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+---
+title: "Arbres quarternaires, compression et Barnes-Hut"
+date: "2023-05-05"
+---
+
+# Le cours précédent (1/2)
+
+## Questions
+
+* Qu'est-ce qu'un arbre quaternaire?
+
+. . .
+
+* Un arbre où chaque nœud a soit **4 enfants** soit **aucun**.
+* A quoi peut servir un arbre quaternaire?
+
+. . .
+
+* Compression
+
+# Le cours précédent (2/2)
+
+## Questions
+
+
+* Structure de données d'un arbre quaternaire?
+
+. . .
+
+```C
+typedef struct _node {
+ int info;
+ struct _node *child[4];
+} node;
+```
+
+. . .
+
+* Dessin d'un nœud d'arbre quaternaire (avec correspondance `node`)?
+
+. . .
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((" "))-->|"child[0]"| id1(("info"));
+ id0-->|"child[1]"| id2(("info"));
+ id0-->|"child[2]"| id3(("info"));
+ id0-->|"child[3]"| id4(("info"));
+```
+
+# Transformations avec un arbre quaternaire
+
+## A faire
+
+* Symétrie axiale (horizontale/verticale).
+* Rotation quart de cercle (gauche/droite).
+* Compression.
+
+# La symétrie verticale
+
+## Que donne la symétrie verticale de
+
+```
+ SG=0 | SD=1
+ 21 | 12 | 4 | 4
+ 9 | 7 | 4 | 4
+-----------------
+ 1 | 1 | 0 | 31
+ 1 | 1 | 3 | 27
+ IG=2 | ID=3
+```
+
+. . .
+
+```
+ SG=0 | SD=1
+ 4 | 4 | 12 | 21
+ 4 | 4 | 7 | 9
+------------------
+ 31 | 0 | 1 | 1
+ 27 | 3 | 1 | 1
+ IG=2 | ID=3
+```
+
+# La symétrie d'axe vertical
+
+## Comment faire sur une matrice (3min, matrix)?
+
+. . .
+
+\footnotesize
+
+```C
+matrice symétrie(matrice)
+ pour i de 0 Ã nb_colonnes(matrice) / 2
+ pour j de 0 Ã nb_lignes(matrice)
+ échanger(matrice[i][j], matrice[nb_colonnes(matrice)-1-i][j])
+ retourne matrice
+```
+
+# La symétrie d'axe vertical
+
+## Comment faire sur un arbre?
+
+* Faire un dessin de l'arbre avant/après (5min, matrix)
+
+ ```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 12 | 21
+ 9 | 7 | 4 | 4 4 | 4 | 7 | 9
+ ----------------- => ----------------
+ 1 | 1 | 0 | 31 31 | 0 | 1 | 1
+ 1 | 1 | 3 | 27 27 | 3 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+ ```
+
+* Écrire le pseudo-code (3min, matrix)
+
+. . .
+
+\footnotesize
+
+```C
+arbre symétrie(arbre)
+ si !est_feuille(arbre)
+ échanger(arbre.enfant[0], arbre.enfant[1])
+ échanger(arbre.enfant[2], arbre.enfant[3])
+ pour i de 0 Ã 3
+ symétrie(arbre.enfant[i])
+ retourne arbre
+```
+
+# La symétrie d'axe horizontal
+
+* Trivial de faire l'axe horizontal (exercice à la maison)
+
+# Rotation d'un quart de cercle
+
+## Comment faire sur un arbre?
+
+* Faire un dessin de l'arbre avant/après (5min, matrix)
+
+ ```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 31 | 27
+ 9 | 7 | 4 | 4 4 | 4 | 0 | 3
+ ----------------- => -----------------
+ 1 | 1 | 0 | 31 12 | 7 | 1 | 1
+ 1 | 1 | 3 | 27 21 | 9 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+
+ ```
+
+* Écrire le pseudo-code (3min, matrix)
+
+. . .
+
+```C
+rien rotation_gauche(arbre)
+ si !est_feuille(arbre)
+ échange_cyclique_gauche(arbre.enfant)
+ pour i de 0 Ã 3
+ rotation_gauche(arbre.enfant[i])
+```
+
+# Rotation d'un quart de cercle
+
+\footnotesize
+
+## Comment faire sur un arbre?
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 4 | 4 | 31 | 27
+ 9 | 7 | 4 | 4 4 | 4 | 0 | 3
+----------------- => -----------------
+ 1 | 1 | 0 | 31 12 | 7 | 1 | 1
+ 1 | 1 | 3 | 27 21 | 9 | 1 | 1
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* Écrire le vrai (5min, matrix)
+
+. . .
+
+```C
+void rotate(node *qt) {
+ if (!is_leaf(qt)) {
+ node *tmp = qt->child[2];
+ qt->child[2] = qt->child[0];
+ qt->child[0] = qt->child[1];
+ qt->child[1] = qt->child[3];
+ qt->child[3] = tmp;
+ for (int i=0;i < 4; i++) {
+ rotate(qt->child[i]);
+ }
+ }
+}
+```
+
+# Compression sans perte (1/N)
+
+## Idée générale
+
+* Regrouper les pixels par valeur
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 4 21 | 12 | 4
+ 9 | 7 | 4 | 4 9 | 7 |
+----------------- => -----------------
+ 1 | 1 | 0 | 31 1 | 0 | 31
+ 1 | 1 | 3 | 27 | 3 | 27
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* Comment faire?
+
+# Compression sans perte (2/N)
+
+## Que devient l'arbre suivant?
+
+
+
+. . .
+
+## Arbre compressé
+
+
+
+# Compression sans perte (3/N)
+
+* Si un nœud a tous ses enfants égaux:
+ * Donner la valeur au nœud,
+ * Supprimer les enfants.
+* Remonter jusqu'Ã la racine.
+
+## Écrire le pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien compression_sans_pertes(arbre)
+ si !est_feuille(arbre)
+ pour i de 0 Ã 3
+ compression_sans_pertes(arbre.enfant[i])
+ si derniere_branche(arbre)
+ valeur, toutes_égales = valeur_enfants(arbre)
+ si toutes_egales
+ arbre.info = valeur
+ detruire_enfants(arbre)
+```
+
+# Compression sans perte (4/N)
+
+\footnotesize
+
+## Écrire le code C (5min, matrix)
+
+. . .
+
+```C
+void lossless_compression(node *qt) {
+ if (!is_leaf(qt)) {
+ for (int i = 0; i < CHILDREN; i++) {
+ lossless_compression(qt->child[i]);
+ }
+ if (is_last_branch(qt)) {
+ int val = -1;
+ if (last_value(qt, &val)) {
+ qt->info = val;
+ for (int i = 0; i < 4; ++i) {
+ free(qt->child[i]);
+ qt->child[i] = NULL;
+ }
+ }
+ }
+ }
+}
+```
+
+# Compression sans perte (5/N)
+
+\footnotesize
+
+```C
+bool is_last_branch(node *qt) {
+ for (int i = 0; i < 4; ++i) {
+ if (!is_leaf(qt)) {
+ return false;
+ }
+ }
+ return true;
+}
+bool last_value(node *qt, int *val) {
+ int info = qt->child[0];
+ for (int i = 1; i < 4; ++i) {
+ if (info != qt->child[i]) {
+ return false;
+ }
+ }
+ *val = info;
+ return true;
+}
+```
+
+
+# Compression avec perte (1/N)
+
+## Idée générale
+
+* Regrouper les pixels par valeur sous certaines conditions
+
+```
+ SG=0 | SD=1 SG=0 | SD=1
+ 21 | 12 | 4 | 3 21 | 12 | 4
+ 9 | 7 | 4 | 4 9 | 7 |
+----------------- => ------------------
+ 1 | 1 | 0 | 31 1 | 0 | 31
+ 2 | 1 | 3 | 27 | 3 | 27
+ IG=2 | ID=3 IG=2 | ID=3
+```
+
+* On enlève si l'écart à la moyenne est "petit"?
+
+# Compression avec perte (2/N)
+
+## Que devient l'arbre suivant si l'écart est petit?
+
+
+
+. . .
+
+## Arbre compressé
+
+
+
+# Compression avec perte (3/N)
+
+## Comment mesurer l'écart à la moyenne?
+
+. . .
+
+* Avec l'écart-type
+
+\begin{equation*}
+\mu = \frac{1}{4}\sum_{i=0}^{3} p[i],\quad \sigma = \sqrt{\frac{1}{4}\sum_{i=0}^3 (\mu-p[i])
+^2} = \sqrt{\frac{1}{4}\left(\sum_{i=0}^3p[i]^2\right)-\mu^2}
+\end{equation*}
+
+## Que devient l'algorithme?
+
+. . .
+
+* Si $\sigma<\theta$, $\theta$ est la **tolérance**:
+ * Remplacer la valeur du pixel par la moyenne des enfants.
+ * Remonter les valeurs dans l'arbre.
+
+## Quelle influence de la valeur de $\theta$ sur la compression?
+
+. . .
+
+* Plus $\theta$ est grand, plus l'image sera compressée.
+
+# Compression avec perte (4/N)
+
+## Que devient l'arbre avec $\theta=0.5$?
+
+
+
+. . .
+
+
+
+# Compression avec perte (6/N)
+
+## Modifications sur la structure de données?
+
+. . .
+
+* On stocke la moyenne, et la moyenne des carrés.
+
+```C
+struct noeud
+ flottant moyenne, moyenne_carre
+ node enfants[4]
+```
+
+* Comment on calcule `moyenne` et `moyenne_carre` sur chaque nœud (pseudo-code)?
+
+# Calcul de la moyenne
+
+## Pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien moyenne(arbre) {
+ si !est_feuille(arbre)
+ pour enfant dans arbre.enfants
+ moyenne(enfant)
+ pour enfant dans arbre.enfants
+ arbre.moyenne += enfant.moyenne
+ arbre.moyenne_carre += enfant.moyenne_carre
+ arbre.moyenne /= 4
+ arbre.moyenne_carre /= 4
+```
+
+# La compression avec pertes
+
+\footnotesize
+
+## Pseudo-code (5min, matrix)
+
+. . .
+
+```C
+rien compression_avec_pertes(arbre, theta)
+ si !est_feuille(arbre)
+ pour i de 0 Ã 3
+ compression_avec_pertes(arbre.enfant[i])
+ si derniere_branche(arbre)
+ si racine(arbre.moyenne_carre - arbre.moyenne^2) < theta
+ detruire_enfants(arbre)
+```
+
+## Le code en entier
+
+```C
+arbre = matrice_Ã _arbre(matrice)
+moyenne(arbre)
+compression_sans_pertes(arbre)
+```
+
+# La dynamique des corps célestes
+
+## Slides très fortement inspirés du cours de J. Latt, Unige
+
+## Simulation du problème à $N$-corps
+
+* Prédiction du mouvement d'un grand nombre de corps célestes.
+* Modélisation:
+ * On se limite aux étoiles;
+ * Chaque étoile est caractérisée par un point (coordonnées) et une masse;
+ * On simule en deux dimensions.
+ * Interactions uniquement par les lois de la gravitation Newtonienne (oui-oui c'est de la **physique**!).
+
+
+# Les équations du mouvement
+
+## Mouvement de la $i$-ème étoile
+
+* Algorithme de Verlet ($t_{n+1}=t_n+\delta t$)
+
+ $$
+ \vec x_i(t_{n+1})= 2\vec x_i(t_n)-\vec x_i(t_{n-1})+\vec a_i(t_n)\delta t^2.
+ $$
+
+## Force de gravitation
+
+* $\vec a_i(t_n)=\vec F_i/m_i$.
+* Sur l'étoile $i$, la force résultante est donnée par
+
+ $$
+ \vec F_i=\sum_{j=1,j\neq i}^N \vec F_{ij}.
+ $$
+ avec
+ $$
+ \vec F_{ij}=\frac{G m_i m_j(\vec x_j-\vec x_i)}{||\vec x_j-\vec x_i||^3}.
+ $$
+
+# Algorithme du problème à $n$-corps
+
+## Pseudo-code: structure de données
+
+```C
+struct étoile
+ flottant m
+ vec x, x_precedent, f
+```
+
+## Pseudo-code: itération temporelle
+
+```C
+rien iteration_temporelle(étoiles, dt)
+ pour étoile_une dans étoiles
+ étoile_une.f = 0
+ pour étoile_deux dans étoiles
+ si (étoile_un != étoile_deux)
+ étoile_une.f +=
+ force(étoile_une, étoile_deux)
+ pour étoile dans étoiles
+ étoile.x, étoile.x_precedent =
+ verlet(étoile.x, étoile.x_precedent,
+ étoile.f / étoile.m, dt)
+```
+
+# Algorithme du problème à $n$-corps
+
+## Complexité
+
+* Complexité de chacune des parties?
+
+. . .
+
+* $\mathcal{O}(N^2)$, $\mathcal{O}(N)$.
+
+## En temps CPU pour **une itération**
+
+\footnotesize
+
+* Si temps pour $N=1$ on calcule en $1\mu s$:
+
++--------+-------+-------+-----------+
+| N | N^2 | t [s] | t [réel] |
++--------+-------+-------+-----------+
+| 10 | 10^2 | 1e-4 | |
++--------+-------+-------+-----------+
+| 10^4 | 10^8 | 1e+2 | ~1min |
++--------+-------+-------+-----------+
+| 10^6 | 10^12 | 1e+6 | ~11j |
++--------+-------+-------+-----------+
+| 10^9 | 10^18 | 1e+12 | ~30k ans |
++--------+-------+-------+-----------+
+| 10^11 | 10^22 | 1e+16 | ~300M ans |
++--------+-------+-------+-----------+
+
+* Typiquement il y a des milliers-millions d'itérations.
+* Il y a $10^{11}$ étoiles dans la galaxie.
+* Houston we have a problem.
+
+# Question
+
+## Comment faire mieux, des idées?
+
+. . .
+
+* 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)
+```
+
+
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+<polygon fill="#000000" stroke="#000000" points="59.114,-30.1439 48.6045,-28.8022 55.9835,-36.4049 59.114,-30.1439"/>
+</g>
+<!-- sd2 -->
+<g id="node7" class="node">
+<title>sd2</title>
+<ellipse fill="none" stroke="#000000" cx="99" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="99" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">12</text>
+</g>
+<!-- sg1->sd2 -->
+<g id="edge6" class="edge">
+<title>sg1->sd2</title>
+<path fill="none" stroke="#000000" d="M155.7307,-74.7307C145.803,-64.803 132.6847,-51.6847 121.5637,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="123.7933,-37.8436 114.2473,-33.2473 118.8436,-42.7933 123.7933,-37.8436"/>
+</g>
+<!-- ig2 -->
+<g id="node8" class="node">
+<title>ig2</title>
+<ellipse fill="none" stroke="#000000" cx="171" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="171" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">9</text>
+</g>
+<!-- sg1->ig2 -->
+<g id="edge7" class="edge">
+<title>sg1->ig2</title>
+<path fill="none" stroke="#000000" d="M171,-71.8314C171,-64.131 171,-54.9743 171,-46.4166"/>
+<polygon fill="#000000" stroke="#000000" points="174.5001,-46.4132 171,-36.4133 167.5001,-46.4133 174.5001,-46.4132"/>
+</g>
+<!-- id2 -->
+<g id="node9" class="node">
+<title>id2</title>
+<ellipse fill="none" stroke="#000000" cx="243" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="243" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">7</text>
+</g>
+<!-- sg1->id2 -->
+<g id="edge8" class="edge">
+<title>sg1->id2</title>
+<path fill="none" stroke="#000000" d="M186.2693,-74.7307C196.197,-64.803 209.3153,-51.6847 220.4363,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="223.1564,-42.7933 227.7527,-33.2473 218.2067,-37.8436 223.1564,-42.7933"/>
+</g>
+<!-- sg3 -->
+<g id="node10" class="node">
+<title>sg3</title>
+<ellipse fill="none" stroke="#000000" cx="315" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="315" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">4</text>
+</g>
+<!-- sd1->sg3 -->
+<g id="edge9" class="edge">
+<title>sd1->sg3</title>
+<path fill="none" stroke="#000000" d="M437.1295,-79.0647C412.7778,-66.8889 373.238,-47.119 345.7715,-33.3858"/>
+<polygon fill="#000000" stroke="#000000" points="347.114,-30.1439 336.6045,-28.8022 343.9835,-36.4049 347.114,-30.1439"/>
+</g>
+<!-- sd3 -->
+<g id="node11" class="node">
+<title>sd3</title>
+<ellipse fill="none" stroke="#000000" cx="387" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="387" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">3</text>
+</g>
+<!-- sd1->sd3 -->
+<g id="edge10" class="edge">
+<title>sd1->sd3</title>
+<path fill="none" stroke="#000000" d="M443.7307,-74.7307C433.803,-64.803 420.6847,-51.6847 409.5637,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="411.7933,-37.8436 402.2473,-33.2473 406.8436,-42.7933 411.7933,-37.8436"/>
+</g>
+<!-- ig3 -->
+<g id="node12" class="node">
+<title>ig3</title>
+<ellipse fill="none" stroke="#000000" cx="459" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="459" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">4</text>
+</g>
+<!-- sd1->ig3 -->
+<g id="edge11" class="edge">
+<title>sd1->ig3</title>
+<path fill="none" stroke="#000000" d="M459,-71.8314C459,-64.131 459,-54.9743 459,-46.4166"/>
+<polygon fill="#000000" stroke="#000000" points="462.5001,-46.4132 459,-36.4133 455.5001,-46.4133 462.5001,-46.4132"/>
+</g>
+<!-- id3 -->
+<g id="node13" class="node">
+<title>id3</title>
+<ellipse fill="none" stroke="#000000" cx="531" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="531" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">4</text>
+</g>
+<!-- sd1->id3 -->
+<g id="edge12" class="edge">
+<title>sd1->id3</title>
+<path fill="none" stroke="#000000" d="M474.2693,-74.7307C484.197,-64.803 497.3153,-51.6847 508.4363,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="511.1564,-42.7933 515.7527,-33.2473 506.2067,-37.8436 511.1564,-42.7933"/>
+</g>
+<!-- sg4 -->
+<g id="node14" class="node">
+<title>sg4</title>
+<ellipse fill="none" stroke="#000000" cx="603" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="603" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->sg4 -->
+<g id="edge13" class="edge">
+<title>ig1->sg4</title>
+<path fill="none" stroke="#000000" d="M659.7307,-74.7307C649.803,-64.803 636.6847,-51.6847 625.5637,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="627.7933,-37.8436 618.2473,-33.2473 622.8436,-42.7933 627.7933,-37.8436"/>
+</g>
+<!-- sd4 -->
+<g id="node15" class="node">
+<title>sd4</title>
+<ellipse fill="none" stroke="#000000" cx="675" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="675" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->sd4 -->
+<g id="edge14" class="edge">
+<title>ig1->sd4</title>
+<path fill="none" stroke="#000000" d="M675,-71.8314C675,-64.131 675,-54.9743 675,-46.4166"/>
+<polygon fill="#000000" stroke="#000000" points="678.5001,-46.4132 675,-36.4133 671.5001,-46.4133 678.5001,-46.4132"/>
+</g>
+<!-- ig4 -->
+<g id="node16" class="node">
+<title>ig4</title>
+<ellipse fill="none" stroke="#000000" cx="747" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="747" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">2</text>
+</g>
+<!-- ig1->ig4 -->
+<g id="edge15" class="edge">
+<title>ig1->ig4</title>
+<path fill="none" stroke="#000000" d="M690.2693,-74.7307C700.197,-64.803 713.3153,-51.6847 724.4363,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="727.1564,-42.7933 731.7527,-33.2473 722.2067,-37.8436 727.1564,-42.7933"/>
+</g>
+<!-- id4 -->
+<g id="node17" class="node">
+<title>id4</title>
+<ellipse fill="none" stroke="#000000" cx="819" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="819" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- ig1->id4 -->
+<g id="edge16" class="edge">
+<title>ig1->id4</title>
+<path fill="none" stroke="#000000" d="M696.8705,-79.0647C721.2222,-66.8889 760.762,-47.119 788.2285,-33.3858"/>
+<polygon fill="#000000" stroke="#000000" points="790.0165,-36.4049 797.3955,-28.8022 786.886,-30.1439 790.0165,-36.4049"/>
+</g>
+<!-- sg5 -->
+<g id="node18" class="node">
+<title>sg5</title>
+<ellipse fill="none" stroke="#000000" cx="891" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="891" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- id1->sg5 -->
+<g id="edge17" class="edge">
+<title>id1->sg5</title>
+<path fill="none" stroke="#000000" d="M947.7307,-74.7307C937.803,-64.803 924.6847,-51.6847 913.5637,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="915.7933,-37.8436 906.2473,-33.2473 910.8436,-42.7933 915.7933,-37.8436"/>
+</g>
+<!-- sd5 -->
+<g id="node19" class="node">
+<title>sd5</title>
+<ellipse fill="none" stroke="#000000" cx="963" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="963" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">31</text>
+</g>
+<!-- id1->sd5 -->
+<g id="edge18" class="edge">
+<title>id1->sd5</title>
+<path fill="none" stroke="#000000" d="M963,-71.8314C963,-64.131 963,-54.9743 963,-46.4166"/>
+<polygon fill="#000000" stroke="#000000" points="966.5001,-46.4132 963,-36.4133 959.5001,-46.4133 966.5001,-46.4132"/>
+</g>
+<!-- ig5 -->
+<g id="node20" class="node">
+<title>ig5</title>
+<ellipse fill="none" stroke="#000000" cx="1035" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="1035" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">3</text>
+</g>
+<!-- id1->ig5 -->
+<g id="edge19" class="edge">
+<title>id1->ig5</title>
+<path fill="none" stroke="#000000" d="M978.2693,-74.7307C988.197,-64.803 1001.3153,-51.6847 1012.4363,-40.5637"/>
+<polygon fill="#000000" stroke="#000000" points="1015.1564,-42.7933 1019.7527,-33.2473 1010.2067,-37.8436 1015.1564,-42.7933"/>
+</g>
+<!-- id5 -->
+<g id="node21" class="node">
+<title>id5</title>
+<ellipse fill="none" stroke="#000000" cx="1107" cy="-18" rx="27" ry="18"/>
+<text text-anchor="middle" x="1107" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">27</text>
+</g>
+<!-- id1->id5 -->
+<g id="edge20" class="edge">
+<title>id1->id5</title>
+<path fill="none" stroke="#000000" d="M984.8705,-79.0647C1009.2222,-66.8889 1048.762,-47.119 1076.2285,-33.3858"/>
+<polygon fill="#000000" stroke="#000000" points="1078.0165,-36.4049 1085.3955,-28.8022 1074.886,-30.1439 1078.0165,-36.4049"/>
+</g>
+</g>
+</svg>
\ No newline at end of file