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+---
+title: "Barnes-Hut et B-arbres"
+date: "2023-05-12"
+---
+
+# Le cours précédent
+
+## A quoi sert l'algorithme de Barnes-Hut?
+
+. . .
+
+* A accélérer la résolution du problème à $n$-corps avec la gravitation,
+* si on peut vivre avec une réduction de précision.
+
+## Sur quelle structure de données se base l'algorithme?
+
+* L'arbre quaternaire.
+
+. . .
+
+## Quelle est l'idée générale?
+
+. . .
+
+* Si un groupe d'étoiles est suffisamment loin, on le modélise comme un corps unique situé en son centre de masse.
+* Exemple: Si on simule plusieurs galaxies, on considère chaque galaxie comme un corps unique!
+* Un arbre quaternaire est une structure parfaite pour regrouper les étoiles.
+
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+::::
+
+:::: {.column width=50%}
+
+## Problématique
+
+* On veut calculer la force sur $1$.
+
+::::
+
+:::
+
+. . .
+
+
+::: columns
+
+:::: {.column width=50%}
+
+## Illustration: le cas à 10 corps
+
+{width=60%}
+
+
+::::
+
+:::: {.column width=50%}
+
+## Résultat
+
+* Calcul et somme des forces venant des $9$ autre corps.
+
+::::
+
+:::
+
+# Le cas à 10 corps
+
+::: columns
+
+:::: {.column width=50%}
+
+## Réduction d'un groupe à un seul corps
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Idée
+
+* On accélère le calcul en traitant un groupe comme un seul corps.
+* Fonctionne uniquement si le groupe est assez loin.
+* Autrement l'approximation est trop grossière.
+
+::::
+
+:::
+
+# Solution: l'arbre quaternaire
+
+## Corps célestes - arbre
+
+
+
+* On omet les nœuds vides pour éviter la surcharge.
+* La numérotation est:
+ * 0: ID
+ * 1: SD
+ * 2: IG
+ * 3: SG
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 1
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant ID.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 2
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est vide, on insère.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (1/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 1
+
+{width=100%}
+
+* Quadrant SD.
+* La feuille est prise par 2.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (2/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 2
+
+{width=100%}
+
+* On crée un nouveau nœud.
+* Deux corps dans le nœud ID.
+* On crée un nouveau nœud.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Insertion corps 3 (3/N)
+
+{width=100%}
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre, niveau 3
+
+{width=100%}
+
+* 2 va dans ID.
+* 3 va dans SG.
+* C'est des feuilles vides, tout va bien.
+
+::::
+
+:::
+
+# Exemple d'insertion
+
+::: columns
+
+:::: {.column width=50%}
+
+## Que fait-on avec les nœuds intérieurs?
+
+* On les utilise pour:
+ * stocker la masse totale;
+ * stocker le centre de masse.
+
+\begin{align}
+m&=m_2+m_3,\\
+\vec x &= \frac{m_2\vec x_2+m_3\vec x_3}{m}.
+\end{align}
+
+## Chaque feuille contient **une étoile**
+
+::::
+
+:::: {.column width=50%}
+
+## Arbre
+
+{width=100%}
+
+::::
+
+:::
+
+# Résumé
+
+* Insertion du corps `c` dans le nœud `n` en partant de la racine.
+* Si le nœud `n`
+ * ne contient pas de corps, on y dépose `c`,
+ * est interne, on met à jour masse et centre de masse. `c` est inséré récursivement dans le bon quadrant.
+ * est externe, on subdivise `n`, on met à jour la masse et centre de masse, on insère récursivement les deux nœuds dans les quadrants appropriés.
+
+## Remarque
+
+* Il faut stocker les coordonnées des quadrants.
+* Un nœud a un comportement différent s'il est interne ou externe.
+
+# Algorithme d'insertion
+
+## Structure de données
+
+```C
+struct node
+ etoile e // externe: pour stocker
+ etoile sup_etoile // interne: pour stocker m, x
+ quadrant q // coordonnées du quadrant
+ node enfants[4]
+```
+
+## Remarque:
+
+* On fait une simplification "moche": `sup_etoile` pourrait juste avoir une masse et une position.
+
+# Algorithme d'insertion
+
+\footnotesize
+
+## Algorithme d'insertion, pseudo-code (15min, matrix)
+
+. . .
+
+```C
+rien insertion_etoile(arbre, e)
+ si (!est_vide(arbre) && dans_le_quadrant(arbre.q, e.x)) {
+ si (est_feuille(arbre))
+ si (!contient_etoile(arbre))
+ arbre.e = e
+ sinon
+ // on crée enfants et arbre.sup_etoile est initialisée
+ subdivision_arbre(arbre, e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, arbre.e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, e)
+ destruction(arbre.e)
+ sinon
+ maj_masse_cdm(arbre.sup_etoile, e)
+ pour enfant dans arbre.enfants
+ insertion_etoile(enfant, e)
+```
+
+# Utilisation de l'arbre
+
+* L'arbre est rempli: comment on calcule la force sur le corps 1?
+* Parcours de l'arbre:
+ * si la distance entre 1 et le centre de masse est suffisante, on utilise la masse totale et centre de masse pour calculer la force.
+ * sinon, on continue le parcours
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Le cadrant ID ne contient que `1`, rien à faire.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Le cadrant SG ne contient `5` corps.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* La distance entre `1` et le centre de masse de SG est `d`.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* La distance entre `1` et le centre de masse de SG est `d`.
+* Est-ce que `d` est assez grand?
+* On va comparer avec la distance `d` avec la taille du quadrant `s`.
+
+# Critère $\theta$
+
+* On compare $d=||\vec x_1-\vec x_{cm}||$ avec $s$ la taille du quadrant.
+* Le domain est assez éloigné si
+
+ $$
+ \frac{s}{d}<\theta,
+ $$
+* $\theta$ est la valeur de seuil.
+* Une valeur typique est $\theta=0.5$, donc la condition devient
+
+ $$
+ d>2s.
+ $$
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Ici $d<2s$, domaine rejeté.
+* ON descend dans l'arbre.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* `s` est plus petit, mais....
+* Cela ne suffit pas $d<2s$, domaine rejeté.
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Les nœuds sont des feuilles, on calcule la force.
+* On ajoute la force qu'ils exercent sur `1`.
+
+# Algorithme pour le calcul de la force
+
+Pour calculer la force sur un corps `c`, on parcourt l'arbre en commençant par la racine:
+
+* Si le nœud `n` est une feuille et n'est pas `c`, on ajoute la force dûe à `n` sur `c`;
+* Sinon si $s/d<\theta$, on traite `n` comme une feuille et on ajoute la force dûe à `n` sur `c`;
+* Sinon on continue sur les enfants récursivement.
+
+
+## Cotinuous notre exemple précédent!
+
+# Calcul de la force
+
+## Calcul de la force sur `1`
+
+
+
+* Il y a deux corps dans le quadrant vert.
+* Quel est le critère pour remplacer les étoiles par leur centre de masse?
+
+. . .
+
+* Et oui! $d>2s$ on peut remplacer les étoiles par leur centre de masse!
+
+# Algorithme du calcul de force
+
+## Écrire le psuedo-code du calcul de la force
+
+\footnotesize
+
+```C
+rien maj_force_sur_etoile(arbre, e, theta)
+ si est_vide(arbre)
+ retourne
+
+ si est_feuille(arbre) && contient_etoile(arbre) && dans_le_quadrant(arbre.q, e.x)
+ maj_force(e, arbre.e)
+ sinon si noeud_assez_loin(arbre, e, theta)
+ maj_force(e, arbre.sup_etoile)
+ sinon
+ pour enfant dans enfants
+ maj_force_sur_etoile(enfant, e, theta)
+```
+
+# Les B-arbres
+
+## Problématique
+
+* Grands jeux de données (en 1970).
+* Stockage dans un arbre, mais l'arbre tiens pas en mémoire.
+* Regrouper les sous-arbres en **pages** qui tiennent en mémoire.
+
+## Exemple
+
+* 100 noeuds par page et l'arbre comporte $10^6$ noeuds:
+ * Recherche B-arbre: $\log_{100}(10^6)=3$;
+ * Recherche ABR: $\log_2(10^6)=20$.
+* Si on doit lire depuis le disque: $10\mathrm{ms}$ par recherche+lecture:
+ * $30\mathrm{ms}$ (lecture beaucoup plus rapide que recherche) vs $200\mathrm{ms}=0.2\mathrm{s}$.
+
+## Remarques
+
+* On sait pas ce que veut dire `B`: Bayer, Boeing, Balanced?
+* Variante plus récente B+-arbres.
+
+# Les B-arbres
+
+## Illustration, arbre divisé en pages de 3 noeuds
+
+
+
+. . .
+
+## Utilisation
+
+* Bases de données (souvent très grandes donc sur le disque);
+* Système de fichier.
+
+# Les B-arbres
+
+## Avantages
+
+* Arbres moins profonds;
+* Diminue les opération de rééquilibrage;
+* Complexité toujours en $\log(N)$;
+
+. . .
+
+## Définition: 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.
+
+# Les B-arbres
+
+## Est-ce un B-arbre?
+
+
+
+. . .
+
+## Oui!
+
+* 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`
+
+
+
+. . .
+
+* 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
+
+## Disclaimer
+
+* Inspiration de <https://en.wikipedia.org/wiki/B-tree>
+
+## Exemples d'insertion: `1`
+
+
+
+. . .
+
+* L'arbre est vide, on insère juste dans la première page.
+
+# Les B-arbres
+
+## Exemples d'insertion: `2`
+
+
+
+. . .
+
+* La première page est pas pleine, on insère dans l'ordre (après 1).
+
+# Les B-arbres
+
+## Exemples d'insertion: `3`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `3`
+
+{width=50%}
+
+. . .
+
+* La page est pleine, on crée deux enfants.
+* On choisit, `2`, la médiane de `1, 2, 3` et il est inséré à la racine.
+* `1` descend à gauche, `3` descend à droite.
+
+# Les B-arbres
+
+## Exemples d'insertion: `4`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `4`
+
+{width=50%}
+
+. . .
+
+* On pourrait insérer à droite de `2`, mais... ça ferait 2 parents pour 2 enfants (mais `m` parents => `m+1` enfants ou `0`);
+* On descend à droite (`4 > 2`);
+* On insère à droite de `3`.
+
+# Les B-arbres
+
+## Exemples d'insertion: `5`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `5`
+
+
+
+. . .
+
+* On descend à droite (on peut pas insérer à la racine comme pour `4`);
+* On dépasse la capacité de l'enfant droite;
+* `4`, médiane de `3, 4, 5`, remonte à la racine;
+* On crée un nouveau noeud à droite de `4`;
+* La règle `m => m+1` est ok.
+
+# Les B-arbres
+
+## Exemples d'insertion: `6`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `6`
+
+
+
+. . .
+
+* `6 > 4` on descend à droite;
+* `6 > 5` et on a à la place à droite, on insère.
+
+# Les B-arbres
+
+## Exemples d'insertion: `7`
+
+{width=50%}
+
+* Comment on insère (1min de réflexion avant de donner une réponse!)?
+
+# Les B-arbres
+
+## Exemples d'insertion: `7`
+
+{width=50%}
+
+. . .
+
+* `7 > 4` on descend à droite;
+* `7 > 6` mais on a dépassé la capacité;
+* `6` est la médiane de `5, 6, 7`, remonte à la racine;
+* `5` reste à gauche, `7` à droite, mais `6` fait dépasser la capacité de la racine;
+* `4` est la médiane de `2, 4, 6`, `4` remonte, `2` reste à gauche, `6` à droite.
+
+# 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, on sépare la page en son milieu :
+ 1. On trouve la médiane, `M`, de la page;
+ 2. On met les éléments `< M` dans la page de gauche de `M` et les `> M` dans la page de droite de `M`;
+ 3. `M` est insérée récursivement dans la page parent.
+
+# Les B-arbres
+
+## Exercice: insérer `22, 45, 50` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `5` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `32, 55, 60` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## Exercice: insérer `41` dans l'arbre d'ordre 2 (3min matrix)
+
+
+
+. . .
+
+
+
+# Les B-arbres
+
+## 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, 32, 41,
+* Dans un B-arbre d'ordre 2.
+
+# Les B-arbres
+
+## Structure de données
+
+* Chaque page a une contrainte de remplissage, par rapport à l'ordre de l'arbre;
+* Un noeud (page) est composé d'un tableau de clés/pointeurs vers les enfants;
+
+```
+P_0 | K_1 | P_1 | K_2 | | P_i | K_{i+1} | | P_{m-1} | K_m | P_m
+```
+
+* `P_0`, ..., `P_m` pointeurs vers enfants;
+* `K_1`, ..., `K_m` les clés.
+* Il y a `m+1` pointeurs mais `m` clés.
+* Comment faire pour gérer l'insertion?
+
+# Les B-arbres
+
+## Faire un dessin de la structure de données (3min matrix)?
+
+. . .
+
+
+
+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`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+{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`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+{width=50%}
+
+# Les B-arbres
+
+## L'insertion cas noeud plein, promotion `3`?
+
+{width=50%}
+
+. . .
+
+## Solution
+
+
+
+# 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
+
+## 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].pg == vide)
+
+entier position(page, valeur)
+ i = 0
+ tant que i < page.nb && val >= page.tab[i+1].clef
+ i += 1
+ retourne i
+
+booléen est_dans_page(page, valeur)
+ i = position(page, valeur)
+ retourne (page.nb > 0 && page.tab[i].val == 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)
+ retourne vide
+ sinon
+ recherche(page.tab[position(page, valeur)], valeur)
+```
+
+# Les B-arbres
+
+## Les fonctions
+
+```C
+page inserer_valeur(page, valeur) // inserer une valeur
+```
+
+. . .
+
+```C
+page inserer_valeur(page, valeur)
+ element = nouvel_element(valeur)
+ // on change element pour savoir s'il faut le remonter
+ inserer_element(page, element)
+ si element.page != vide && page.nb > 2*page.ordre
+ // si on atteint le sommet!
+ page = ajouter_niveau(page, element)
+ 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)
+ // un element a été promu
+ 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)
+ pos = position(page, element.clé)
+ pour i de 2*page.ordre à pos+1
+ page.tab[i+1] = page.tab[i]
+ page.tab[pos+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 Ã 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 -->
+
+<!-- ## 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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