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slides/cours_22.md
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......@@ -15,12 +15,6 @@ patat:
backend: auto
---
# Le cours précédent
* Questions:
# Les B-arbres
## Problématique
......@@ -293,7 +287,7 @@ patat:
# Les B-arbres
## Exercice
## Exercice (matrix, 15min)
* Insérer 20, 40, 10, 30, 15, 35, 7, 26, 18, 22, 5, 42, 13, 46, 27, 8, 32, 38, 24, 45, 25, 2, 14, 28, 29, 41,
* Dans un B-arbre d'ordre 2.
......@@ -322,6 +316,66 @@ P_0 | K_1 | P_1 | K_2 | | P_i | K_{i+1} | | P_{m-1} | K_m | P_m
![Strcture d'une page de B-arbre d'ordre 2.](figs/barbres_struct.png)
1. On veut un tableau de `P_i, K_i => struct`;
2. `K_0` va être en "trop";
3. Pour simplifier l'insertion dans une page, on ajoute un élément de plus.
# Les B-arbres
## L'insertion cas noeud pas plein, insertion `4`?
![](figs/barbres_insert_easy.svg){width=50%}
. . .
## Solution
![](figs/barbres_insert_easy_after.svg){width=50%}
# Les B-arbres
## L'insertion cas noeud pas plein, insertion `N`
* On décale les éléments plus grand que `N`;
* On insère `N` dans la place "vide";
* Si la page n'est pas pleine, on a terminé.
# Les B-arbres
## L'insertion cas noeud plein, insertion `2`?
![](figs/barbres_insert_hard_before.svg){width=50%}
. . .
## Solution
![](figs/barbres_insert_hard_during.svg){width=50%}
# Les B-arbres
## L'insertion cas noeud plein, promotion `3`?
![](figs/barbres_insert_hard_during.svg){width=50%}
. . .
## Solution
![](figs/barbres_insert_hard_after.svg)
# Les B-arbres
## L'insertion cas noeud plein, insertion `N`
* On décale les éléments plus grand que `N`;
* On insère `N` dans la place "vide";
* Si la page est pleine:
* On trouve la valent médiance `M` de la page (quel indice?);
* On crée une nouvelle page de droite;
* On copie les valeur à droite de `M` dans la nouvelle page;
* On promeut `M` dans la page du dessus;
* On connecte le pointeur de gauche de `M` et de droite de `M` avec l'ancienne et la nouvelle page respectivement.
# Les B-arbres
......@@ -331,7 +385,7 @@ P_0 | K_1 | P_1 | K_2 | | P_i | K_{i+1} | | P_{m-1} | K_m | P_m
```C
struct page
int ordre, nb
entier ordre, nb
element tab[2*ordre + 2]
```
......@@ -343,24 +397,200 @@ struct element
# 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
## Structure de données en C (3min, matrix)
## Les fonctions (5min matrix)
```C
rien scinder(page, element) // casser une page et remonter
```
. . .
```C
typedef struct _page {
int order, nb;
struct _element *tab;
} page;
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
typedef struct element {
int key;
struct _page *pg;
} element;
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 -->
<!-- ## 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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