From 485f82b257e0e3fe2981cad693bc5f325d0575d4 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Tue, 8 Nov 2022 11:20:37 +0100
Subject: [PATCH] updated to 2022

---
 slides/cours_6.md                        | 489 +++++++++++++++++++++++
 slides/figs/Binary_search_complexity.svg | 318 +++++++++++++++
 2 files changed, 807 insertions(+)
 create mode 100644 slides/cours_6.md
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diff --git a/slides/cours_6.md b/slides/cours_6.md
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+---
+title: "Récursivité et complexité"
+date: "2021-11-10"
+---
+
+# 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 $x^n$
+
+* Algorithme naîf et impératif
+
+    ```C
+    int pow(x, n) {
+        if (0 == n) {
+            return 1;
+        }
+        for (int i = 1; i < n; ++i) {
+            x *= x;
+        }
+        return x;
+    }
+    ```
+
+. . .
+
+* Complexité? Combien de multiplication en fonction de `n`?
+
+# Exponentiation rapide ou indienne (2/4)
+
+* 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
+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
+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/figs/Binary_search_complexity.svg b/slides/figs/Binary_search_complexity.svg
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