From 0ea6fbe2951e9cda532c339d3dc2ab06bb4cc049 Mon Sep 17 00:00:00 2001
From: "paul.albuquer" <paul.albuquerque@hesge.ch>
Date: Mon, 18 Oct 2021 13:55:25 +0200
Subject: [PATCH] added lesson 6, fig. 8-queens, version 2020-21, recursivity
codes
---
lessons/contenu_cours_6.md | 97 ++++++++++++++++++
lessons/figures/fig_recursivite_8_reines.png | Bin 0 -> 11151 bytes
source_codes/recursivity/lissage.c | 66 ++++++++++++
source_codes/recursivity/rec_binary.c | 23 +++++
source_codes/recursivity/rec_factorielle.c | 19 ++++
source_codes/recursivity/rec_fibonacci.c | 22 ++++
source_codes/recursivity/rec_pgcd.c | 23 +++++
source_codes/recursivity/rec_ppcm.c | 24 +++++
.../recursivity/rec_puissance_indienne.c | 26 +++++
source_codes/recursivity/rec_reines.c | 75 ++++++++++++++
source_codes/recursivity/rec_reines_skel.c | 54 ++++++++++
11 files changed, 429 insertions(+)
create mode 100644 lessons/contenu_cours_6.md
create mode 100644 lessons/figures/fig_recursivite_8_reines.png
create mode 100644 source_codes/recursivity/lissage.c
create mode 100644 source_codes/recursivity/rec_binary.c
create mode 100644 source_codes/recursivity/rec_factorielle.c
create mode 100644 source_codes/recursivity/rec_fibonacci.c
create mode 100644 source_codes/recursivity/rec_pgcd.c
create mode 100644 source_codes/recursivity/rec_ppcm.c
create mode 100644 source_codes/recursivity/rec_puissance_indienne.c
create mode 100644 source_codes/recursivity/rec_reines.c
create mode 100644 source_codes/recursivity/rec_reines_skel.c
diff --git a/lessons/contenu_cours_6.md b/lessons/contenu_cours_6.md
new file mode 100644
index 0000000..0f59c42
--- /dev/null
+++ b/lessons/contenu_cours_6.md
@@ -0,0 +1,97 @@
+# Algorithmique et structures de données 2020-21
+
+Contenu du cours 6 du 28.10.2020
+
+*****
+
+## Récursivité
+
+- Exemple de la factorielle : n ! = n·(n-1)·(n-2)·... ·3·2·1 = n·(n-1) !
+ Donc fact(n) = n·fact(n-1) (récursivité)
+ et fact(1) = 1 (condition d'arrêt)
+```C
+ int fact(int n) {
+ if (n > 1) {
+ return n*fact(n1);
+ } else {
+ return 1;
+ }
+ }
+ void main() {
+ int f = fact(4);
+ }
+```
+
+
+
+- Exemple du PGCD
+ Algorithme d'Euclide pour le PGCD de 42 et 27
+
+> 42 = 27·1 + 15
+> 27 = 15·1 + 12
+> 15 = 12·1 + 3
+> 12 = 3·4 + 0
+ PGCD(42,27)=PGCD(27,15)=PGCD(15,12)=PGCD(12,3)=3
+```C
+ int pgcd(int n,int m) {
+ if (n%m > 0) {
+ return pgcd(m,n%m);
+ } else {
+ return m;
+ }
+ }
+```
+
+
+
+- Exemple de l'écriture binaire
+```C
+ void binaire(int n) {
+ printf("%d",n%2);
+ if (n/2 != 0) {
+ binaire(n/2);
+ } else {
+ printf("\n");
+ }
+ // printf("%d",n%2);
+ }
+
+ Binaire(13); // affiche 1 0 1 1 puis un retour à la ligne`
+```
+> > > $\hspace*{36mm} 2^0 2^1 2^2 2^3$
+
+
+
+- Que se passe-t-il si on décommente le deuxième `printf` ?
+
+## Exemples et exercices de récursivité
+
+- Algorithme du PPCM de deux nombres `n` et `m`
+ - `ppcm(mult_n,mult_m) = ppcm(mult_n + n, mult_m)`
+ si `mult_n < mult_m` (récursivité)
+ - `ppcm(mult_n,mult_m) = ppcm(mult_n, mult_m + m)`
+ si `mult_n > mult_m` (récursivité)
+ - `ppcm(mult_n,mult_m) = mult_n`
+ si `mult_n = mult_m` (condition d’arrêt)
+
+- Puissance indienne
+\begin{align*}
+a^b & = a^{b/2}\cdot a^{b/2} & \textrm{ si $b$ est pair (récursivité)} \\
+a^b & = a^{b-1}\cdot a & \textrm{ si $b$ est impair (récursivité)} \\
+a^0 & = 1 & \textrm{ (condition d’arrêt)}
+\end{align*}
+
+- Suite de Fibonacci
+\begin{align*}
+a_n & = a_{n-1} + a_{n-2} & \textrm{ (récursivité)} \\
+a_1 & = 1, a_0 = 0 & \textrm{ (condition d’arrêt)}
+\end{align*}
+
+## Problème des 8-reines
+
+- Le but du problème des 8 reines est de placer 8 reines d'un jeu d'échecs sur un échiquier de $8 \times 8$ cases sans que les reines ne puissent se menacer mutuellement, conformément aux règles du jeu d'échecs.
+- Ainsi, deux reines ne devraient jamais partager la même rangée, colonne, ou diagonale.
+- Le problème se généralise au placement de N reines sur un échiquier de $N \times N$ cases. Pour $N=8$, il y a 92 solutions
+- Il s'agit d'un exemple classique de problème de backtracking qui se programme avec la récursivité.
+
+
\ No newline at end of file
diff --git a/lessons/figures/fig_recursivite_8_reines.png b/lessons/figures/fig_recursivite_8_reines.png
new file mode 100644
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HcmV?d00001
diff --git a/source_codes/recursivity/lissage.c b/source_codes/recursivity/lissage.c
new file mode 100644
index 0000000..f501ca0
--- /dev/null
+++ b/source_codes/recursivity/lissage.c
@@ -0,0 +1,66 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+const int SIZE = 64;
+
+// Moyenne des pixels de l'image <im>
+double moyenne(int n,int im[n][n]) {
+ double size = n*n;
+ double moy = 0.0;
+ for (int i=0;i<n;i++) {
+ for (int j=0;j<n;j++) {
+ moy += im[i][j]/size;
+ }
+ }
+ return moy;
+}
+
+// Nombre de pixels de l'image <im> dont la différence
+// à la valeur <val> est supérieure à une tolérance <tol>
+int nb_pixels_diff(int n,int im[n][n],double val,double tol) {
+ // à compléter
+ return 0;
+}
+
+// Affectation de la valeur <val> à tous les pixels de l’image <im>
+void copy(int n,int im[n][n],int val) {
+ // à compléter
+}
+
+void copy_slice(int decal_x,int decal_y,int n,int im1[n][n],int k,int im2[k][k]) {
+ // à compléter
+}
+
+void copy_back(int decal_x,int decal_y,int n,int im1[n][n],int k,int im2[k][k]) {
+ // à compléter
+}
+
+//Lissage récursif de l'image <im> en remplaçant les pixels
+// d'un quadrant par leur moyenne
+void lissage(int n,int im[n][n],double tol,double seuil) {
+ double moy = moyenne(n,im);
+ double size = n*n;
+ if (nb_pixels_diff(n,im,moy,tol)/size < seuil) {
+ copy(n,im,moy);
+ } else { // si on ne peut pas lisser l'image, alors on essaie sur les quadrants
+ int quadrant[n/2][n/2];
+ copy_slice(0,0,n,im,n/2,quadrant);
+ lissage(n/2,quadrant,tol,seuil);
+ copy_back(0,0,n,im,n/2,quadrant);
+ copy_slice(n/2,0,n,im,n/2,quadrant);
+ lissage(n,quadrant,tol,seuil);
+ copy_back(n/2,0,n,im,n/2,quadrant);
+ copy_slice(0,n/2,n,im,n/2,quadrant);
+ lissage(n/2,quadrant,tol,seuil);
+ copy_back(0,n/2,n,im,n/2,quadrant);
+ copy_slice(n/2,n/2,n,im,n/2,quadrant);
+ lissage(n,quadrant,tol,seuil);
+ copy_back(n/2,n/2,n,im,n/2,quadrant);
+ }
+}
+
+void main() {
+ int image_gris[SIZE][SIZE];
+ copy(SIZE,image_gris,234);
+ lissage(SIZE,image_gris,0.1,10.0);
+}
diff --git a/source_codes/recursivity/rec_binary.c b/source_codes/recursivity/rec_binary.c
new file mode 100644
index 0000000..2eb856b
--- /dev/null
+++ b/source_codes/recursivity/rec_binary.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+// Ecriture binaire
+void binaire(int n);
+
+int main() {
+ int n;
+ printf("n=");
+ scanf("%d",&n);
+ binaire(n);
+ printf("\n");
+}
+
+// Ecriture binaire
+void binaire(int n) {
+ //printf("%d",n%2);
+ if (n/2 != 0) {
+ binaire(n/2);
+ }
+ printf("%d",n%2);
+}
+
diff --git a/source_codes/recursivity/rec_factorielle.c b/source_codes/recursivity/rec_factorielle.c
new file mode 100644
index 0000000..e5513d1
--- /dev/null
+++ b/source_codes/recursivity/rec_factorielle.c
@@ -0,0 +1,19 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+int fact(int n);
+
+void main() {
+ int n;
+ printf("n=");
+ scanf("%d",&n);
+ printf("%d\n",fact(n));
+}
+
+int fact(int n) {
+ if (1 == n) {
+ return 1;
+ } else {
+ return n*fact(n-1);
+ }
+}
diff --git a/source_codes/recursivity/rec_fibonacci.c b/source_codes/recursivity/rec_fibonacci.c
new file mode 100644
index 0000000..0c8a487
--- /dev/null
+++ b/source_codes/recursivity/rec_fibonacci.c
@@ -0,0 +1,22 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+// Suite de Fibonacci: a_n = a_{n-1} + a_{n-2}, a_0 = 0, a_1 = 1
+int fib(int n);
+
+int main() {
+ int n;
+ printf("n=");
+ scanf("%d",&n);
+ printf("%d\n",fib(n));
+}
+
+// Suite de Fibonacci: a_n = a_{n-1} + a_{n-2}, a_0 = 0, a_1 = 1
+int fib(int n) {
+ switch(n) {
+ case 0: return 0;
+ case 1: return 1;
+ default: return fib(n-1)+fib(n-2);
+ }
+}
+
diff --git a/source_codes/recursivity/rec_pgcd.c b/source_codes/recursivity/rec_pgcd.c
new file mode 100644
index 0000000..4c6b6a4
--- /dev/null
+++ b/source_codes/recursivity/rec_pgcd.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+int pgcd(int n,int m);
+
+void main() {
+ int n,m;
+ printf("n=");
+ scanf("%d",&n);
+ printf("m=");
+ scanf("%d",&m);
+ printf("%d\n",pgcd(n,m));
+}
+
+int pgcd(int n,int m) {
+ if (n%m > 0) {
+ return pgcd(m,n%m);
+ } else {
+ return m;
+ }
+}
+
+
diff --git a/source_codes/recursivity/rec_ppcm.c b/source_codes/recursivity/rec_ppcm.c
new file mode 100644
index 0000000..ef505ee
--- /dev/null
+++ b/source_codes/recursivity/rec_ppcm.c
@@ -0,0 +1,24 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+int ppcm(int mult_n,int mult_m,int n,int m);
+
+void main() {
+ int n,m;
+ printf("n=");
+ scanf("%d",&n);
+ printf("m=");
+ scanf("%d",&m);
+ printf("%d\n",ppcm(n,m,n,m));
+}
+
+int ppcm(int mult_n,int mult_m,int n,int m) {
+ if (mult_n < mult_m) {
+ return ppcm(n+mult_n,mult_m,n,m);
+ } else if (mult_n > mult_m) {
+ return ppcm(mult_n,m+mult_m,n,m);
+ } else {
+ return mult_n;
+ }
+}
+
diff --git a/source_codes/recursivity/rec_puissance_indienne.c b/source_codes/recursivity/rec_puissance_indienne.c
new file mode 100644
index 0000000..ceff6ba
--- /dev/null
+++ b/source_codes/recursivity/rec_puissance_indienne.c
@@ -0,0 +1,26 @@
+#include <stdio.h>
+#include <stdlib.h>
+
+// Puissance indienne
+int puissance(int a,int b);
+
+int main() {
+ int n,m;
+ printf("n=");
+ scanf("%d",&n);
+ printf("m=");
+ scanf("%d",&m);
+ printf("%d\n",puissance(n,m));
+}
+
+// Puissance indienne
+int puissance(int a,int b) {
+ if (0 == b) {
+ return 1;
+ } else if (0 == b%2) {
+ return puissance(a,b/2)*puissance(a,b/2);
+ } else {
+ return puissance(a,b-1)*a;
+ }
+}
+
diff --git a/source_codes/recursivity/rec_reines.c b/source_codes/recursivity/rec_reines.c
new file mode 100644
index 0000000..5f54088
--- /dev/null
+++ b/source_codes/recursivity/rec_reines.c
@@ -0,0 +1,75 @@
+// Problème des N-reines
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+
+bool** board_alloc(int n) {
+ bool** tab = malloc(n*sizeof(bool*));
+ for (int i=0;i<n;i++) {
+ tab[i] = malloc(n*sizeof(bool));
+ for (int j=0;j<n;j++) {
+ tab[i][j] = true;
+ }
+ }
+ return tab;
+}
+
+void board_free(int n,bool** tab) {
+ for (int i=0;i<n;i++) {
+ free(tab[i]);
+ }
+ free(tab);
+}
+
+bool** clone(int n,bool** board) {
+ bool** tab = board_alloc(n);
+ for (int i=0;i<n;i++) {
+ for (int j=0;j<n;j++) {
+ tab[i][j] = board[i][j];
+ }
+ }
+ return tab;
+}
+
+// Retourne une copie du tableau <board> complété avec les positions
+// prises sur la droite droite par une reine placée en <board(li,co)>
+bool** prises_devant(int n,bool** board,int li,int co) {
+ bool** cases_prises = clone(n,board);
+ cases_prises[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) {
+ cases_prises[li-j][co+j] = false;
+ }
+ cases_prises[li][co+j] = false;
+ if (li+j < n) {
+ cases_prises[li+j][co+j] = false;
+ }
+ }
+ return cases_prises;
+}
+
+// Calcule le nombre de solutions au problème des <N> reines
+void nb_sol(int n,bool** board,int co, int* ptr_cpt) {
+ for (int li=0;li<n;li++) {
+ if (board[li][co]) {
+ if (co < n-1) {
+ bool** tab = prises_devant(n,board,li,co);
+ nb_sol(n,tab,co+1,ptr_cpt);
+ board_free(n,tab);
+ } else {
+ *ptr_cpt = (*ptr_cpt)+1;
+ }
+ }
+ }
+}
+
+void main() {
+ int n = 8;
+ // échiquier où placer les reines
+ bool** board = board_alloc(n);
+ // compteur du nombre de solutions au problème des <N> reines
+ int cpt = 0;
+ nb_sol(n,board,0,&cpt);
+ printf("Nombre de solutions: %d\n",cpt);
+}
diff --git a/source_codes/recursivity/rec_reines_skel.c b/source_codes/recursivity/rec_reines_skel.c
new file mode 100644
index 0000000..7b2b056
--- /dev/null
+++ b/source_codes/recursivity/rec_reines_skel.c
@@ -0,0 +1,54 @@
+// Problème des N-reines
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+
+// Alloue un tableau de dimension nxn de booléens dont le contenu est true
+bool** board_alloc(int n) {
+ // à compléter!!!!
+ return NULL;
+}
+
+// Désalloue un tableau de dimension nxn de booléens
+void board_free(int n,bool** tab) {
+ // à compléter!!!!
+}
+
+// Retourne un clone de <board>
+bool** clone(int n,bool** board) {
+ // à compléter!!!!
+ return NULL;
+}
+
+// Retourne une copie du tableau <board> complété avec les positions
+// prises sur la droite droite par une reine placée en <board(li,co)>
+bool** prises_devant(int n,bool** board,int li,int co) {
+ bool** cases_prises = clone(n,board);
+ // à compléter!!!!
+ return cases_prises;
+}
+
+// Calcule le nombre de solutions au problème des <N> reines
+void nb_sol(int n,bool** board,int co, int* ptr_cpt) {
+ for (int li=0;li<n;li++) {
+ if (board[li][co]) {
+ if (co < n-1) {
+ bool** tab = prises_devant(n,board,li,co);
+ nb_sol(n,tab,co+1,ptr_cpt);
+ board_free(n,tab);
+ } else {
+ *ptr_cpt = (*ptr_cpt)+1;
+ }
+ }
+ }
+}
+
+void main() {
+ int n = 8;
+ // échiquier où placer les reines
+ bool** board = board_alloc(n);
+ // compteur du nombre de solutions au problème des <N> reines
+ int cpt = 0;
+ nb_sol(n,board,0,&cpt);
+ printf("Nombre de solutions: %d\n",cpt);
+}
--
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