slides_2022/cours_5.md

+---
+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 $0=0000.0000$ à $15.9375=1111.1111$.
+* 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*.
+
+![Nombres à virgule flottante 32 bits. Source:
+[Wikipedia](https://en.wikipedia.org/wiki/Single-precision_floating-point_format#/media/File:Float_example.svg)](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)
+
+![Un exercice de nombres à virgule flottante 32 bits. Source:
+[Wikipedia](https://en.wikipedia.org/wiki/Single-precision_floating-point_format#/media/File:Float_example.svg)](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` à `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 à 
+$$
+2^{-23}=1.19\cdot 10^{-7},
+$$
+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!

slides_2022/cours_6.md

+---
+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
+
+![Source: [Wikipédia](https://upload.wikimedia.org/wikipedia/commons/a/aa/Binary_search_complexity.svg)](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
+
+$$
+\mathcal{O}(N^2\mbox{ comparaisons}) + \mathcal{O}(N\mbox{swaps})=\mathcal{O}(N^2).
+$$
+
+

slides_2022/cours_7.md

+---
+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 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 (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
+```
+
+![Un exemple de quicksort.](figs/quicksort.svg)
+
+# 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
+
+
+
+
+
+
+
+
+
+
+
+
+
+```
+
+
+

slides_2022/cours_8.md

+---
+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 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
+rien tri_a_bulles(entier tableau[])
+    pour i de longueur(tableau)-1 à 1:
+        trié = vrai
+        pour j de 0 à 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.
+
+![Problème des 8-reines. Source:
+[wikipedia](https://fr.wikipedia.org/wiki/Problème_des_huit_dames)](./figs/fig_recursivite_8_reines.png){width=35%}
+
+# Problème des 2-reines
+
+![Le problème des 2 reines n'a pas de solution.](figs/2reines.svg){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'à 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
+
+![Le problème des 3 reines n'a pas de solution non plus.](figs/3reines.svg)
+
+# Problème des 4-reines
+
+![Le problème des 4 reines a une solution.](figs/4reines.svg)
+
+# Problème des 4-reines, symétrie
+
+![Le problème des 4 reines a une autre solution (symétrie
+horizontale).](figs/4reines_sym.svg)
+
+# 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;
+        }
+    }
+}
+```
+

slides_2022/cours_9.md

+---
+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*).
+
+![Une pile où on ajoute A, puis B avant de les retirer. Source:
+[Wikipedia](https://upload.wikimedia.org/wikipedia/commons/e/e1/Stack_%28data_structure%29_LIFO.svg)](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 à partir de 1 à 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
+
+![Un exemple de tri à deux piles](figs/tri_piles.svg){width=70%}
+
+# Le tri à deux piles (2/3)
+
+## Exercice: formaliser l'algorithme
+
+. . .
+
+## Algorithme de tri nécessitant 2 piles (G, D)
+
+Soit `tab` le tableau à trier:
+
+```C
+pour i de 0 à N-1
+    tant que (tab[i] > que le sommet de G)
+        dépiler G dans D
+    tant que (tab[i] < que le sommet de D)
+        dépiler de D dans G
+    empiler tab[i] sur G
+dépiler tout D dans G
+tab est trié dans G
+```
+
+# Le tri à deux piles (3/3)
+
+## Exercice: trier le tableau `[2, 10, 5, 20, 15]`
+
+```C
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+```

slides_2022/docker-compose.yml

+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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slides_2022/figs/Float_example_bare.svg

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