From bceee86fff38a91dee0e3951b87155d4d9dc58f1 Mon Sep 17 00:00:00 2001
From: Robin VAN DE MERGHEL <robin.van_de_merghel@etu.uca.fr>
Date: Fri, 28 Apr 2023 18:03:14 +0200
Subject: [PATCH] Adding course of algo in class

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 Algorithmique/TD/TD2.md  | 392 +++++++++++++++++++++++++++++++++++++++
 Algorithmique/TD/TD2.pdf | Bin 0 -> 68023 bytes
 2 files changed, 392 insertions(+)
 create mode 100644 Algorithmique/TD/TD2.md
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diff --git a/Algorithmique/TD/TD2.md b/Algorithmique/TD/TD2.md
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+---
+title: TD2 Listes
+author: VAN DE MERGHEL Robin
+date: 2023
+lang: fr
+geometry: margin=2cm
+---
+
+# Exercice 2.1
+
+## Question 1
+
+> **Proposer une implémentation des listes chaînées avec un tableau (le suivant dans la liste n’est pas forcément le suivant dans le tableau). Pensez à gérer la liste des cases libres.**
+
+| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+|---|---|---|---|---|---|---|---|---|---|
+|   |   |   |   | 5 | 2 | 8 | 9 |   |   |
+
+1. Suivant dans la liste c'est le suivant dans le tableau
+2. Suivant dans la liste n'est pas forcément le suivant dans le tableau
+
+
+| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+|---|---|---|---|---|---|---|---|---|---|
+| 9 |   | 5 |   |   | 8 |   |   |   | 5 |
+| -1|   | 9 |   |   | 2 |   |   |   | 8 |
+
+Il faut gérer dans la liste l'ensemble des cases du tableau qui ne contiennent pas un élément de la liste libre : $ \{ 1, 3 , 6, 7 8 \} $
+
+Ex : 
+1. `insererDebut(l, 20)`
+
+- On commence par chercher un index libre dans la liste des cases libres
+
+
+2. Si un élément est supprimé de la liste, son index dans le tableau doit être inséré dans l'ensemble libres
+
+**Fonctions linéaire :**
+
+- `creerListe()`
+- `debutListe(l)`
+- `estVide(l)`
+- `ajouterDebut(l, e)`
+- `ajouterFin(l, e)`
+- `supprimerDebut(l)`
+- `supprimerFin(l)`
+
+3. Faire de même avec les cellules (que l'on demande au système), la liste n'est pas stockée dans un tableau.
+
+On implémente les deux méthodes :
+
+### Méthode 1 : Tableau dans un intervalle
+
+```ruby
+function creerListe() {
+    n = 42
+    Liste l
+    tab = new int[n]
+    l.tab = tab
+    l.taille = 0
+    l.debut = 0
+    l.fin = 0
+    return l
+}
+
+function estVide(l) {
+    return l.taille == 0
+}
+
+
+function debutListe(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    return l.debut
+}
+
+function ajouterDebut(l, e) {
+    if (l.taille == n) {
+        return -1
+    }
+    l.tab[l.debut] = e
+    l.debut = (l.debut + 1) % n
+    l.taille = l.taille + 1
+    return 0
+}
+
+function ajouterFin(l, e) {
+    if (l.taille == n) {
+        return -1
+    }
+    l.tab[l.fin] = e
+    l.fin = (l.fin - 1) % n
+    l.taille = l.taille + 1
+    return 0
+}
+
+function supprimerDebut(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    l.debut = (l.debut - 1) % n
+    l.taille = l.taille - 1
+    return 0
+}
+
+function supprimerFin(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    l.fin = (l.fin + 1) % n
+    l.taille = l.taille - 1
+    return 0
+}
+```
+
+### Méthode 2 : Tableau où l'on cherche les cases libres
+
+```ruby
+function creerListe() {
+    n = 42
+    Liste l = new Liste()
+    tab = new int[n]
+    l.tab = tab
+    File f = new File()
+    for (i = 0; i < n; i++) {
+        ajouterFin(f, i)
+    }
+    l.libres = f
+    l.taille = 0
+    l.debut = 0
+    l.fin = 0
+    return l
+}
+
+function estVide(l) {
+    return l.taille == 0
+}
+
+function debutListe(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    return l.debut
+}
+
+function ajouterDebut(l, e) {
+    if (estVide(l.libres)) {
+        return -1
+    }
+    i = supprimerDebut(l.libres)
+    l.tab[i] = e
+    l.debut = i
+    l.taille = l.taille + 1
+    return 0
+}
+
+function ajouterFin(l, e) {
+    if (estVide(l.libres)) {
+        return -1
+    }
+    i = supprimerDebut(l.libres)
+    l.tab[i] = e
+    l.fin = i
+    l.taille = l.taille + 1
+    return 0
+}
+
+function supprimerDebut(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    ajouterFin(l.libres, l.debut)
+    l.debut = l.tab[l.debut]
+    l.taille = (l.taille - 1) % n
+    return 0
+}
+
+function supprimerFin(l) {
+    if (estVide(l)) {
+        return -1
+    }
+    ajouterFin(l.libres, l.fin)
+    l.fin = l.tab[l.fin]
+    l.taille = (l.taille - 1) % n
+    return 0
+}
+```
+
+
+## Question 2
+
+> **Faire de même avec les pointeurs.**
+
+On redéfini la structure d'une cellule, on a un pointeur sur son suivant :
+
+```ruby
+enregistrement Cellule {
+    T val
+    Cellule suiv
+}
+```
+
+Le système a deux fonctions :
+
+- `allocation()` : alloue une cellule (`malloc` en C, `new` en Java)
+- `liberation(c)` : libère une cellule (`free` en C, automatique en Java)
+
+On créera une liste avec un pointeur sur sa première cellule :
+
+```ruby
+enregistrement Liste {
+    Cellule debutListe
+}
+```
+
+Pour le code concret, on doit créer une cellule constante qui représentera la fin de la liste (comme le `null` en Java, ou le `None` en Python) :
+
+```ruby
+Cellule cVide = ... # Dépend du langage
+
+# En C on peut faire :
+# Cellule cVide = NULL
+
+# En Java on peut faire :
+# Cellule cVide = null
+
+# En Python on peut faire :
+# cVide = None
+```
+
+Maintenant, on peut implémenter les fonctions :
+
+```ruby
+Liste creerListe() {
+    Liste l = new Liste()
+    l.debutListe = cVide
+    return l
+}
+
+bool estVideListe(Liste l) {
+    return l.debutListe == cVide
+}
+
+Cellule debutListe(Liste l) {
+    if (estVideListe(l)) {
+        return cVide
+    }
+    return l.debutListe
+}
+
+void ajouterDebut(Liste l, T e) {
+    Cellule c = allocation()
+
+    # En C on vérifie si l'allocation a réussi
+    # if (c == NULL) {
+    #     printf("Erreur d'allocation\n")
+    #     exit(1)
+    # }
+
+    c.val = e
+    c.suiv = l.debutListe
+    l.debutListe = c
+}
+
+void ajouterFin(Liste l, T e) {
+    Cellule c = allocation()
+    c.val = e
+    c.suiv = cVide
+
+    if (estVideListe(l)) {
+        l.debutListe = c
+    } else {
+        Cellule c2 = l.debutListe
+        while (c2.suiv != cVide) {
+            c2 = c2.suiv
+        }
+        c2.suiv = c
+    }
+
+}
+
+Liste supprimerDebut(Liste l) {
+    if (estVideListe(l)) {
+        return l
+    }
+    Cellule c = l.debutListe
+    l.debutListe = c.suiv
+    liberation(c)
+    return l
+}
+
+Liste supprimerFin(Liste l) {
+    if (estVideListe(l)) {
+        return l
+    }
+    if (l.debutListe.suiv == cVide) {
+        liberation(l.debutListe)
+        l.debutListe = cVide
+        return l
+    }
+    Cellule c = l.debutListe
+    while (c.suiv.suiv != cVide) {
+        c = c.suiv
+    }
+    liberation(c.suiv)
+    c.suiv = cVide
+    return l
+}
+
+# On pourrait faire une fonction auxiliaire pour aller au dernier élément
+# Et l'appeler dans ajouterFin et supprimerFin, car le code est le même
+```
+
+## Question 3
+
+> **Comment modifier vos implémentations pour implémenter une liste chaînée bi-directionnelle ? Une liste circulaire ?**
+
+Pour une liste chaînée bi-directionnelle, on peut modifier l'enregistrement `Cellule` pour avoir un pointeur sur la cellule précédente :
+
+```ruby
+enregistrement Cellule {
+    T val
+    Cellule suiv
+    Cellule prec
+}
+```
+
+Un exemple de représentation :
+
+```mermaid
+graph LR
+    c1 --> c2[Cellule]
+    c2 --> c1[Début]
+    c2 --> c3[Cellule]
+    c3 --> c2[Cellule]
+    c3 --> c4[Fin]
+    c4 --> c3[Cellule]
+```
+
+Pour une liste circulaire, peut modifier l'enregistrement `Liste` pour avoir un pointeur sur la dernière cellule :
+
+```ruby
+enregistrement Liste {
+    Cellule debutListe
+    Cellule finListe
+}
+```
+
+Un exemple de représentation :
+
+```mermaid
+graph LR
+    c1 --> c2[Cellule]
+    c2 --> c3[Cellule]
+    c3 --> c4[Fin]
+    c4 --> c1[Début]
+```
+
+
+# Exercice 2.3
+
+## Question 1
+
+> **Écrire une procédure de concaténation de deux listes en temps $O(1)$**
+
+On rappelle notre implémentation : on a une liste chaînée avec pointeur sur le début et la fin de la liste.
+
+```ruby
+enregistrement Liste {
+    Cellule debutListe
+    Cellule finListe
+}
+```
+
+On peut donc concaténer deux listes en faisant pointer la fin de la première liste sur le début de la deuxième liste :
+
+```ruby
+Liste concatener(Liste l1, Liste l2) {
+    if (estVideListe(l1)) {
+        return l2
+    }
+    if (estVideListe(l2)) {
+        return l1
+    }
+    l1.finListe.suiv = l2.debutListe
+    l1.finListe = l2.finListe
+    return l1
+}
+```
+
+Le temps est bien en $O(1)$ car on ne fait que modifier les pointeurs.
\ No newline at end of file
diff --git a/Algorithmique/TD/TD2.pdf b/Algorithmique/TD/TD2.pdf
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-- 
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