update for 7th week

slides/cours_7.md

+---
+title: "Tris"
+date: "2022-16-11"
+---
+
+# 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 moyenne = 0.0;
+for (int i = 0; i < N; ++i) { // N assignations
+    moyenne += tab[i];        // N additions
+}
+moyenne /= 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 moyenne = moyenne(N, tab); // O(N)
+double ecart = 0.0;
+for (int i = 0; i < N; ++i) {
+    ecart += pow(tab[i] - moyenne, 2); // N tours
+}
+ecart = sqrt(ecart); ecart /= 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 déjà
+triés du tableau.
+
+![Tri par insertion d'un tableau d'entiers](figs/tri_insertion.svg)
+
+# Tri par insertion (2/3)
+
+## Exercice: Proposer un algorithme
+
+. . .
+
+```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)
+
+Deux variables sont primordiales:
+
+```C
+int low, high; // les indices min/max des tableaux à trier
+```
+
+![Un exemple de quicksort.](figs/quicksort.svg)
+
+# Tri rapide ou quicksort (4/8)
+
+Deux variables sont primordiales:
+
+```C
+int low, high; // les indices min/max des tableaux à trier
+```
+
+## Pseudocode: quicksort
+
+```C
+void quicksort(array, low, high) {
+    if (more than 1 elems) {
+        pivot_ind = partition(array, low, high);
+        if (something left of pivot)
+            quicksort(array, low, pivot_ind - 1);
+        if (something right of pivot)
+            quicksort(array, pivot_ind + 1, high);
+    }
+}
+```
+
+# Tri rapide ou quicksort (5/8)
+
+## Pseudocode: partition
+
+```C
+int partition(array, low, high) {
+    pivot = array[high]; // choix arbitraire
+    i = low;
+    j = high-1;
+    while i < j {
+      en remontant i trouver le premier élément > pivot;
+      en descendant j trouver le premier élément < pivot;
+      swap(array[i], array[j]);
+      // les plus grands à droite
+      // mettre les plus petits à gauche
+    }
+    // on met le pivot "au milieu"
+    swap(array[i], array[pivot]);     
+    return 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);
+        }
+    }
+}
+```
+
+# L'algorithme à la main
+
+## Exercice *sur papier*
+
+* Trier par tri rapide le tableau `[5, -2, 1, 3, 10, 15, 7, 4]`
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+
+
+# 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++;
+        } while (array[i] < pivot && i < j);
+        do {
+            j--;
+        } 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: $\mathcal{O}(N\cdot \log_2(N))$.
+
+
+
+# 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 à
+  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 bulles d'un tableau d'entiers](figs/tri_bulles.svg)
+
+
+# Tri à bulle (3/4)
+
+## Exercice: écrire l'algorithme (poster le résultat sur matrix)
+
+. . .
+
+```C
+void bubble_sort(int size, int array[]) {
+    for i in [size-1, 1] {
+        sorted = true;
+        for j in [0, i-1] {
+            if (array[j] > array[j+1]) {
+                swap(array[j], array[j+1]);
+                sorted = false;
+            }
+        }
+        if (sorted) {
+            return;
+        }
+    }
+}
+```
+
+# 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
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+

slides/figs/quicksort.svg

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