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---
title: "B-Arbres"
date: "2022-04-13"
patat:
eval:
tai:
command: fish
fragment: false
replace: true
ccc:
command: fish
fragment: false
replace: true
images:
backend: auto
---
# Rappel: Les B-arbres
## Pourquoi utiliser un B-arbre?
. . .
## À quoi ressemble un B-arbre?
. . .
## Qu'est-ce qu'un B-arbre d'ordre $n$
* Chaque page d'un arbre contient au plus $2\cdot n$ *clés*;
* Chaque page (excepté la racine) contient au moins $n$ clés;
* Chaque page qui contient $m$ clés contient soit:
* $0$ descendants;
* $m+1$ descendants.
* Toutes les pages terminales apparaissent au même niveau.
# Rappel: Les B-arbres
## Quelques propriétés
* Dans chaque noeud les clés sont **triées**.
* Chaque page contient au plus $n$ noeuds: check;
* Chaque noeud avec $m$ clés a $m+1$ descendants;
* Toutes les feuilles apparaissent au même niveau.
# Les B-arbres
## Exemple de recherche: trouver `32`
![B-arbre d'ordre 2.](figs/barbres_exemple.png)
. . .
* Si `n` plus petit que la 1e clé ou plus grand que la dernière descendre.
* Sinon parcourir (par bissection ou séquentiellement) jusqu'à trouver ou descendre entre 2 éléments.
# Les B-arbres
## La recherche de la clé `C` algorithme
0. En partant de la racine.
1. Si on est dans une feuille:
* Si la `C` est dans une page, retourner la page;
* Sinon c'est perdu.
2. Sinon:
* Tant que `C > page` passer à la page suivante
* Descendre
# Les B-arbres
## Exercice: insérer `22, 45, 50, 5, 32, 55, 60, 41` dans l'arbre d'ordre 2
![](figs/barbres_ex1.png)
. . .
![](figs/barbres_ex5.png)
# Les B-arbres
## L'algorithme d'insertion
0. Rechercher la feuille (la page a aucun enfant) où insérer;
1. Si la page n'est pas pleine insérer dans l'ordre croissant.
2. Si la page est pleine:
1. On décale les éléments plus grand que `N`;
2. On insère `N` dans la place "vide";
3. On trouve la valeur médiane `M` de la page (quel indice?);
4. On crée une nouvelle page de droite;
5. On copie les valeur à droite de `M` dans la nouvelle page;
6. On promeut `M` dans la page du dessus;
7. On connecte le pointeur de gauche de `M` et de droite de `M` avec l'ancienne et la nouvelle page respectivement.
# Les B-arbres
## Pseudo-code structure de données (3min, matrix)?
. . .
```C
struct page
entier ordre, nb
element tab[2*ordre + 2]
```
```C
struct element
int clé
page pg
```
# Les B-arbres
\footnotesize
## Les fonctions utilitaires (5min matrix)
```C
booléen est_feuille(page) // la page est elle une feuille?
entier position(page, valeur) // à quelle indice on insère?
booléen est_dans_page(page, valeur) // la valeur est dans la page
```
. . .
```C
booléen est_feuille(page)
retourne (page.tab[0] == vide)
entier position(page, valeur)
i = 0
tant que i < page.nb && val >= page.tab[i]
i += 1
retourne i
booléen est_dans_page(page, valeur)
i = position(page, valeur)
retourne (i > 0 && page.tab[i] == valeur)
```
# Les B-arbres
\footnotesize
## Les fonctions utilitaires (5min matrix)
```C
page nouvelle_page(ordre) // creer une page
rien liberer_memoire(page) // liberer tout un arbre!
```
. . .
```C
page nouvelle_page(ordre)
page = allouer(page)
page.ordre = ordre
page.nb = 0
page.tab = allouer(2*ordre+2)
retourner page
rien liberer_memoire(page)
si est_feuille(page)
liberer(page.tab)
liberer(page)
sinon
pour fille dans page.tab
liberer_memoire(fille)
liberer(page.tab)
liberer(page)
```
# Les B-arbres
## Les fonctions (5min matrix)
```C
page recherche(page, valeur) // retourner la page contenant
// la valeur ou vide
```
. . .
```C
page recherche(page, valeur)
si est_dans_page(page, valeur)
retourne page
sinon si est_feuille(page) && !est_dans_page(page, valeur)
retourne vide
sinon
recherche(page.tab[position(page, valeur)], valeur)
```
# Les B-arbres
## Les fonctions
```C
page inserer(page, valeur) // inserer une valeur
```
. . .
```C
page inserer(page, valeur)
element = nouvel_element(valeur)
inserer_element(page, element) // on change elmement pour savoir s'il faut le remonter
si element.page != vide
page = ajouter_niveau(page, element) // si on atteint le sommet!
retourne page
```
# Les B-arbres
## Les fonctions
```C
rien inserer_element(page, element) // inserer un element et voir s'il remonte
```
. . .
```C
rien inserer_element(page, element)
si est_feuille(page)
placer(page, element)
sinon
sous_page = page.tab[position(page, element)].page
inserer_element(sous_page, element)
si element.page != vide
placer(page, element)
```
# Les B-arbres
## Les fonctions (5min matrix)
```C
rien placer(page, element) // inserer un élément
```
. . .
```C
rien placer(page, element)
i = position(page, element.clé)
pour i de 2*page.ordre à i+1
page.tab[i+1] = page.tab[i]
page.tab[i+1] = element
page.nb += 1
si page.nb > 2*page.ordre
scinder(page, element)
```
# Les B-arbres
## Les fonctions (5min matrix)
```C
rien scinder(page, element) // casser une page et remonter
```
. . .
```C
rien scinder(page, element)
new_page = new_page(page.ordre)
new_page.nb = page.ordre
pour i de 0 à ordre inclu
new_page.tab[i] = page.tab[i+ordre+1]
element.clé = page.tab[ordre+1].clé
element.page = new_page
```
# Les B-arbres
## Les fonctions (5min matrix)
```C
page ajouter_niveau(page, element) // si on remonte à la racine...
// on doit créer une nouvelle racine
```
. . .
```C
page ajouter_niveau(page, element)
tmp = nouvelle_page(page.ordre)
tmp.tab[0].page = page
tmp.tab[1].clé = element.clé
tmp.tab[1].page = element.page
retourne tmp
```
# Les B-arbres: suppression
## Cas simplissime
![Suppression de 25.](figs/barbres_ordre2_supp1.svg){width=80%}
. . .
![25 supprimé, on décale juste 27.](figs/barbres_ordre2_supp2.svg){width=80%}
<!-- # Les B-arbres -->
<!-- ## Structure de données en C (3min, matrix) -->
<!-- . . . -->
<!-- ```C -->
<!-- typedef struct _page { -->
<!-- int order, nb; -->
<!-- struct _element *tab; -->
<!-- } page; -->
<!-- ``` -->
<!-- ```C -->
<!-- typedef struct element { -->
<!-- int key; -->
<!-- struct _page *pg; -->
<!-- } element; -->
<!-- ``` -->
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