From 76e6a7e614ce333794632b9754d97b9557a2a007 Mon Sep 17 00:00:00 2001
From: "paul.albuquer" <paul.albuquerque@hesge.ch>
Date: Sun, 14 Nov 2021 13:10:05 +0100
Subject: [PATCH] added transformation infixe-postfixe with stack

---
 lessons/contenu_cours_applications_piles.md   |  74 ++++++++++++++++++
 .../figures/fig_transf_infixe_postfixe.png    | Bin 0 -> 71361 bytes
 2 files changed, 74 insertions(+)
 create mode 100644 lessons/contenu_cours_applications_piles.md
 create mode 100644 lessons/figures/fig_transf_infixe_postfixe.png

diff --git a/lessons/contenu_cours_applications_piles.md b/lessons/contenu_cours_applications_piles.md
new file mode 100644
index 0000000..a8fe3f2
--- /dev/null
+++ b/lessons/contenu_cours_applications_piles.md
@@ -0,0 +1,74 @@
+# Algorithmes et structures de données
+
+# Deux applications des piles
+
+*****
+
+##  La transformation infixe $\Leftarrow$ postfixe
+
+Exemples:
+
+| Expression infixe     |                  | Expression postfixe |
+|-----------------------|------------------|---------------------|
+| A + B * C             |  transformé en   | A B C * +           |
+| A + (B / C – D ^ E )  |  transformé en   | A B C / D E ^ – +   |
+
+
+A chaque opérateur, on associe une priorité, par exemple:
+
+| Opérateur | Priorité |
+|-----------|----------|
+|     ^     |    3     |
+|    *, /   |    2     |
+|    +, -   |    1     |
+
+
+L'algorithme utilise une pile dans laquelle on stocke les opérateurs et les parenthèses ouvertes 
+rencontrées dans l'expression.
+
+Les étapes sont les suivantes :
+
+1. Examiner le prochain caractère dans l'expression donnée.
+2. Si c'est un opérande,  le mettre directement dans l'expression du résultat.
+3. Si c'est une parenthèse ouverte, la mettre dans la pile avec la priorité zéro.
+4. Si c'est un opérateur, comparer sa priorité avec celle de l'élément au sommet de la pile: 
+    * Si la priorité de l'opérateur est plus grande que l'opérateur du sommet de la pile, empiler ce nouvel opérateur. 
+    * Sinon prendre l'opérateur du sommet de la pile, le mettre dans l'expression du résultat et répéter la comparaison avec le nouveau sommet de la pile, jusqu'à ce qu'un opérateur de priorité inférieure à celle de l'opérateur de l'expression apparaisse au sommet de la pile ou jusqu'à ce que la pile soit vide.
+5. Si c'est une parenthèse fermée, enlever de la pile les opérateurs et les placer dans l'expression du résultat jusqu'à ce qu'une parenthèse ouverte  apparaisse au sommet de la pile, se défaire de cette parenthèse.
+6. Si l'expression donnée est vide, enlever de la pile les opérateurs restants et les placer dans l'expression du résultat.
+
+La table ci-dessous montre la réalisation d'une transformation infixe - postfixe sur une expression algébrique (supposée syntaxiquement correcte):
+
+![Exemple de transformation infixe-postfixe](./figures/fig_transf_infixe_postfixe.png)
+
+**Remarque:** On utilise aussi d'autres structures de données pour traiter ce problème. Entre autres les arbres.
+
+## L'évaluation d'une expression en notation postfixée
+
+La manière la plus simple de procéder est d'utiliser une pile contenant à chaque instant les valeurs des opérandes déjà calculés.
+
+Dans une expression donnée sous forme postfixée, chaque opérateur  porte  sur  les  opérandes qui  le  précédent immédiatement, ceux ci doivent toujours se trouver en tête de pile au  moment de l'opération. Le résultat d'une opération est un nouvel opérande qui est remis au sommet de la pile pour la suite de l'évaluation.
+
+Les règles d'évaluation sont les suivantes:
+
+* la valeur d'un opérande est toujours empilée.
+* Un opérateur s'applique toujours aux 2 opérandes en tête de pile.
+* Le résultat de l'opération est remis en tête de pile.
+
+Exemple:
+
+Soit l'expression postfixe à évaluer:    A B C + * D -
+
+| Caractères lus | Pile des opérandes (nombres) |
+|----------------|------------------------------|
+|       A        | A                            |
+|       B        | A, B                         |
+|       C        | A, B, C                      |
+|       +        | A, B+C                       |
+|       *        | A*(B+C)                      |
+|       D        | A*(B+C), D                   |
+|       A        | *(B+C)-D                     |
+
+              La pile contient finalement la valeur de l'expression.
+
+Ce traitement exige naturellement que l'expression à évaluer soit syntaxiquement correcte.
\ No newline at end of file
diff --git a/lessons/figures/fig_transf_infixe_postfixe.png b/lessons/figures/fig_transf_infixe_postfixe.png
new file mode 100644
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literal 0
HcmV?d00001

-- 
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