+# Tri par sélection (1/2)
+
+## Problématique
+
+Trier un tableau par ordre croissant.
+
+## Idée d'algorithme
+
+```C
+ind = 0
+tant que (ind < SIZE-1)
+ Trouver le minimum du tableau, tab_min[ind:SIZE].
+ Échanger tab_min avec tab[ind]
+ ind += 1
+```
+
+# Tri par sélection (2/2)
+
+## Implémentation par groupe de 3
+
+* Initialiser aléatoirement un tableau de `double` de taille 10;
+* Afficher le tableau;
+* Trier par sélection le tableau;
+* Afficher le résultat trié;
+* Vérifier algorithmiquement que le résultat est bien trié.
+
diff --git a/slides_2022/cours_4.md b/slides_2022/cours_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..773109671559e3109ef35f25675fb3ead50b6163
--- /dev/null
+++ b/slides_2022/cours_4.md
@@ -0,0 +1,851 @@
+---
+title: "Introduction aux algorithmes"
+date: "2022-10-19"
+---
+
+# Rappel
+
+## Quel est l'algorithme du tri par sélection?
+
+. . .
+
+1. Soit un tableau d'entiers, `tab[0:SIZE-1]` et `i=0`.
+2. Trouver l'indice, `j`, de `tab[i:SIZE-2]` où la valeur est minimale.
+3. Échanger `tab[i]` et `tab[j]`.
+4. `i+=1` et revenir à 2, tant que `j < SIZE-2`.
+
+# Un type de tableau particulier
+
+## Les chaînes de caractères
+
+```C
+string = tableau + char + magie noire
+```
+
+# Le type `char`{.C}
+
+- Le type `char`{.C} est utilisé pour représenter un caractère.
+- C'est un entier 8 bits signé.
+- En particulier:
+ - Écrire
+
+ ```C
+ char c = 'A';
+ ```
+ - Est équivalent à :
+
+ ```C
+ char c = 65;
+ ```
+- Les fonctions d'affichage interprètent le nombre comme sa valeur ASCII.
+
+# Chaînes de caractères (strings)
+
+- Chaîne de caractère `==` tableau de caractères **terminé par la valeur** `'\0'`{.C} ou `0`{.C}.
+
+## Exemple
+
+```C
+char *str = "HELLO !";
+char str[] = "HELLO !";
+```
+
+Est représenté par
+
+| `char` | `H` | `E` | `L` | `L` | `O` | | `!` | `\0`|
+|---------|------|------|------|------|------|------|------|-----|
+| `ASCII` | `72` | `69` | `76` | `76` | `79` | `32` | `33` | `0` |
+
+. . .
+
+## A quoi sert le `\0`?
+
+. . .
+
+Permet de connaître la fin de la chaîne de caractères (pas le cas des autres
+sortes de tableaux).
+
+# Syntaxe
+
+```C
+char name[5];
+name[0] = 'P'; // = 70;
+name[1] = 'a'; // = 97;
+name[2] = 'u'; // = 117;
+name[3] = 'l'; // = 108;
+name[4] = '\0'; // = 0;
+char name[] = {'P', 'a', 'u', 'l', '\0'};
+char name[5] = "Paul";
+char name[] = "Paul";
+char name[100] = "Paul is not 100 characters long.";
+```
+
+# Fonctions
+
+- Il existe une grande quantités de fonction pour la manipulation de chaînes de caractères dans `string.h`.
+- Fonctions principales:
+
+ ```C
+ // longueur de la chaîne (sans le \0)
+ size_t strlen(char *str);
+ // copie jusqu'Ã un \0
+ char *strcpy(char *dest, const char *src);
+ // copie len char
+ char *strncpy(char *dest, const char *src, size_t len);
+ // compare len chars
+ int strncmp(char *str1, char *str2, size_t len);
+ // compare jusqu'Ã un \0
+ int strcmp(char *str1, char *str2);
+ ```
+
+- Pour avoir la liste complète: `man string`.
+
+. . .
+
+## Quels problèmes peuvent se produire avec `strlen`, `strcpy`, `strcmp`?
+
+# Les anagrammes
+
+## Définition
+
+Deux mots sont des anagrammes l'un de l'autre quand ils contiennent les mêmes
+lettres mais dans un ordre différent.
+
+## Exemple
+
+| `t` | `u` | `t` | `u` | `t` | `\0` | ` ` | ` ` |
+|------|------|------|------|------|------|------|-----|
+| `t` | `u` | `t` | `t` | `u` | `\0` | ` ` | ` ` |
+
+
+## Problème: Trouvez un algorithme pour déterminer si deux mots sont des anagrammes.
+
+# Les anagrammes
+
+## Il suffit de:
+
+1. Trier les deux mots.
+2. Vérifier s'ils contiennent les mêmes lettres.
+
+## Implémentation ensemble
+
+```C
+int main() { // pseudo C
+ tri(mot1);
+ tri(mot2);
+ if egalite(mot1, mot2) {
+ // anagrammes
+ } else {
+ // pas anagrammes
+ }
+}
+```
+
+<!-- TODO: Live implémentation hors des cours? -->
+
+# Les palindromes
+
+Mot qui se lit pareil de droite à gauche que de gauche à droite:
+
+. . .
+
+* rotor, kayak, ressasser, ...
+
+## Problème: proposer un algorithme pour détecter un palindrome
+
+. . .
+
+## Solution 1
+
+```C
+while (first_index < last_index {
+ if (mot[first_index] != mot [last_index]) {
+ return false;
+ }
+ first_index += 1;
+ last_index -= 1;
+}
+return true;
+```
+
+. . .
+
+## Solution 2
+
+```C
+mot_tmp = revert(mot);
+return mot == mot_tmp;
+```
+
+# Crible d'Ératosthène
+
+Algorithme de génération de nombres premiers.
+
+## Exercice
+
+* À l'aide d'un tableau de booléens,
+* Générer les nombres premiers plus petits qu'un nombre $N$
+
+## Pseudo-code
+
+* Par groupe de trois, réfléchir à un algorithme.
+
+## Programme en C
+
+* Implémenter l'algorithme et le poster sur le salon `Element`.
+
+# Crible d'Ératosthène: solution
+
+\footnotesize
+
+```C
+#include <stdio.h>
+#include <stdbool.h>
+#define SIZE 51
+int main() {
+ bool tab[SIZE];
+ for (int i=0;i<SIZE;i++) {
+ tab[i] = true;
+ }
+ for (int i = 2; i < SIZE; i++) {
+ if (tab[i]) {
+ printf("%d ", i);
+ int j = i;
+ while (j < SIZE) {
+ j += i;
+ tab[j] = false;
+ }
+ }
+ }
+ printf("\n");
+}
+```
+
+
+# Réusinage de code (refactoring)
+
+## Le réusinage est?
+
+. . .
+
+* le processus de restructuration d'un programme:
+ * en modifiant son design,
+ * en modifiant sa structure,
+ * en modifiant ses algorithmes
+ * mais en **conservant ses fonctionalités**.
+
+. . .
+
+## Avantages?
+
+. . .
+
+* Amélioration de la lisibilité,
+* Amélioration de la maintenabilité,
+* Réduction de la complexité.
+
+. . .
+
+## "Make it work, make it nice, make it fast", Kent Beck.
+
+. . .
+
+## Exercice:
+
+* Réusiner le code se trouvant sur
+ [Cyberlearn](https://cyberlearn.hes-so.ch/mod/resource/view.php?id=1627712).
+
+# Tableau à deux dimensions (1/4)
+
+## Mais qu'est-ce donc?
+
+. . .
+
+* Un tableau où chaque cellule est un tableau.
+
+## Quels cas d'utilisation?
+
+. . .
+
+* Tableau à double entrée;
+* Image;
+* Écran (pixels);
+* Matrice (mathématique);
+
+# Tableau à deux dimensions (2/4)
+
+## Exemple: tableau à 3 lignes et 4 colonnes d'entiers
+
++-----------+-----+-----+-----+-----+
+| `indices` | `0` | `1` | `2` | `3` |
++-----------+-----+-----+-----+-----+
+| `0` | `7` | `4` | `7` | `3` |
++-----------+-----+-----+-----+-----+
+| `1` | `2` | `2` | `9` | `2` |
++-----------+-----+-----+-----+-----+
+| `2` | `4` | `8` | `8` | `9` |
++-----------+-----+-----+-----+-----+
+
+## Syntaxe
+
+```C
+int tab[3][4]; // déclaration d'un tableau 4x3
+tab[2][1]; // accès à la case 2, 1
+tab[2][1] = 14; // assignation de 14 Ã la position 2, 1
+```
+
+# Tableau à deux dimensions (3/4)
+
+## Exercice: déclarer et initialiser aléatoirement un tableau `50x100`
+
+. . .
+
+```C
+#define NX 50
+#define NY 100
+int tab[NX][NY];
+for (int i = 0; i < NX; ++i) {
+ for (int j = 0; j < NY; ++j) {
+ tab[i][j] = rand() % 256; // 256 niveaux de gris
+ }
+}
+```
+
+## Exercice: afficher le tableau
+
+. . .
+
+```C
+for (int i = 0; i < NX; ++i) {
+ for (int j = 0; j < NY; ++j) {
+ printf("%d ", tab[i][j]);
+ }
+ printf("\n");
+}
+```
+
+# Tableau à deux dimensions (4/4)
+
+## Attention
+
+* Les éléments ne sont **jamais** initialisés.
+* Les bornes ne sont **jamais** vérifiées.
+
+ ```C
+ int tab[3][2] = { {1, 2}, {3, 4}, {5, 6} };
+ printf("%d\n", tab[2][1]); // affiche?
+ printf("%d\n", tab[10][9]); // affiche?
+ printf("%d\n", tab[3][1]); // affiche?
+ ```
+
+# La couverture de la reine
+
+* Aux échecs la reine peut se déplacer horizontalement et verticalement
+* Pour un échiquier `5x6`, elle *couvre* les cases comme ci-dessous
+
++-----+-----+-----+-----+-----+-----+-----+
+| ` ` | `0` | `1` | `2` | `3` | `4` | `5` |
++-----+-----+-----+-----+-----+-----+-----+
+| `0` | `*` | ` ` | `*` | ` ` | `*` | ` ` |
++-----+-----+-----+-----+-----+-----+-----+
+| `1` | ` ` | `*` | `*` | `*` | ` ` | ` ` |
++-----+-----+-----+-----+-----+-----+-----+
+| `2` | `*` | `*` | `R` | `*` | `*` | `*` |
++-----+-----+-----+-----+-----+-----+-----+
+| `3` | ` ` | `*` | `*` | `*` | ` ` | ` ` |
++-----+-----+-----+-----+-----+-----+-----+
+| `4` | `*` | ` ` | `*` | ` ` | `*` | ` ` |
++-----+-----+-----+-----+-----+-----+-----+
+
+## Exercice
+
+* En utilisant les conditions, les tableaux à deux dimensions, et des
+ `char` uniquement.
+* Implémenter un programme qui demande à l'utilisateur d'entrer les
+ coordonnées de la reine et affiche un tableau comme ci-dessus pour un
+ échiquier `8x8`.
+
+## Poster le résultat sur `Element`
+
+# Types énumérés (1/2)
+
+* Un **type énuméré**: ensemble de *variantes* (valeurs constantes).
+* En `C` les variantes sont des entiers numérotés à partir de 0.
+
+ ```C
+ enum days {
+ monday, tuesday, wednesday,
+ thursday, friday, saturday, sunday
+ };
+ ```
+* On peut aussi donner des valeurs "custom"
+
+ ```C
+ enum days {
+ monday = 2, tuesday = 8, wednesday = -2,
+ thursday = 1, friday = 3, saturday = 12, sunday = 9
+ };
+ ```
+* S'utilise comme un type standard et un entier
+
+ ```C
+ enum days d = monday;
+ (d + 2) == tuesday + tuesday; // true
+ ```
+
+# Types énumérés (2/2)
+
+* Très utile dans les `switch ... case`{.C}
+
+ ```C
+ enum days d = monday;
+ switch (d) {
+ case monday:
+ // trucs
+ break;
+ case tuesday:
+ printf("0 ou 1\n");
+ break;
+ }
+ ```
+* Le compilateur vous prévient qu'il en manque!
+
+# Utilisation des types énumérés
+
+## Réusiner votre couverture de la reine avec des `enum`
+
+# Représentation des nombres (1/2)
+
+* Le nombre `247`.
+
+## Nombres décimaux: Les nombres en base 10
+
++--------+--------+--------+
+| $10^2$ | $10^1$ | $10^0$ |
++--------+--------+--------+
+| `2` | `4` | `7` |
++--------+--------+--------+
+
+$$
+247 = 2\cdot 10^2 + 4\cdot 10^1 + 7\cdot 10^0.
+$$
+
+# Représentation des nombres (2/2)
+
+* Le nombre `11110111`.
+
+## Nombres binaires: Les nombres en base 2
+
++-------+-------+-------+-------+-------+-------+-------+-------+
+| $2^7$ | $2^6$ | $2^5$ | $2^4$ | $2^3$ | $2^2$ | $2^1$ | $2^0$ |
++-------+-------+-------+-------+-------+-------+-------+-------+
+| `1` | `1` | `1` | `1` | `0` | `1` | `1` | `1` |
++-------+-------+-------+-------+-------+-------+-------+-------+
+
+$$
+1\cdot 2^7 + 1\cdot 2^6 +1\cdot 2^5 +1\cdot 2^4 +0\cdot 2^3 +1\cdot 2^2
++1\cdot 2^1 +1\cdot 2^0
+$$
+
+. . .
+
+$$
+= 247.
+$$
+
+# Conversion de décimal à binaire (1/2)
+
+## Convertir 11 en binaire?
+
+. . .
+
+* On décompose en puissances de 2 en partant de la plus grande possible
+
+ ```
+ 11 / 8 = 1, 11 % 8 = 3
+ 3 / 4 = 0, 3 % 4 = 3
+ 3 / 2 = 1, 3 % 2 = 1
+ 1 / 1 = 1, 1 % 1 = 0
+ ```
+* On a donc
+
+ $$
+ 1011 \Rightarrow 1\cdot 2^3 + 0\cdot 2^2 + 1\cdot 2^1 + 1\cdot
+ 2^0=11.
+ $$
+
+# Conversion de décimal à binaire (2/2)
+
+## Convertir un nombre arbitraire en binaire: 247?
+
+* Par groupe établir un algorithme.
+
+. . .
+
+## Algorithme
+
+1. Initialisation
+
+ ```C
+ num = 247
+ while (2^N < num) {
+ N += 1
+ }
+ ```
+
+. . .
+
+2. Boucle
+
+ ```C
+ while (N >= 0) {
+ bit = num / 2^N
+ num = num % 2^N
+ N += 1
+ }
+ ```
+
+# Les additions en binaire
+
+Que donne l'addition `1101` avec `0110`?
+
+* L'addition est la même que dans le système décimal
+
+ ```
+ 1101 8+4+0+1 = 13
+ + 0110 + 0+4+2+0 = 6
+ ------- -----------------
+ 10011 16+0+0+2+1 = 19
+ ```
+* Les entiers sur un ordinateur ont une précision **fixée** (ici 4 bits).
+* Que se passe-t-il donc ici?
+
+. . .
+
+## Dépassement de capacité: le nombre est "tronqué"
+
+* `10011 (19) -> 0011 (3)`.
+* On fait "le tour"."
+
+# Entier non-signés minimal/maximal
+
+* Quel est l'entier non-signé maximal représentable avec 4 bit?
+
+. . .
+
+$$
+(1111)_2 = 8+4+2+1 = 15
+$$
+
+* Quel est l'entier non-signé minimal représentable avec 4 bit?
+
+. . .
+
+$$
+(0000)_2 = 0+0+0+0 = 0
+$$
+
+* Quel est l'entier non-signé min/max représentable avec N bit?
+
+. . .
+
+$$
+0\mbox{ et }2^N-1.
+$$
+
+* Donc `uint32_t?` maximal est?
+
+. . .
+
+$$
+4294967295
+$$
+
+
+# Les multiplications en binaire (1/2)
+
+Que donne la multiplication de `1101` avec `0110`?
+
+* L'addition est la même que dans le système décimal
+
+ ```
+ 1101 13
+ * 0110 * 6
+ --------- --------------
+ 0000 78
+ 11010
+ 110100
+ + 0000000
+ --------- --------------
+ 1001110 64+0+0+8+4+2+0
+ ```
+
+# Les multiplications en binaire (2/2)
+
+## Que fait la multiplication par 2?
+
+. . .
+
+* Décalage de un bit vers la gauche!
+
+ ```
+ 0110
+ * 0010
+ ---------
+ 0000
+ + 01100
+ ---------
+ 01100
+ ```
+
+. . .
+
+## Que fait la multiplication par $2^N$?
+
+. . .
+
+* Décalade de $N$ bits vers la gauche!
+
+# Entiers signés (1/2)
+
+Pas de nombres négatifs encore...
+
+## Comment faire?
+
+. . .
+
+## Solution naïve:
+
+* On ajoute un bit de signe (le bit de poids fort):
+
+ ```
+ 00000010: +2
+ 10000010: -2
+ ```
+
+## Problèmes?
+
+. . .
+
+* Il y a deux zéros (pas trop grave): `10000000` et `00000000`
+* Les additions différentes que pour les non-signés (très grave)
+
+ ```
+ 00000010 2
+ + 10000100 + -4
+ ---------- ----
+ 10000110 = -6 != -2
+ ```
+
+# Entiers signés (2/2)
+
+## Beaucoup mieux
+
+* Complément à un:
+ * on inverse tous les bits: `1001 => 0110`.
+
+## Encore un peu mieux
+
+* Complément à deux:
+ * on inverse tous les bits,
+ * on ajoute 1 (on ignore les dépassements).
+
+. . .
+
+* Comment écrit-on `-4` en 8 bits?
+
+. . .
+
+```
+ 4 = 00000100
+ ________
+ -4 => 00000100
+
+ 11111011
+ + 00000001
+ ----------
+ 11111100
+```
+
+# Le complément à 2 (1/2)
+
+## Questions:
+
+* Comment on écrit `+0` et `-0`?
+* Comment calcule-t-on `2 + (-4)`?
+* Quel est le complément à 2 de `1000 0000`?
+
+. . .
+
+## Réponses
+
+* Comment on écrit `+0` et `-0`?
+
+ ```
+ +0 = 00000000
+ -0 = 11111111 + 00000001 = 100000000 => 00000000
+ ```
+* Comment calcule-t-on `2 + (-4)`?
+
+ ```
+ 00000010 2
+ + 11111100 + -4
+ ---------- -----
+ 11111110 -2
+ ```
+* En effet
+
+ ```
+ 11111110 => 00000001 + 00000001 = 00000010 = 2.
+ ```
+
+# Le complément à 2 (1/2)
+
+## Quels sont les entiers représentables en 8 bits?
+
+. . .
+
+```
+01111111 => 127
+10000000 => -128 // par définition
+```
+
+## Quels sont les entiers représentables sur $N$ bits?
+
+. . .
+
+$$
+-2^{N-1} ... 2^{N-1}-1.
+$$
+
+## Remarque: dépassement de capacité en `C`
+
+* Comportement indéfini!
+
+
+<!-- # TODO --
+
+<!-- ## Entiers, entiers non-signés -->
+
+<!-- ## Complément à 1, 2 -->
+
+<!-- ## Nombres à virgule flottante, simple/double précision -->
+
+# Types composés: `struct`{.C} (1/6)
+
+## Fractions
+
+* Numérateur: `int num`;
+* Dénominateur: `int denom`.
+
+## Addition
+
+```C
+int num1 = 1, denom1 = 2;
+int num2 = 1, denom2 = 3;
+int num3 = num1 * denom2 + num2 * denom1;
+int denom3 = denom1 * denom2;
+```
+
+## Pas super pratique....
+
+# Types composés: `struct`{.C} (2/6)
+
+## On peut faire mieux
+
+* Plusieurs variables qu'on aimerait regrouper dans un seul type: `struct`{.C}.
+
+```C
+struct fraction { // déclaration du type
+ int32_t num, denom;
+}
+
+struct fraction frac; // déclaration de frac
+```
+
+# Types composés: `struct`{.C} (3/6)
+
+## Simplifications
+
+- `typedef`{.C} permet de définir un nouveau type.
+
+ ```C
+ typedef unsinged int uint;
+ typedef struct fraction fraction_t;
+ typedef struct fraction {
+ int32_t num, denom;
+ } fraction_t;
+ ```
+- L'initialisation peut aussi se faire avec
+
+ ```C
+ fraction_t frac = {1, -2}; // num = 1, denom = -2
+ fraction_t frac = {.denom = 1, .num = -2};
+ fraction_t frac = {.denom = 1}; // argl! .num non initialisé
+ fraction_t frac2 = frac; // copie
+ ```
+
+# Types composés: `struct`{.C} (4/6)
+
+## Pointeurs
+
+- Comme pour tout type, on peut avoir des pointeurs vers un `struct`{.C}.
+- Les champs sont accessible avec le sélecteur `->`{.C}
+
+ ```C
+ fraction_t *frac; // on crée un pointeur
+ frac->num = 1; // seg fault...
+ frac->denom = -1; // mémoire pas allouée.
+ ```
+
+{width=50%}
+
+# Types composés: `struct`{.C} (5/6)
+
+## Initialisation
+
+- Avec le passage par **référence** on peut modifier un struct *en place*.
+- Les champs sont accessible avec le sélecteur `->`{.C}
+
+ ```C
+ void fraction_init(fraction_t *frac,
+ int32_t re, int32_t im)
+ {
+ // frac a déjà été allouée
+ frac->num = frac;
+ frac->denom = denom;
+ }
+ int main() {
+ fraction_t frac; // on alloue une fraction
+ fraction_init(&frac, 2, -1); // on l'initialise
+ }
+ ```
+
+# Types composés: `struct`{.C} (6/6)
+
+## Initialisation version copie
+
+* On peut allouer une fraction, l'initialiser et le retourner.
+* La valeur retournée peut être copiée dans une nouvelle structure.
+
+ ```C
+ fraction_t fraction_create(int32_t re, int32_t im) {
+ fraction_t frac;
+ frac.num = re;
+ frac.denom = im;
+ return frac;
+ }
+ int main() {
+ // on crée une fraction et on l'initialise
+ // en copiant la fraction créé par fraction_create
+ // deux allocation et une copie
+ fraction_t frac = fraction_create(2, -1);
+ }
+ ```
+
+<!-- # TODO jusqu'aux vacances -->
+
+<!-- * Refactorisation -->
+<!-- * Tris et complexité -->
+<!-- * Récursivité -->
diff --git a/slides_2022/cours_5.md b/slides_2022/cours_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..da216103b1004e9f3683adaf07143dce7dd004f9
--- /dev/null
+++ b/slides_2022/cours_5.md
@@ -0,0 +1,834 @@
+---
+title: "Représentation des nombres et récursivité"
+date: "2022-11-02"
+---
+
+# Types énumérés (1/2)
+
+* Un **type énuméré**: ensemble de *variantes* (valeurs constantes).
+* En `C` les variantes sont des entiers numérotés à partir de 0.
+
+ ```C
+ enum days {
+ monday, tuesday, wednesday,
+ thursday, friday, saturday, sunday
+ };
+ ```
+* On peut aussi donner des valeurs "custom"
+
+ ```C
+ enum days {
+ monday = 2, tuesday = 8, wednesday = -2,
+ thursday = 1, friday = 3, saturday = 12, sunday = 9
+ };
+ ```
+* S'utilise comme un type standard et un entier
+
+ ```C
+ enum days d = monday;
+ (d + 2) == monday + monday; // true
+ ```
+
+# Types énumérés (2/2)
+
+* Très utile dans les `switch ... case`{.C}
+
+ ```C
+ enum days d = monday;
+ switch (d) {
+ case monday:
+ // trucs
+ break;
+ case tuesday:
+ printf("0 ou 1\n");
+ break;
+ }
+ ```
+* Le compilateur vous prévient qu'il en manque!
+
+# Utilisation des types énumérés
+
+## Réusiner votre couverture de la reine avec des `enum`
+
+A faire à la maison comme exercice!
+
+# Représentation des nombres (1/2)
+
+* Le nombre `247`.
+
+## Nombres décimaux: Les nombres en base 10
+
++--------+--------+--------+
+| $10^2$ | $10^1$ | $10^0$ |
++--------+--------+--------+
+| `2` | `4` | `7` |
++--------+--------+--------+
+
+$$
+247 = 2\cdot 10^2 + 4\cdot 10^1 + 7\cdot 10^0.
+$$
+
+# Représentation des nombres (2/2)
+
+* Le nombre `11110111`.
+
+## Nombres binaires: Les nombres en base 2
+
++-------+-------+-------+-------+-------+-------+-------+-------+
+| $2^7$ | $2^6$ | $2^5$ | $2^4$ | $2^3$ | $2^2$ | $2^1$ | $2^0$ |
++-------+-------+-------+-------+-------+-------+-------+-------+
+| `1` | `1` | `1` | `1` | `0` | `1` | `1` | `1` |
++-------+-------+-------+-------+-------+-------+-------+-------+
+
+$$
+1\cdot 2^7 + 1\cdot 2^6 +1\cdot 2^5 +1\cdot 2^4 +0\cdot 2^3 +1\cdot 2^2
++1\cdot 2^1 +1\cdot 2^0
+$$
+
+. . .
+
+$$
+= 247.
+$$
+
+# Conversion de décimal à binaire (1/2)
+
+## Convertir 11 en binaire?
+
+. . .
+
+* On décompose en puissances de 2 en partant de la plus grande possible
+
+ ```
+ 11 / 8 = 1, 11 % 8 = 3
+ 3 / 4 = 0, 3 % 4 = 3
+ 3 / 2 = 1, 3 % 2 = 1
+ 1 / 1 = 1, 1 % 1 = 0
+ ```
+* On a donc
+
+ $$
+ 1011 \Rightarrow 1\cdot 2^3 + 0\cdot 2^2 + 1\cdot 2^1 + 1\cdot
+ 2^0=11.
+ $$
+
+# Conversion de décimal à binaire (2/2)
+
+## Convertir un nombre arbitraire en binaire: 247?
+
+* Par groupe établir un algorithme.
+
+. . .
+
+## Algorithme
+
+1. Initialisation
+
+ ```C
+ num = 247
+ tant que (2^N < num) {
+ N += 1
+ }
+ ```
+
+. . .
+
+2. Boucle
+
+ ```C
+ tant que (N >= 0) {
+ bit = num / 2^N
+ num = num % 2^N
+ N -= 1
+ }
+ ```
+
+# Les additions en binaire
+
+Que donne l'addition `1101` avec `0110`?
+
+* L'addition est la même que dans le système décimal
+
+ ```
+ 1101 8+4+0+1 = 13
+ + 0110 + 0+4+2+0 = 6
+ ------- -----------------
+ 10011 16+0+0+2+1 = 19
+ ```
+* Les entiers sur un ordinateur ont une précision **fixée** (ici 4 bits).
+* Que se passe-t-il donc ici?
+
+. . .
+
+## Dépassement de capacité: le nombre est "tronqué"
+
+* `10011 (19) -> 0011 (3)`.
+* On fait "le tour"."
+
+# Entier non-signés minimal/maximal
+
+* Quel est l'entier non-signé maximal représentable avec 4 bit?
+
+. . .
+
+$$
+(1111)_2 = 8+4+2+1 = 15
+$$
+
+* Quel est l'entier non-signé minimal représentable avec 4 bit?
+
+. . .
+
+$$
+(0000)_2 = 0+0+0+0 = 0
+$$
+
+* Quel est l'entier non-signé min/max représentable avec N bit?
+
+. . .
+
+$$
+0\mbox{ et }2^N-1.
+$$
+
+* Donc `uint32_t?` maximal est?
+
+. . .
+
+$$
+2^{32}-1=4'294'967'295
+$$
+
+
+# Les multiplications en binaire (1/2)
+
+Que donne la multiplication de `1101` avec `0110`?
+
+* La mutliplication est la même que dans le système décimal
+
+ ```
+ 1101 13
+ * 0110 * 6
+ --------- --------------
+ 0000 78
+ 11010
+ 110100
+ + 0000000
+ --------- --------------
+ 1001110 64+0+0+8+4+2+0
+ ```
+
+# Les multiplications en binaire (2/2)
+
+## Que fait la multiplication par 2?
+
+. . .
+
+* Décalage de un bit vers la gauche!
+
+ ```
+ 0110
+ * 0010
+ ---------
+ 0000
+ + 01100
+ ---------
+ 01100
+ ```
+
+. . .
+
+## Que fait la multiplication par $2^N$?
+
+. . .
+
+* Décalade de $N$ bits vers la gauche!
+
+# Entiers signés (1/2)
+
+Pas de nombres négatifs encore...
+
+## Comment faire?
+
+. . .
+
+## Solution naïve:
+
+* On ajoute un bit de signe (le bit de poids fort):
+
+ ```
+ 00000010: +2
+ 10000010: -2
+ ```
+
+## Problèmes?
+
+. . .
+
+* Il y a deux zéros (pas trop grave): `10000000` et `00000000`
+* Les additions différentes que pour les non-signés (très grave)
+
+ ```
+ 00000010 2
+ + 10000100 + -4
+ ---------- ----
+ 10000110 = -6 != -2
+ ```
+
+# Entiers signés (2/2)
+
+## Beaucoup mieux
+
+* Complément à un:
+ * on inverse tous les bits: `1001 => 0110`.
+
+## Encore un peu mieux
+
+* Complément à deux:
+ * on inverse tous les bits,
+ * on ajoute 1 (on ignore les dépassements).
+
+. . .
+
+* Comment écrit-on `-4` en 8 bits?
+
+. . .
+
+```
+ 4 = 00000100
+ ________
+ -4 => 00000100
+
+ 11111011
+ + 00000001
+ ----------
+ 11111100
+```
+
+# Le complément à 2 (1/2)
+
+## Questions:
+
+* Comment on écrit `+0` et `-0`?
+* Comment calcule-t-on `2 + (-4)`?
+* Quel est le complément à 2 de `1000 0000`?
+
+. . .
+
+## Réponses
+
+* Comment on écrit `+0` et `-0`?
+
+ ```
+ +0 = 00000000
+ -0 = 11111111 + 00000001 = 100000000 => 00000000
+ ```
+* Comment calcule-t-on `2 + (-4)`?
+
+ ```
+ 00000010 2
+ + 11111100 + -4
+ ---------- -----
+ 11111110 -2
+ ```
+* En effet
+
+ ```
+ 11111110 => 00000001 + 00000001 = 00000010 = 2.
+ ```
+
+# Le complément à 2 (2/2)
+
+## Quels sont les entiers représentables en 8 bits?
+
+. . .
+
+```
+01111111 => 127
+10000000 => -128 // par définition
+```
+
+## Quels sont les entiers représentables sur $N$ bits?
+
+. . .
+
+$$
+-2^{N-1} ... 2^{N-1}-1.
+$$
+
+## Remarque: dépassement de capacité en `C`
+
+* Comportement indéfini!
+
+
+# Nombres à virgule (1/3)
+
+## Comment manipuler des nombres à virgule?
+
+$$
+0.1 + 0.2 = 0.3.
+$$
+
+Facile non?
+
+. . .
+
+## Et ça?
+
+```C
+#include <stdio.h>
+#include <stdlib.h>
+int main(int argc, char *argv[]) {
+ float a = atof(argv[1]);
+ float b = atof(argv[2]);
+ printf("%.10f\n", (double)(a + b));
+}
+```
+
+. . .
+
+## Que se passe-t-il donc?
+
+# Nombres à virgule (2/3)
+
+## Nombres à virgule fixe
+
++-------+-------+-------+-------+-----+----------+----------+----------+----------+
+| $2^3$ | $2^2$ | $2^1$ | $2^0$ | `.` | $2^{-1}$ | $2^{-2}$ | $2^{-3}$ | $2^{-4}$ |
++-------+-------+-------+-------+-----+----------+----------+----------+----------+
+| `1` | `0` | `1` | `0` | `.` | `0` | `1` | `0` | `1` |
++-------+-------+-------+-------+-----+----------+----------+----------+----------+
+
+## Qu'est-ce ça donne en décimal?
+
+. . .
+
+$$
+2^3+2^1+\frac{1}{2^2}+\frac{1}{2^4} = 8+2+0.5+0.0625=10.5625.
+$$
+
+## Limites de cette représentation?
+
+. . .
+
+
+* Tous les nombres `> 16`.
+* Tous les nombres `< 0.0625`.
+* Tous les nombres dont la décimale est pas un multiple de `0.0625`.
+
+# Nombres à virgule (3/3)
+
+## Nombres à virgule fixe
+
+* Nombres de $0=0000.0000$ Ã $15.9375=1111.1111$.
+* Beaucoup de "trous" (au moins $0.0625$) entre deux nombres.
+
+## Solution partielle?
+
+. . .
+
+* Rajouter des bits.
+* Bouger la virgule.
+
+# Nombres à virgule flottante (1/2)
+
+## Notation scientifique
+
+* Les nombres sont représentés en terme:
+ * Une mantisse
+ * Une base
+ * Un exposant
+
+$$
+\underbrace{22.1214}_{\mbox{nombre}}=\underbrace{221214}_{\mbox{mantisse}}\cdot
+{\underbrace{10}_{\mbox{base}}}{\overbrace{^{-4}}^{\mbox{exp.}}},
+$$
+
+. . .
+
+On peut donc séparer la représentation en 2:
+
+* La mantisse
+* L'exposant
+
+# Nombres à virgule flottante (2/2)
+
+## Quel est l'avantage?
+
+. . .
+
+On peut représenter des nombres sur énormément d'ordres de grandeur avec un
+nombre de bits fixés.
+
+## Différence fondamentale avec la virgule fixe?
+
+. . .
+
+La précision des nombres est **variable**:
+
+* On a uniquement un nombre de chiffres **significatifs**.
+$$
+123456\cdot 10^{23}+ 123456\cdot 10^{-23}.
+$$
+
+## Quel inconvénient y a-t-il?
+
+. . .
+
+Ce mélange d'échelles entraîne un **perte de précision**.
+
+# Nombres à virgule flottante simple précision (1/4)
+
+Aussi appelés *IEEE 754 single-precision binary floating point*.
+
+](figs/Float_example_bare.svg)
+
+## Spécification
+
+* 1 bit de signe,
+* 8 bits d'exposant,
+* 23 bits de mantisse.
+
+$$
+(-1)^{b_{31}}\cdot 2^{(b_{30}b_{29}\dots b_{23})_{2}-127}\cdot (1.b_{22}b_{21}\dots b_{0})_{2},
+$$
+
+## Calculer la valeur décimale du nombre ci-dessus
+
+# Nombres à virgule flottante simple précision (2/4)
+
+](figs/Float_example.svg)
+
+. . .
+
+\begin{align}
+\mbox{exposant}&=\sum_{i=0}^7 b_{23+i}2^i=2^2+2^3+2^4+2^5+2^6=124-127,\\
+\mbox{mantisse}&=1+\sum_{i=1}^{23}b_{23-i}2^{-i}=1+2^{-2}=1.25,\\
+&\Rightarrow (-1)^0\cdot 2^{-3}\cdot 1.25=0.15625
+\end{align}
+
+# Nombres à virgule flottante simple précision (3/4)
+
+## Quel nombre ne peux pas être vraiment représenté?
+
+. . .
+
+## Zéro: exception pour l'exposant
+
+* Si l'exposant est `00000000` (zéro)
+$$
+(-1)^{\mbox{sign}}\cdot 2^{-126}\cdot 0.\mbox{mantisse},
+$$
+* Sinon si l'exposant est `00000001` Ã `11111110`
+$$
+\mbox{valeur normale},
+$$
+* Sinon `11111111` donne `NaN`.
+
+# Nombres à virgule flottante simple précision (4/4)
+
+## Quels sont les plus petits/grands nombres positifs représentables?
+
+. . .
+
+\begin{align}
+0\ 0\dots0\ 0\dots01&=2^{-126}\cdot 2^{-23}=1.4...\cdot
+10^{-45},\\
+0\ 1\dots10\ 1\dots1&=2^{127}\cdot (2-2^{-23})=3.4...\cdot
+10^{38}.
+\end{align}
+
+## Combien de chiffres significatifs en décimal?
+
+. . .
+
+* 24 bits ($23 + 1$) sont utiles pour la mantisse, soit $2^{24}-1$:
+ * La mantisse fait $\sim2^{24}\sim 10^7$,
+ * Ou encore $\sim \log_{10}(2^{24})\sim 7$.
+* Environ **sept** chiffres significatifs.
+
+# Nombres à virgule flottante double précision (64bits)
+
+## Spécification
+
+* 1 bit de signe,
+* 11 bits d'exposant,
+* 52 bits de mantisse.
+
+. . .
+
+## Combien de chiffres significatifs?
+
+* La mantisse fait $\sim 2^{53}\sim10^{16}$,
+* Ou encore $\sim \log_{10}(2^{53})\sim 16$,
+* Environ **seize** chiffres significatifs.
+
+## Plus petit/plus grand nombre représentable?
+
+. . .
+
+* Plus petite mantisse et exposant: $\sim 2^{-52}\cdot 2^{-1022}\sim 4\cdot 10^{-324}$,
+* Plus grande mantisse et exposant: $\sim 2\cdot 2^{1023}\sim \cdot 1.8\cdot 10^{308}$.
+
+# Précision finie (1/3)
+
+## Erreur de représentation
+
+* Les nombres réels ont potentiellement un **nombre infini** de décimales
+ * $1/3=0.\overline{3}$,
+ * $\pi=3.1415926535...$.
+* Les nombres à virgule flottante peuvent en représenter qu'un **nombre
+ fini**.
+ * $1/3\cong 0.33333$, erreur $0.00000\overline{3}$.
+ * $\pi\cong3.14159$, erreur $0.0000026535...$.
+
+On rencontre donc des **erreurs de représentation** ou **erreurs
+d'arrondi**.
+
+. . .
+
+## Et quand on calcule?
+
+* Avec deux chiffres significatifs
+\begin{align}
+&8.9+(0.02+0.04)=8.96=9.0,\\
+&(8.9+0.02)+0.04=8.9+0.04=8.9.
+\end{align}
+
+. . .
+
+## Même pas associatif!
+
+# Précision finie (2/3)
+
+## Erreur de représentation virgule flottante
+
+$$
+(1.2)_{10} = 1.\overline{0011}\cdot 2^0\Rightarrow 0\ 01111111\
+00110011001100110011010.
+$$
+Erreur d'arrondi dans les deux derniers bits et tout ceux qui viennent
+ensuite
+$$
+\varepsilon_2 = (00000000000000000000011)_2.
+$$
+Ou en décimal
+$$
+\varepsilon_{10} = 4.76837158203125\cdot 10^{-8}.
+$$
+
+# Précision finie (3/3)
+
+## Comment définir l'égalité de 2 nombres à virgule flottante?
+
+. . .
+
+Ou en d'autres termes, pour quel $\varepsilon>0$ (appelé `epsilon-machine`) on a
+$$
+1+\varepsilon = 1,
+$$
+pour un nombre à virgule flottante?
+
+. . .
+
+Pour un `float` (32 bits) la différence est Ã
+$$
+2^{-23}=1.19\cdot 10^{-7},
+$$
+Soit la précision de la mantisse.
+
+## Comment le mesurer (par groupe)?
+
+. . .
+
+```C
+float eps = 1.0;
+while ((float)1.0 + (float)0.5 * eps != (float)1.0) {
+ eps = (float)0.5 * eps;
+}
+printf("eps = %g\n", eps);
+```
+
+# Erreurs d'arrondi
+
+Et jusqu'ici on a encore pas fait d'arithmétique!
+
+## Multiplication avec deux chiffres significatifs, décimal
+
+$$
+(1.1)_{10}\cdot (1.1)_{10}=(1.21)_{10}=(1.2)_{10}.
+$$
+En continuant ce petit jeu:
+$$
+\underbrace{1.1\cdot 1.1\cdots 1.1}_{\mbox{10 fois}}=2.0.
+$$
+Alors qu'en réalité
+$$
+1.1^{10}=2.5937...
+$$
+Soit une erreur de près de 1/5e!
+
+. . .
+
+## Le même phénomène se produit (à plus petite échelle) avec les `float` ou `double`.
+
+# And now for something completely different
+
+\Huge La récursivité
+
+# Exemple de récursivité (1/2)
+
+## La factorielle
+
+```C
+int factorial(int n) {
+ if (n > 1) {
+ return n * factorial(n - 1);
+ } else {
+ return 1;
+ }
+}
+```
+
+. . .
+
+## Que se passe-t-il quand on fait `factorial(4)`?
+
+. . .
+
+## On empile les appels
+
++----------------+----------------+----------------+----------------+
+| | | | `factorial(1)` |
++----------------+----------------+----------------+----------------+
+| | | `factorial(2)` | `factorial(2)` |
++----------------+----------------+----------------+----------------+
+| | `factorial(3)` | `factorial(3)` | `factorial(3)` |
++----------------+----------------+----------------+----------------+
+| `factorial(4)` | `factorial(4)` | `factorial(4)` | `factorial(4)` |
++----------------+----------------+----------------+----------------+
+
+# Exemple de récursivité (2/2)
+
+## La factorielle
+
+```C
+int factorial(int n) {
+ if (n > 1) {
+ return n * factorial(n - 1);
+ } else {
+ return 1;
+ }
+}
+```
+
+. . .
+
+## Que se passe-t-il quand on fait `factorial(4)`?
+
+. . .
+
+## On dépile les calculs
+
++----------------+----------------+----------------+----------------+
+| `1` | | | |
++----------------+----------------+----------------+----------------+
+| `factorial(2)` | `2 * 1 = 2` | | |
++----------------+----------------+----------------+----------------+
+| `factorial(3)` | `factorial(3)` | `3 * 2 = 6` | |
++----------------+----------------+----------------+----------------+
+| `factorial(4)` | `factorial(4)` | `factorial(4)` | `4 * 6 = 24` |
++----------------+----------------+----------------+----------------+
+
+# La récursivité (1/4)
+
+## Formellement
+
+* Une condition de récursivité - qui *réduit* les cas successifs vers...
+* Une condition d'arrêt - qui retourne un résultat
+
+## Pour la factorielle, qui est qui?
+
+```C
+int factorial(int n) {
+ if (n > 1) {
+ return n * factorial(n - 1);
+ } else {
+ return 1;
+ }
+}
+```
+
+# La récursivité (2/4)
+
+## Formellement
+
+* Une condition de récursivité - qui *réduit* les cas successifs vers...
+* Une condition d'arrêt - qui retourne un résultat
+
+## Pour la factorielle, qui est qui?
+
+```C
+int factorial(int n) {
+ if (n > 1) { // Condition de récursivité
+ return n * factorial(n - 1);
+ } else { // Condition d'arrêt
+ return 1;
+ }
+}
+```
+
+# La récursivité (3/4)
+
+## Exercice: trouver l'$\varepsilon$-machine pour un `double`
+
+. . .
+
+Rappelez-vous vous l'avez fait en style **impératif** plus tôt.
+
+. . .
+
+```C
+double epsilon_machine(double eps) {
+ if (1.0 + eps != 1.0) {
+ return epsilon_machine(eps / 2.0);
+ } else {
+ return eps;
+ }
+}
+```
+
+# La récursivité (4/4)
+
+\footnotesize
+
+## Exercice: que fait ce code récursif?
+
+```C
+void recurse(int n) {
+ printf("%d ", n % 2);
+ if (n / 2 != 0) {
+ recurse(n / 2);
+ } else {
+ printf("\n");
+ }
+}
+recurse(13);
+```
+
+. . .
+
+```C
+recurse(13): n = 13, n % 2 = 1, n / 2 = 6,
+ recurse(6): n = 6, n % 2 = 0, n / 2 = 3,
+ recurse(3): n = 3, n % 2 = 1, n / 2 = 1,
+ recurse(1): n = 1, n % 2 = 1, n / 2 = 0.
+
+// affiche: 1 1 0 1
+```
+
+. . .
+
+Affiche la représentation binaire d'un nombre!
diff --git a/slides_2022/cours_6.md b/slides_2022/cours_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..fcb044df3c12c317246c477637d0ff0da47d4e85
--- /dev/null
+++ b/slides_2022/cours_6.md
@@ -0,0 +1,488 @@
+---
+title: "Récursivité et complexité"
+date: "2022-11-09"
+---
+
+# La récursivité (1/2)
+
+* Code récursif
+
+ ```C
+ int factorial(int n) {
+ if (n > 1) { // Condition de récursivité
+ return n * factorial(n - 1);
+ } else { // Condition d'arrêt
+ return 1;
+ }
+ }
+```
+
+. . .
+
+* Code impératif
+
+ ```C
+ int factorial(int n) {
+ int f = 1;
+ for (int i = 1; i < n; ++i) {
+ f *= i;
+ }
+ return f;
+ }
+ ```
+
+# Exercice: réusinage et récursivité (1/4)
+
+## Réusiner le code du PGCD avec une fonction récursive
+
+## Étudier l'exécution
+
+```C
+42 = 27 * 1 + 15
+27 = 15 * 1 + 12
+15 = 12 * 1 + 3
+12 = 3 * 4 + 0
+```
+
+# Exercice: réusinage et récursivité (2/4)
+
+## Réusiner le code du PGCD avec une fonction récursive
+
+## Étudier l'exécution
+
+```C
+42 = 27 * 1 + 15 | PGCD(42, 27)
+27 = 15 * 1 + 12 | PGCD(27, 15)
+15 = 12 * 1 + 3 | PGCD(15, 12)
+12 = 3 * 4 + 0 | PGCD(12, 3)
+```
+
+# Exercice: réusinage et récursivité (3/4)
+
+## Réusiner le code du PGCD avec une fonction récursive
+
+## Étudier l'exécution
+
+```C
+42 = 27 * 1 + 15 | PGCD(42, 27)
+27 = 15 * 1 + 12 | PGCD(27, 15)
+15 = 12 * 1 + 3 | PGCD(15, 12)
+12 = 3 * 4 + 0 | PGCD(12, 3)
+```
+
+## Effectuer l'empilage - dépilage
+
+. . .
+
+```C
+PGCD(12, 3) | 3
+PGCD(15, 12) | 3
+PGCD(27, 15) | 3
+PGCD(42, 27) | 3
+```
+
+# Exercice: réusinage et récursivité (4/4)
+
+## Écrire le code
+
+. . .
+
+```C
+int pgcd(int n, int m) {
+ if (n % m > 0) {
+ return pgcd(m, n % m);
+ } else {
+ return m;
+ }
+}
+```
+
+# La suite de Fibonacci (1/2)
+
+## Règle
+
+$$
+\mathrm{Fib}(n) = \mathrm{Fib}(n-1) + \mathrm{Fib}(n-2),\quad
+\mathrm{Fib}(0)=0,\quad \mathrm{Fib}(1)=1.
+$$
+
+## Exercice: écrire la fonction $\mathrm{Fib}$ en récursif et impératif
+
+. . .
+
+## En récursif (6 lignes)
+
+```C
+int fib(int n) {
+ if (n > 1) {
+ return fib(n - 1) + fib(n - 2);
+ }
+ return n;
+}
+```
+
+# La suite de Fibonacci (2/2)
+
+## Et en impératif (11 lignes)
+
+```C
+int fib_imp(int n) {
+ int fib0 = 1;
+ int fib1 = 1;
+ int fib = n == 0 ? 0 : fib1;
+ for (int i = 2; i < n; ++i) {
+ fib = fib0 + fib1;
+ fib0 = fib1;
+ fib1 = fib;
+ }
+ return fib;
+}
+```
+
+# Exponentiation rapide ou indienne (1/4)
+
+## But: Calculer $x^n$
+
+* Quel est l'algorithmie le plus simple que vous pouvez imaginer?
+
+. . .
+
+```C
+int pow(int x, int n) {
+ if (0 == n) {
+ return 1;
+ }
+ for (int i = 1; i < n; ++i) {
+ x = x * x; // x *= x
+ }
+ return x;
+}
+```
+
+* Combien de multiplication et d'assignations en fonction de `n`?
+
+. . .
+
+* `n` assignations et `n` multiplications.
+
+# Exponentiation rapide ou indienne (2/4)
+
+* Proposez un algorithme naïf et récursif
+
+. . .
+
+```C
+int pow(x, n) {
+ if (n != 0) {
+ return x * pow(x, n-1);
+ } else {
+ return 1;
+ }
+}
+```
+
+# Exponentiation rapide ou indienne (3/4)
+
+## Exponentiation rapide ou indienne de $x^n$
+
+* Écrivons $n=\sum_{i=0}^{d-1}b_i 2^i,\ b_i=\{0,1\}$ (écriture binaire sur $d$ bits, avec
+$d\sim\log_2(n)$).
+*
+$$
+x^n={x^{2^0}}^{b_0}\cdot {x^{2^1}}^{b_1}\cdots {x^{2^{d-1}}}^{b_{d-1}}.
+$$
+* On a besoin de $d$ calculs pour les $x^{2^i}$.
+* On a besoin de $d$ calculs pour évaluer les produits de tous les termes.
+
+## Combien de calculs en terme de $n$?
+
+. . .
+
+* $n$ est représenté en binaire avec $d$ bits $\Rightarrow d\sim\log_2(n)$.
+* il y a $2\log_2(n)\sim \log_2(n)$ calculs.
+
+# Exponentiation rapide ou indienne (4/4)
+
+## Le vrai algorithme
+
+* Si n est pair: calculer $\left(x^{n/2}\right)^2$,
+* Si n est impair: calculer $x \cdot \left(x^{(n-1)/2}\right)^2$.
+
+## Exercice: écrire l'algorithme récursif correspondant
+
+. . .
+
+```C
+double pow(double x, int n) {
+ if (1 == n) {
+ return x;
+ } else if (n % 2 == 0) {
+ return pow(x, n / 2) * pow(x, n/2);
+ } else {
+ return x * pow(x, (n-1));
+ }
+}
+```
+
+
+# Efficacité d'un algorithmique
+
+Comment mesurer l'efficacité d'un algorithme?
+
+. . .
+
+* Mesurer le temps CPU,
+* Mesurer le temps d'accès à la mémoire,
+* Mesurer la place prise mémoire,
+
+. . .
+
+Dépendant du **matériel**, du **compilateur**, des **options de compilation**, etc!
+
+## Mesure du temps CPU
+
+```C
+#include <time.h>
+struct timespec tstart={0,0}, tend={0,0};
+clock_gettime(CLOCK_MONOTONIC, &tstart);
+// some computation
+clock_gettime(CLOCK_MONOTONIC, &tend);
+printf("computation about %.5f seconds\n",
+ ((double)tend.tv_sec + 1e-9*tend.tv_nsec) -
+ ((double)tstart.tv_sec + 1e-9*tstart.tv_nsec));
+```
+
+# Programme simple: mesure du temps CPU
+
+## Preuve sur un [petit exemple](../source_codes/complexity/sum.c)
+
+```bash
+source_codes/complexity$ make bench
+RUN ONCE -O0
+the computation took about 0.00836 seconds
+RUN ONCE -O3
+the computation took about 0.00203 seconds
+RUN THOUSAND TIMES -O0
+the computation took about 0.00363 seconds
+RUN THOUSAND TIMES -O3
+the computation took about 0.00046 seconds
+```
+
+Et sur votre machine les résultats seront **différents**.
+
+. . .
+
+## Conclusion
+
+* Nécessité d'avoir une mesure indépendante du/de la
+ matériel/compilateur/façon de mesurer/météo.
+
+# Analyse de complexité algorithmique (1/4)
+
+* On analyse le **temps** pris par un algorithme en fonction de la **taille de
+ l'entrée**.
+
+## Exemple: recherche d'un élément dans une liste triée de taille N
+
+```C
+int sorted_list[N];
+bool in_list = is_present(N, sorted_list, elem);
+```
+
+* Plus `N` est grand, plus l'algorithme prend de temps sauf si...
+
+. . .
+
+* l'élément est le premier de la liste (ou à une position toujours la même).
+* ce genre de cas pathologique ne rentre pas en ligne de compte.
+
+# Analyse de complexité algorithmique (2/4)
+
+## Recherche linéaire
+
+```C
+bool is_present(int n, int tab[], int elem) {
+ for (int i = 0; i < n; ++i) {
+ if (tab[i] == elem) {
+ return true;
+ } else if (elem < tab[i]) {
+ return false;
+ }
+ }
+ return false;
+}
+```
+
+* Dans le **meilleurs des cas** il faut `1` comparaison.
+* Dans le **pire des cas** (élément absent p.ex.) il faut `n` comparaisons.
+
+. . .
+
+La **complexité algorithmique** est proportionnelle à `N`: on double la taille
+du tableau $\Rightarrow$ on double le temps pris par l'algorithme.
+
+# Analyse de complexité algorithmique (3/4)
+
+## Recherche dichotomique
+
+```C
+bool is_present_binary_search(int n, int tab[], int elem) {
+ int left = 0;
+ int right = n - 1;
+ while (left <= right) {
+ int mid = (right + left) / 2;
+ if (tab[mid] < elem) {
+ left = mid + 1;
+ } else if (tab[mid] > elem) {
+ right = mid - 1;
+ } else {
+ return true;
+ }
+ }
+ return false;
+}
+```
+
+# Analyse de complexité algorithmique (4/4)
+
+## Recherche dichotomique
+
+](figs/Binary_search_complexity.svg){width=80%}
+
+. . .
+
+* Dans le **meilleurs de cas** il faut `1` comparaison.
+* Dans le **pire des cas** il faut $\log_2(N)+1$ comparaisons
+
+. . .
+
+## Linéaire vs dichotomique
+
+* $N$ vs $\log_2(N)$ comparaisons logiques.
+* Pour $N=1000000$: `1000000` vs `21` comparaisons.
+
+# Notation pour la complexité
+
+## Constante de proportionnalité
+
+* Pour la recherche linéaire ou dichotomique, on a des algorithmes qui sont $\sim N$ ou $\sim \log_2(N)$
+* Qu'est-ce que cela veut dire?
+
+. . .
+
+* Temps de calcul est $t=C\cdot N$ (où $C$ est le temps pris pour une comparaisons sur une machine/compilateur donné)
+* La complexité ne dépend pas de $C$.
+
+## Le $\mathcal{O}$ de Leibnitz
+
+* Pour noter la complexité d'un algorithme on utilise le symbole $\mathcal{O}$ (ou "grand Ô de").
+* Les complexités les plus couramment rencontrées sont
+
+. . .
+
+$$
+\mathcal{O}(1),\quad \mathcal{O}(\log(N)),\quad \mathcal{O}(N),\quad
+\mathcal{O}(\log(N)\cdot N), \quad \mathcal{O}(N^2), \quad
+\mathcal{O}(N^3).
+$$
+
+# Ordres de grandeur
+
+\begin{table}[!h]
+\begin{center}
+\caption{Valeurs approximatives de quelques fonctions usuelles de complexité.}
+\medskip
+\begin{tabular}{|c|c|c|c|c|}
+\hline
+$\log_2(N)$ & $\sqrt{N}$ & $N$ & $N\log_2(N)$ & $N^2$ \\
+\hline\hline
+$3$ & $3$ & $10$ & $30$ & $10^2$ \\
+\hline
+$6$ & $10$ & $10^2$ & $6\cdot 10^2$ & $10^4$ \\
+\hline
+$9$ & $31$ & $10^3$ & $9\cdot 10^3$ & $10^6$ \\
+\hline
+$13$ & $10^2$ & $10^4$ & $1.3\cdot 10^5$ & $10^8$ \\
+\hline
+$16$ & $3.1\cdot 10^2$ & $10^5$ & $1.6\cdot 10^6$ & $10^{10}$ \\
+\hline
+$19$ & $10^3$ & $10^6$ & $1.9\cdot 10^7$ & $10^{12}$ \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+
+# Quelques exercices (1/3)
+
+## Complexité de l'algorithme de test de primalité naïf?
+
+```C
+for (i = 2; i < sqrt(N); ++i) {
+ if (N % i == 0) {
+ return false;
+ }
+}
+return true;
+```
+
+. . .
+
+## Réponse
+
+$$
+\mathcal{O}(\sqrt{N}).
+$$
+
+# Quelques exercices (2/3)
+
+## Complexité de trouver le minimum d'un tableau?
+
+```C
+int min = MAX;
+for (i = 0; i < N; ++i) {
+ if (tab[i] < min) {
+ min = tab[i];
+ }
+}
+return min;
+```
+
+. . .
+
+## Réponse
+
+$$
+\mathcal{O}(N).
+$$
+
+# Quelques exercices (3/3)
+
+## Complexité du tri par sélection?
+
+```C
+int ind = 0
+while (ind < SIZE-1) {
+ min = find_min(tab[ind:SIZE]);
+ swap(min, tab[ind]);
+ ind += 1
+}
+```
+
+. . .
+
+## Réponse
+
+### `min = find_min`
+
+$$
+(N-1)+(N-2)+...+2+1=\sum_{i=1}^{N-1}i=N\cdot(N-1)/2=\mathcal{O}(N^2).
+$$
+
+## Finalement
+
+$$
+\mathcal{O}(N^2\mbox{ comparaisons}) + \mathcal{O}(N\mbox{swaps})=\mathcal{O}(N^2).
+$$
+
+
diff --git a/slides_2022/cours_7.md b/slides_2022/cours_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..1ee5638f4ae27c2bd1d27ba24246ba3a9d2c4593
--- /dev/null
+++ b/slides_2022/cours_7.md
@@ -0,0 +1,295 @@
+---
+title: "Tris"
+date: "2022-11-16"
+---
+
+# Rappel: Complexité
+
+\footnotesize
+
+Soit `tab` un tableau de longueur `N` de `double`.
+
+1. Comment déclare-t-on un tel tableau?
+
+. . .
+
+```C
+double tab[N];
+```
+
+2. Quelle est la complexité du calcul de la moyenne de `tab`?
+
+. . .
+
+```C
+double mean = 0.0;
+for (int i = 0; i < N; ++i) { // N assignations
+ mean += tab[i]; // N additions
+}
+mean /= N; // O(N)
+```
+
+. . .
+
+3. Quelle est la complexité du calcul de l'écart type de `tab` ($\sigma=\sqrt{\sum_i (t_i - m)^2}/N$?
+
+. . .
+
+```C
+double mean = moyenne(N, tab); // O(N)
+double dev = 0.0;
+for (int i = 0; i < N; ++i) {
+ dev += pow(tab[i] - moyenne, 2); // N tours
+}
+dev = sqrt(dev); dev /= N; // O(2*N) = O(N)
+```
+
+# Tri par insertion (1/3)
+
+## But
+
+* trier un tableau par ordre croissant
+
+## Algorithme
+
+Prendre un élément du tableau et le mettre à sa place parmis les éléments déjÃ
+triés du tableau.
+
+
+
+# Tri par insertion (2/3)
+
+## Exercice: Proposer un algorithme (en C)
+
+. . .
+
+```C
+void tri_insertion(int N, int tab[N]) {
+ for (int i = 1; i < N; i++) {
+ int tmp = tab[i];
+ int pos = i;
+ while (pos > 0 && tab[pos - 1] > tmp) {
+ tab[pos] = tab[pos - 1];
+ pos = pos - 1;
+ }
+ tab[pos] = tmp;
+ }
+}
+```
+
+# Tri par insertion (3/3)
+
+## Question: Quelle est la complexité?
+
+. . .
+
+* Parcours de tous les éléments ($N-1$ passages dans la boucle)
+ * Placer: en moyenne $i$ comparaisons et affectations à l'étape $i$
+* Moyenne: $\mathcal{O}(N^2)$
+
+. . .
+
+* Pire des cas, liste triée à l'envers: $\mathcal{O}(N^2)$
+* Meilleurs des cas, liste déjà triée: $\mathcal{O}(N)$
+
+# L'algorithme à la main
+
+## Exercice *sur papier*
+
+* Trier par insertion le tableau `[5, -2, 1, 3, 10]`
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+# Tri rapide ou quicksort (1/8)
+
+## Idée: algorithme `diviser pour régner` (`divide-and-conquer`)
+
+* Diviser: découper un problème en sous problèmes;
+* Régner: résoudre les sous-problèmes (souvent récursivement);
+* Combiner: à partir des sous problèmes résolu, calculer la solution.
+
+## Le pivot
+
+* Trouver le **pivot**, un élément qui divise le tableau en 2, tels que:
+ 1. Éléments à gauche sont **plus petits** que le pivot.
+ 2. Élements à droite sont **plus grands** que le pivot.
+
+# Tri rapide ou quicksort (2/8)
+
+## Algorithme `quicksort(tableau)`
+
+1. Choisir le pivot et l'amener à sa place:
+ * Les éléments à gauche sont plus petits que le pivot.
+ * Les éléments à droite sont plus grand que le pivot.
+2. `quicksort(tableau_gauche)` en omettant le pivot.
+3. `quicksort(tableau_droite)` en omettant le pivot.
+4. S'il y a moins de deux éléments dans le tableau, le tableau est trié.
+
+. . .
+
+Compris?
+
+. . .
+
+Non c'est normal, faisons un exemple.
+
+# Tri rapide ou quicksort (3/8)
+
+\footnotesize
+
+Deux variables sont primordiales:
+
+```C
+entier ind_min, ind_max; // les indices min/max des tableaux à trier
+```
+
+
+
+# Tri rapide ou quicksort (4/8)
+
+\footnotesize
+
+Deux variables sont primordiales:
+
+```C
+entier ind_min, ind_max; // les indices min/max des tableaux à trier
+```
+
+## Pseudocode: quicksort
+
+```python
+rien quicksort(entier tableau[], entier ind_min, entier ind_max)
+ si (longueur(tab) > 1)
+ ind_pivot = partition(tableau, ind_min, ind_max)
+ si (longueur(tableau[ind_min:ind_pivot-1]) != 0)
+ quicksort(tableau, ind_min, pivot_ind - 1)
+ si (longueur(tableau[ind_pivot+1:ind_max-1]) != 0)
+ quicksort(tableau, ind_pivot + 1, ind_max)
+```
+
+# Tri rapide ou quicksort (5/8)
+
+\footnotesize
+
+## Pseudocode: partition
+
+```C
+entier partition(entier tableau[], entier ind_min, entier ind_max)
+ pivot = tableau[ind_max] // choix arbitraire
+ i = ind_min
+ j = ind_max-1
+ tant que i < j:
+ en remontant i trouver le premier élément > pivot
+ en descendant j trouver le premier élément < pivot
+ échanger(tableau[i], tableau[j])
+ // les plus grands à droite
+ // mettre les plus petits à gauche
+
+ // on met le pivot "au milieu"
+ échanger(tableau[i], tableau[ind_max])
+ retourne i // on retourne l'indice pivot
+```
+
+# Tri rapide ou quicksort (6/8)
+
+## Exercice: implémenter les fonctions `quicksort` et `partition`
+
+. . .
+
+```C
+void quicksort(int size, int array[size], int first,
+ int last)
+{
+ if (first < last) {
+ int midpoint = partition(size, array, first, last);
+ if (first < midpoint - 1) {
+ quicksort(size, array, first, midpoint - 1);
+ }
+ if (midpoint + 1 < last) {
+ quicksort(size, array, midpoint + 1, last);
+ }
+ }
+}
+```
+
+
+# Tri rapide ou quicksort (7/8)
+
+\footnotesize
+
+## Exercice: implémenter les fonctions `quicksort` et `partition`
+
+```C
+int partition(int size, int array[size], int first, int last) {
+ int pivot = array[last];
+ int i = first - 1, j = last;
+ do {
+ do {
+ i += 1;
+ } while (array[i] < pivot && i < j);
+ do {
+ j -= 1;
+ } while (array[j] > pivot && i < j);
+ if (j > i) {
+ swap(&array[i], &array[j]);
+ }
+ } while (j > i);
+ swap(&array[i], &array[last]);
+ return i;
+}
+```
+
+# Tri rapide ou quicksort (8/8)
+
+## Quelle est la complexité du tri rapide?
+
+. . .
+
+* Pire des cas plus: $\mathcal{O}(N^2)$
+ * Quand le pivot sépare toujours le tableau de façon déséquilibrée ($N-1$
+ éléments d'un côté $1$ de l'autre).
+ * $N$ boucles et $N$ comparaisons $\Rightarrow N^2$.
+* Meilleur des cas (toujours le meilleur pivot): $\mathcal{O}(N\cdot \log_2(N))$.
+ * Chaque fois le tableau est séparé en $2$ parties égales.
+ * On a $\log_2(N)$ partitions, et $N$ boucles $\Rightarrow N\cdot
+ \log_2(N)$.
+* En moyenne: $\mathcal{O}(N\cdot \log_2(N))$.
+
+# L'algorithme à la main
+
+## Exercice *sur papier*
+
+* Trier par tri rapide le tableau `[5, -2, 1, 3, 10, 15, 7, 4]`
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+
+
diff --git a/slides_2022/cours_8.md b/slides_2022/cours_8.md
new file mode 100644
index 0000000000000000000000000000000000000000..2cfa2a3b115ca5eda52941bc0d446a703ad90b67
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+++ b/slides_2022/cours_8.md
@@ -0,0 +1,281 @@
+---
+title: "Backtracking et piles"
+date: "2022-11-23"
+---
+
+# Tri à bulle (1/4)
+
+## Algorithme
+
+* Parcours du tableau et comparaison des éléments consécutifs:
+ - Si deux éléments consécutifs ne sont pas dans l'ordre, ils sont échangés.
+* On recommence depuis le début du tableau jusqu'à avoir plus d'échanges Ã
+ faire.
+
+## Que peut-on dire sur le dernier élément du tableau après un parcours?
+
+. . .
+
+* Le plus grand élément est **à la fin** du tableau.
+ * Plus besoin de le traiter.
+* A chaque parcours on s'arrête un élément plus tôt.
+
+# Tri à bulle (2/4)
+
+## Exemple
+
+
+
+
+# Tri à bulle (3/4)
+
+## Exercice: écrire l'algorithme (poster le résultat sur matrix)
+
+. . .
+
+```C
+rien tri_a_bulles(entier tableau[])
+ pour i de longueur(tableau)-1 Ã 1:
+ trié = vrai
+ pour j de 0 Ã i-1:
+ si (tableau[j] > tableau[j+1])
+ échanger(array[j], array[j+1])
+ trié = faux
+
+ si trié
+ retourner
+```
+
+# Tri à bulle (4/4)
+
+## Quelle est la complexité du tri à bulles?
+
+. . .
+
+* Dans le meilleurs des cas:
+ * Le tableau est déjà trié: $\mathcal{O}(N)$ comparaisons.
+* Dans le pire des cas, $N\cdot (N-1)/2\sim\mathcal{O}(N^2)$:
+$$
+\sum_{i=1}^{N-1}i\mbox{ comparaison et }3\sum_{i=1}^{N-1}i \mbox{ affectations
+(swap)}\Rightarrow \mathcal{O}(N^2).
+$$
+* En moyenne, $\mathcal{O}(N^2)$ ($N^2/2$ comparaisons).
+
+# L'algorithme à la main
+
+## Exercice *sur papier*
+
+* Trier par tri à bulles le tableau `[5, -2, 1, 3, 10, 15, 7, 4]`
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+
+# Problème des 8-reines
+
+* Placer 8 reines sur un échiquier de $8 \times 8$.
+* Sans que les reines ne puissent se menacer mutuellement (92 solutions).
+
+## Conséquence
+
+* Deux reines ne partagent pas la même rangée, colonne, ou diagonale.
+* Donc chaque solution a **une** reine **par colonne** ou **ligne**.
+
+## Généralisation
+
+* Placer $N$ reines sur un échiquier de $N \times
+ N$.
+- Exemple de **backtracking** (retour en arrière) $\Rightarrow$ récursivité.
+
+](./figs/fig_recursivite_8_reines.png){width=35%}
+
+# Problème des 2-reines
+
+{width=50%}
+
+# Comment trouver les solutions?
+
+* On pose la première reine sur la première case disponible.
+* On rend inaccessibles toutes les cases menacées.
+* On pose la reine suivante sur la prochaine case non-menacée.
+* Jusqu'Ã ce qu'on puisse plus poser de reine.
+* On revient alors en arrière jusqu'au dernier coup où il y avait plus qu'une
+ possibilité de poser une reine.
+* On recommence depuis là .
+
+. . .
+
+* Le jeu prend fin quand on a énuméré *toutes* les possibilités de poser les
+ reines.
+
+# Problème des 3-reines
+
+
+
+# Problème des 4-reines
+
+
+
+# Problème des 4-reines, symétrie
+
+
+
+# Problème des 5 reines
+
+## Exercice: Trouver une solution au problème des 5 reines
+
+* Faire une capture d'écran / une photo de votre solution et la poster sur
+ matrix.
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+# Quelques observations sur le problème
+
+* Une reine par colonne au plus.
+* On place les reines sur des colonnes successives.
+* On a pas besoin de "regarder en arrière" (on place "devant" uniquement).
+* Trois étapes:
+ * On place une reine dans une case libre.
+ * On met à jour le tableau.
+ * Quand on a plus de cases libres on "revient dans le temps" ou c'est qu'on
+ a réussi.
+
+# Le code du problème des 8 reines (1/N)
+
+## Quelle structure de données?
+
+. . .
+
+Une matrice de booléens fera l'affaire:
+
+```C
+bool board[n][n];
+```
+
+## Quelles fonctionnalités?
+
+. . .
+
+```C
+// Pour chaque ligne placer la reine sur toutes les colonnes
+// et compter les solutions
+void nbr_solutions(board, column, counter);
+// Copier un tableau dans un autre
+void copy(board_in, board_out);
+// Placer la reine à li, co et rendre inaccessible devant
+void placer_devant(board, li, co);
+```
+
+# Le code du problème des 8 reines (2/N)
+
+## Le calcul du nombre de solutions
+
+```C
+// Calcule le nombre de solutions au problème des <n> reines
+nbr_solutions(board, column, count)
+ // pour chaque ligne
+ // si la case libre
+ // si column < n - 1
+ // copier board dans un "new" board,
+ // y poser une reine
+ // et mettre à jour ce "new" board
+ // nbr_solutions(new_board, column+1, count)
+ // sinon
+ // on a posé la n-ème et on a gagné
+ // count += 1
+```
+
+# Le code du problème des 8 reines (3/N)
+
+## Le calcul du nombre de solutions
+
+```C
+// Placer une reine et mettre à jour
+placer_devant(board, ligne, colonne)
+ // board est occupé à ligne/colonne
+ // toutes les cases des colonnes
+ // suivantes sont mises à jour
+```
+
+# Le code du problème des 8 reines (4/N)
+
+## Compris? Alors écrivez le code et postez le!
+
+. . .
+
+## Le nombre de solutions
+
+\footnotesize
+
+```C
+// Calcule le nombre de solutions au problème des <n> reines
+void nb_sol(int n, bool board[n][n], int co, int *ptr_cpt) {
+ for (int li = 0; li < n; li++) {
+ if (board[li][co]) {
+ if (co < n-1) {
+ bool new_board[n][n]; // alloué à chaque nouvelle tentative
+ copy(n, board, new_board);
+ prises_devant(n, new_board, li, co);
+ nb_sol(n, new_board, co+1, ptr_cpt);
+ } else {
+ *ptr_cpt = (*ptr_cpt)+1;
+ }
+ }
+ }
+}
+```
+
+
+# Le code du problème des 8 reines (5/N)
+
+\footnotesize
+
+## Placer devant
+
+```C
+// Retourne une copie du tableau <board> complété avec les positions
+// prises sur la droite droite par une reine placée en <board(li,co)>
+void prises_devant(int n, bool board[n][n], int li, int co) {
+ board[li][co] = false; // position de la reine
+ for (int j = 1; j < n-co; j++) {
+ // horizontale et diagonales à droite de la reine
+ if (j <= li) {
+ board[li-j][co+j] = false;
+ }
+ board[li][co+j] = false;
+ if (li+j < n) {
+ board[li+j][co+j] = false;
+ }
+ }
+}
+```
+
diff --git a/slides_2022/cours_9.md b/slides_2022/cours_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..d3b4a02e555f48f0fe4a396ba0e752985045ae74
--- /dev/null
+++ b/slides_2022/cours_9.md
@@ -0,0 +1,304 @@
+---
+title: "Piles"
+date: "2022-12-07"
+---
+
+# Les piles (1/5)
+
+## Qu'est-ce donc?
+
+* Structure de données abstraite...
+
+. . .
+
+* de type `LIFO` (*Last in first out*).
+
+](figs/Stack.svg){width=70%}
+
+## Des exemples de la vraie vie
+
+. . .
+
+* Pile d'assiettes, de livres, ...
+* Adresses visitées par un navigateur web.
+* Les calculatrices du passé (en polonaise inverse).
+* Les boutons *undo* de vos éditeurs de texte (aka *u* dans vim).
+
+# Les piles (2/5)
+
+## Fonctionnalités
+
+. . .
+
+1. Empiler (push): ajouter un élément sur la pile.
+2. Dépiler (pop): retirer l'élément du sommet de la pile et le retrouner.
+3. Liste vide? (is_empty?).
+
+. . .
+
+4. Jeter un oeil (peek): retourner l'élément du sommet de la pile (sans le dépiler).
+5. Nombre d'éléments (length).
+
+## Comment faire les 4,5 Ã partir de 1 Ã 3?
+
+. . .
+
+4. Dépiler l'élément, le copier, puis l'empiler à nouveau.
+5. Dépiler jusqu'à ce que la pile soit vide, puis empiler à nouveau.
+
+. . .
+
+## Existe en deux goûts
+
+* Pile avec ou sans limite de capacité (à concurrence de la taille de la
+mémoire).
+
+# Les piles (3/5)
+
+## Implémentation
+
+* Jusqu'ici on n'a pas du tout parlé d'implémentation (d'où le nom de structure
+ abstraite).
+* Pas de choix unique d'implémentation.
+
+## Quelle structure de données allons nous utiliser?
+
+. . .
+
+Et oui vous avez deviné: un tableau!
+
+## La structure: de quoi avons-nous besoin (pile de taille fixe)?
+
+. . .
+
+```C
+#define MAX_CAPACITY 500
+typedef struct _stack {
+ int data[MAX_CAPACITY]; // les données
+ int top; // indice du sommet
+} stack;
+```
+
+# Les piles (4/5)
+
+## Initialisation
+
+. . .
+
+```C
+void stack_init(stack *s) {
+ s->top = -1;
+}
+```
+
+## Est vide?
+
+. . .
+
+```C
+bool stack_is_empty(stack s) {
+ return s.top == -1;
+}
+```
+
+## Empiler (ajouter un élément au sommet)
+
+. . .
+
+```C
+void stack_push(stack *s, int val) {
+ s->top += 1;
+ s->data[s->top] = val;
+}
+```
+
+# Les piles (5/5)
+
+## Dépiler (enlever l'élément du sommet)
+
+. . .
+
+```C
+int stack_pop(stack *s) {
+ s->top -= 1;
+ return s->data[s->top+1];
+}
+```
+
+## Jeter un oeil (regarder le sommet)
+
+. . .
+
+```C
+int stack_peek(stack *s) {
+ return s->data[s->top];
+}
+```
+
+## Quelle est la complexité de ces opérations?
+
+. . .
+
+## Voyez-vous des problèmes potentiels avec cette implémentation?
+
+. . .
+
+* Empiler avec une pile pleine.
+* Dépiler avec une pile vide.
+* Jeter un oeil au sommet d'une pile vide.
+
+# Gestion d'erreur, level 0
+
+* Il y a plusieurs façon de traiter les erreur:
+ * Ne rien faire (laisser la responsabilité à l'utilisateur).
+ * Faire paniquer le programme (il plante plus ou moins violemment).
+ * Utiliser des codes d'erreurs.
+
+## La panique
+
+* En C, on a les `assert()` pour faire paniquer un programme.
+
+
+# Assertions (1/3)
+
+```C
+#include <assert.h>
+void assert(int expression);
+```
+
+## Qu'est-ce donc?
+
+- Macro permettant de tester une condition lors de l'exécution d'un programme:
+ - Si `expression == 0`{.C} (condition fausse), `assert()`{.C} affiche un message d'erreur sur `stderr`{.C} et termine l'exécution du programme.
+ - Sinon l'exécution se poursuit normalement.
+ - Peuvent être désactivés à la compilation avec `-DNDEBUG` (équivalent à `#define
+ NDEBUG`)
+
+## À quoi ça sert?
+
+- Permet de réaliser des tests unitaires.
+- Permet de tester des conditions catastrophiques d'un programme.
+- **Ne permet pas** de gérer les erreurs.
+
+# Assertions (2/3)
+
+<!-- \footnotesize -->
+
+## Exemple
+
+```C
+#include <assert.h>
+void stack_push(stack *s, int val) {
+ assert(s->top < MAX_CAPACITY-1);
+ s->top += 1;
+ s->data[s->top] = val;
+}
+int stack_pop(stack *s) {
+ assert(s->top >= 0);
+ s->top -= 1;
+ return s->data[s->top+1];
+}
+int stack_peek(stack *s) {
+ assert(s->top >= 0);
+ return s->data[s->top];
+}
+```
+
+# Assertions (3/3)
+
+## Cas typiques d'utilisation
+
+- Vérification de la validité des pointeurs (typiquement `!= NULL`{.C}).
+- Vérification du domaine des indices (dépassement de tableau).
+
+## Bug vs. erreur de *runtime*
+
+- Les assertions sont là pour détecter les bugs (erreurs d'implémentation).
+- Les assertions ne sont pas là pour gérer les problèmes externes au programme (allocation mémoire qui échoue, mauvais paramètre d'entrée passé par l'utilisateur, ...).
+
+. . .
+
+- Mais peuvent être pratiques quand même pour ça...
+- Typiquement désactivées dans le code de production.
+
+# La pile dynamique
+
+## Comment modifier le code précédent pour avoir une taille dynamique?
+
+. . .
+
+```C
+// alloue une zone mémoire de size octets
+void *malloc(size_t size);
+// change la taille allouée à size octets (contiguïté garantie)
+void *realloc(void *ptr, size_t size);
+```
+
+## Et maintenant?
+
+. . .
+
+```C
+stack_create(); // crée une pile avec une taille par défaut
+// vérifie si la pile est pleine et réalloue si besoin
+stack_push();
+// vérifie si la pile est vide/trop grande
+// et réalloue si besoin
+stack_pop();
+```
+
+## Exercice: ouvrir un repo/issues pour l'implémentation
+
+* Oui-oui cela est une introduction au développement collaboratif (et
+ hippie).
+
+# 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 Ã 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
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
diff --git a/slides_2022/docker-compose.yml b/slides_2022/docker-compose.yml
new file mode 100644
index 0000000000000000000000000000000000000000..57a6d7b3025ba7379c862fc0d18656055efdd270
--- /dev/null
+++ b/slides_2022/docker-compose.yml
@@ -0,0 +1,10 @@
+version: "3.3"
+services:
+ slides:
+ image: omalaspinas/pandoc:latest
+ user: 1000:1000
+ container_name: slides
+ volumes:
+ - ./:/data
+ entrypoint: ["make"]
+ working_dir: /data
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+<text text-anchor="middle" x="279" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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+<!-- sd1->ig2 -->
+<g id="edge7" class="edge">
+<title>sd1->ig2</title>
+<path fill="none" stroke="#000000" d="M306.2854,-144.5708C302.0403,-136.0807 296.8464,-125.6929 292.1337,-116.2674"/>
+<polygon fill="#000000" stroke="#000000" points="295.237,-114.6477 287.6343,-107.2687 288.976,-117.7782 295.237,-114.6477"/>
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+<!-- id2 -->
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+<text text-anchor="middle" x="351" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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+<!-- sd1->id2 -->
+<g id="edge8" class="edge">
+<title>sd1->id2</title>
+<path fill="none" stroke="#000000" d="M323.7146,-144.5708C327.9597,-136.0807 333.1536,-125.6929 337.8663,-116.2674"/>
+<polygon fill="#000000" stroke="#000000" points="341.024,-117.7782 342.3657,-107.2687 334.763,-114.6477 341.024,-117.7782"/>
+</g>
+<!-- sg3 -->
+<g id="node10" class="node">
+<title>sg3</title>
+<ellipse fill="none" stroke="#000000" cx="423" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="423" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->sg3 -->
+<g id="edge9" class="edge">
+<title>ig1->sg3</title>
+<path fill="none" stroke="#000000" d="M479.7307,-146.7307C469.803,-136.803 456.6847,-123.6847 445.5637,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="447.7933,-109.8436 438.2473,-105.2473 442.8436,-114.7933 447.7933,-109.8436"/>
+</g>
+<!-- sd3 -->
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+<title>sd3</title>
+<ellipse fill="none" stroke="#000000" cx="495" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="495" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->sd3 -->
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+<title>ig1->sd3</title>
+<path fill="none" stroke="#000000" d="M495,-143.8314C495,-136.131 495,-126.9743 495,-118.4166"/>
+<polygon fill="#000000" stroke="#000000" points="498.5001,-118.4132 495,-108.4133 491.5001,-118.4133 498.5001,-118.4132"/>
+</g>
+<!-- ig3 -->
+<g id="node12" class="node">
+<title>ig3</title>
+<ellipse fill="none" stroke="#000000" cx="567" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="567" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- ig1->ig3 -->
+<g id="edge11" class="edge">
+<title>ig1->ig3</title>
+<path fill="none" stroke="#000000" d="M510.2693,-146.7307C520.197,-136.803 533.3153,-123.6847 544.4363,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="547.1564,-114.7933 551.7527,-105.2473 542.2067,-109.8436 547.1564,-114.7933"/>
+</g>
+<!-- id3 -->
+<g id="node13" class="node">
+<title>id3</title>
+<ellipse fill="none" stroke="#000000" cx="639" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="639" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->id3 -->
+<g id="edge12" class="edge">
+<title>ig1->id3</title>
+<path fill="none" stroke="#000000" d="M516.8705,-151.0647C541.2222,-138.8889 580.762,-119.119 608.2285,-105.3858"/>
+<polygon fill="#000000" stroke="#000000" points="610.0165,-108.4049 617.3955,-100.8022 606.886,-102.1439 610.0165,-108.4049"/>
+</g>
+<!-- sg4 -->
+<g id="node14" class="node">
+<title>sg4</title>
+<ellipse fill="none" stroke="#000000" cx="711" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="711" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- id1->sg4 -->
+<g id="edge13" class="edge">
+<title>id1->sg4</title>
+<path fill="none" stroke="#000000" d="M767.7307,-146.7307C757.803,-136.803 744.6847,-123.6847 733.5637,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="735.7933,-109.8436 726.2473,-105.2473 730.8436,-114.7933 735.7933,-109.8436"/>
+</g>
+<!-- sd4 -->
+<g id="node15" class="node">
+<title>sd4</title>
+<ellipse fill="none" stroke="#000000" cx="783" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="783" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- id1->sd4 -->
+<g id="edge14" class="edge">
+<title>id1->sd4</title>
+<path fill="none" stroke="#000000" d="M783,-143.8314C783,-136.131 783,-126.9743 783,-118.4166"/>
+<polygon fill="#000000" stroke="#000000" points="786.5001,-118.4132 783,-108.4133 779.5001,-118.4133 786.5001,-118.4132"/>
+</g>
+<!-- ig4 -->
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+<title>ig4</title>
+<ellipse fill="none" stroke="#000000" cx="855" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="855" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- id1->ig4 -->
+<g id="edge15" class="edge">
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+<path fill="none" stroke="#000000" d="M798.2693,-146.7307C808.197,-136.803 821.3153,-123.6847 832.4363,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="835.1564,-114.7933 839.7527,-105.2473 830.2067,-109.8436 835.1564,-114.7933"/>
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+<!-- id4 -->
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+<text text-anchor="middle" x="927" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
+</g>
+<!-- id1->id4 -->
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+<title>id1->id4</title>
+<path fill="none" stroke="#000000" d="M804.8705,-151.0647C829.2222,-138.8889 868.762,-119.119 896.2285,-105.3858"/>
+<polygon fill="#000000" stroke="#000000" points="898.0165,-108.4049 905.3955,-100.8022 894.886,-102.1439 898.0165,-108.4049"/>
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+<!-- sg5 -->
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+<ellipse fill="none" stroke="#000000" cx="27" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="27" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- sg2->sg5 -->
+<g id="edge17" class="edge">
+<title>sg2->sg5</title>
+<path fill="none" stroke="#000000" d="M115.6918,-77.1278C98.6445,-65.763 73.5981,-49.0654 54.4656,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="56.4031,-33.3956 46.1411,-30.7607 52.5201,-39.2199 56.4031,-33.3956"/>
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+<!-- sd5 -->
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+<text text-anchor="middle" x="99" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- sg2->sd5 -->
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+<title>sg2->sd5</title>
+<path fill="none" stroke="#000000" d="M126.2854,-72.5708C122.0403,-64.0807 116.8464,-53.6929 112.1337,-44.2674"/>
+<polygon fill="#000000" stroke="#000000" points="115.237,-42.6477 107.6343,-35.2687 108.976,-45.7782 115.237,-42.6477"/>
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+<!-- ig5 -->
+<g id="node20" class="node">
+<title>ig5</title>
+<ellipse fill="none" stroke="#000000" cx="171" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="171" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- sg2->ig5 -->
+<g id="edge19" class="edge">
+<title>sg2->ig5</title>
+<path fill="none" stroke="#000000" d="M143.7146,-72.5708C147.9597,-64.0807 153.1536,-53.6929 157.8663,-44.2674"/>
+<polygon fill="#000000" stroke="#000000" points="161.024,-45.7782 162.3657,-35.2687 154.763,-42.6477 161.024,-45.7782"/>
+</g>
+<!-- id5 -->
+<g id="node21" class="node">
+<title>id5</title>
+<ellipse fill="none" stroke="#000000" cx="243" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="243" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- sg2->id5 -->
+<g id="edge20" class="edge">
+<title>sg2->id5</title>
+<path fill="none" stroke="#000000" d="M154.3082,-77.1278C171.3555,-65.763 196.4019,-49.0654 215.5344,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="217.4799,-39.2199 223.8589,-30.7607 213.5969,-33.3956 217.4799,-39.2199"/>
+</g>
+<!-- sg6 -->
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+<title>sg6</title>
+<ellipse fill="none" stroke="#000000" cx="819" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="819" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- id4->sg6 -->
+<g id="edge21" class="edge">
+<title>id4->sg6</title>
+<path fill="none" stroke="#000000" d="M907.6918,-77.1278C890.6445,-65.763 865.5981,-49.0654 846.4656,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="848.4031,-33.3956 838.1411,-30.7607 844.5201,-39.2199 848.4031,-33.3956"/>
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+<!-- sd6 -->
+<g id="node23" class="node">
+<title>sd6</title>
+<ellipse fill="none" stroke="#000000" cx="891" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="891" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- id4->sd6 -->
+<g id="edge22" class="edge">
+<title>id4->sd6</title>
+<path fill="none" stroke="#000000" d="M918.2854,-72.5708C914.0403,-64.0807 908.8464,-53.6929 904.1337,-44.2674"/>
+<polygon fill="#000000" stroke="#000000" points="907.237,-42.6477 899.6343,-35.2687 900.976,-45.7782 907.237,-42.6477"/>
+</g>
+<!-- ig6 -->
+<g id="node24" class="node">
+<title>ig6</title>
+<ellipse fill="none" stroke="#000000" cx="963" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="963" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- id4->ig6 -->
+<g id="edge23" class="edge">
+<title>id4->ig6</title>
+<path fill="none" stroke="#000000" d="M935.7146,-72.5708C939.9597,-64.0807 945.1536,-53.6929 949.8663,-44.2674"/>
+<polygon fill="#000000" stroke="#000000" points="953.024,-45.7782 954.3657,-35.2687 946.763,-42.6477 953.024,-45.7782"/>
+</g>
+<!-- id6 -->
+<g id="node25" class="node">
+<title>id6</title>
+<ellipse fill="none" stroke="#000000" cx="1035" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="1035" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- id4->id6 -->
+<g id="edge24" class="edge">
+<title>id4->id6</title>
+<path fill="none" stroke="#000000" d="M946.3082,-77.1278C963.3555,-65.763 988.4019,-49.0654 1007.5344,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="1009.4799,-39.2199 1015.8589,-30.7607 1005.5969,-33.3956 1009.4799,-39.2199"/>
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+</g>
+</svg>
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diff --git a/slides_2022/figs/quad_img_simple.svg b/slides_2022/figs/quad_img_simple.svg
new file mode 100644
index 0000000000000000000000000000000000000000..f9fea0d611cafcfc47bc0bdc6f3dd00f607c7f38
--- /dev/null
+++ b/slides_2022/figs/quad_img_simple.svg
@@ -0,0 +1,257 @@
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+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 184)">
+<title>G</title>
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+<text text-anchor="middle" x="567" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
+</g>
+<!-- sg1 -->
+<g id="node2" class="node">
+<title>sg1</title>
+<ellipse fill="none" stroke="#000000" cx="171" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="171" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">SG</text>
+</g>
+<!-- 0->sg1 -->
+<g id="edge1" class="edge">
+<title>0->sg1</title>
+<path fill="none" stroke="#000000" d="M540.7107,-157.2201C471.8543,-144.7008 286.1298,-110.9327 207.2397,-96.589"/>
+<polygon fill="#000000" stroke="#000000" points="207.7449,-93.1236 197.28,-94.7782 206.4926,-100.0107 207.7449,-93.1236"/>
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+<title>sd1</title>
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+<text text-anchor="middle" x="459" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">SD</text>
+</g>
+<!-- 0->sd1 -->
+<g id="edge2" class="edge">
+<title>0->sd1</title>
+<path fill="none" stroke="#000000" d="M547.6918,-149.1278C530.6445,-137.763 505.5981,-121.0654 486.4656,-108.3104"/>
+<polygon fill="#000000" stroke="#000000" points="488.4031,-105.3956 478.1411,-102.7607 484.5201,-111.2199 488.4031,-105.3956"/>
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+<!-- ig1 -->
+<g id="node4" class="node">
+<title>ig1</title>
+<ellipse fill="none" stroke="#000000" cx="675" cy="-90" rx="27" ry="18"/>
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+<polygon fill="#000000" stroke="#000000" points="649.4799,-111.2199 655.8589,-102.7607 645.5969,-105.3956 649.4799,-111.2199"/>
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+<!-- id1 -->
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+<title>id1</title>
+<ellipse fill="none" stroke="#000000" cx="963" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="963" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">ID</text>
+</g>
+<!-- 0->id1 -->
+<g id="edge4" class="edge">
+<title>0->id1</title>
+<path fill="none" stroke="#000000" d="M593.2893,-157.2201C662.1457,-144.7008 847.8702,-110.9327 926.7603,-96.589"/>
+<polygon fill="#000000" stroke="#000000" points="927.5074,-100.0107 936.72,-94.7782 926.2551,-93.1236 927.5074,-100.0107"/>
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+<!-- sg2 -->
+<g id="node6" class="node">
+<title>sg2</title>
+<ellipse fill="none" stroke="#000000" cx="27" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="27" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">21</text>
+</g>
+<!-- sg1->sg2 -->
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+<title>sg1->sg2</title>
+<path fill="none" stroke="#000000" d="M149.1295,-79.0647C124.7778,-66.8889 85.238,-47.119 57.7715,-33.3858"/>
+<polygon fill="#000000" stroke="#000000" points="59.114,-30.1439 48.6045,-28.8022 55.9835,-36.4049 59.114,-30.1439"/>
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+<ellipse fill="none" stroke="#000000" cx="99" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="99" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">12</text>
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+<path fill="none" stroke="#000000" d="M155.7307,-74.7307C145.803,-64.803 132.6847,-51.6847 121.5637,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="123.7933,-37.8436 114.2473,-33.2473 118.8436,-42.7933 123.7933,-37.8436"/>
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+<polygon fill="#000000" stroke="#000000" points="174.5001,-46.4132 171,-36.4133 167.5001,-46.4133 174.5001,-46.4132"/>
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diff --git a/slides_2022/intro.md b/slides_2022/intro.md
new file mode 100644
index 0000000000000000000000000000000000000000..67c328113a63b4af40f28257fd4fbec66415a316
--- /dev/null
+++ b/slides_2022/intro.md
@@ -0,0 +1,53 @@
+---
+title: "Introduction générale"
+date: "2022-09-20"
+---
+
+# La hotline
+
+Nom Mél Bureau
+-------------------- ------------------------------ --------------------
+Paul Albuquerque paul.albuquerque@hesge.ch B410
+Orestis Malaspinas orestis.malaspinas@hesge.ch A401
+-------------------- ------------------------------ --------------------
+
+* Utilisez le libre service (l'horaire sera fixé prochainement).
+* On va intensivement utiliser *Element*, installez le et utilisez le!
+
+* Espace de discussion [Matrix](https://matrix.to/#/!aKYVlcclmPGYXQFxAK:matrix.org?via=matrix.org), installez [element.io](https://element.io).
+
+ {width=20%}
+
+# Cyberlearn
+
+Tout le contenu de ce qu'on raconte se trouve sur cyberlearn:
+
+- Algorithmes et structures de données
+ - <https://cyberlearn.hes-so.ch/course/view.php?id=13941>
+ - Clé d'inscription: algo_2021_22
+
+- Programmation Sequentielle en C
+ - <https://cyberlearn.hes-so.ch/course/view.php?id=12399>
+ - Clé d'inscription: prog_seq_2021_22
+
+
+# Organisation du module
+
+* Deux cours, 50% chacun.
+1. Algorithmes et structures de données:
+ * 1er semestre:
+ * bases de programmation en C jusqu'à Noël.
+ * algorithmique jusqu'Ã fin janvier.
+ * 2e semestre:
+ * algorithmique.
+ * Deux évaluations écrites par semestre (1er: novembre et janvier).
+2. Programmation séquentielle en C
+ * Familiarisation avec l'environnement Linux.
+ * Travaux pratiques en C.
+ * Apprentissage du gestionnaire de versions: git.
+ * Plusieurs exercices illustrant les concepts d'algorithmique.
+ * Évaluations:
+ * Deux évaluations machine (1er semestre).
+ * Probablement, une évaluation machine et un projet (2e semestre).
+
+
diff --git a/slides_2022/metadata.yaml b/slides_2022/metadata.yaml
new file mode 100644
index 0000000000000000000000000000000000000000..f81b6ade406b88c0f59fe07a59d026650798571a
--- /dev/null
+++ b/slides_2022/metadata.yaml
@@ -0,0 +1,9 @@
+---
+subtitle: "Algorithmique et structures de données, 2022-2023"
+author: "P. Albuquerque (B410), P. Künzli et O. Malaspinas (A401), ISC, HEPIA"
+institute: En partie inspirés des supports de cours de P. Albuquerque
+lang: fr-CH
+revealjs-url: /reveal.js
+mathjaxurl: "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"
+---
+
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