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---
title: "Arbres quarternaires"
date: "2023-04-28"
---
# Les arbres quaternaires
## Définition
Arbre dont chaque nœud a 4 enfants ou aucun.
![Un exemple de quadtree.](figs/quad_ex.svg)
# Les arbres quaternaires
## Cas d'utilisation
Typiquement utilisés pour représenter des données bidimensionnelles.
Son équivalent tri-dimensionnel est l'octree (chaque nœud a 8 enfants ou aucun).
## Cas d'utilisation: images
* Stockage: compression.
* Transformations: symétries, rotations, etc.
## Cas d'utilisation: simulation
* Indexation spatiale.
* Détection de collisions.
* Simulation de galaxies, Barnes-Hut.
# Exemple de compression
::: columns
:::: {.column width=30%}
## Comment représenter l'image
![Image noir/blanc.](figs/board_blacked_parts.svg)
::::
:::: {.column width=70%}
## Sous la forme d'un arbre quaternaire?
. . .
![L'arbre quaternaire correspondant.](figs/quad_img.svg)
**Économie?**
. . .
Image 64 pixels, arbre 25 neouds.
::::
:::
# Structure de données
::: columns
:::: {.column width=50%}
## Pseudocode?
. . .
```python
struct node
info
node sup_gauche, sup_droit,
inf_gauche, inf_droit
```
![Un nœud d'arbre quaternaire.](figs/quad_struct.svg)
::::
:::: {.column width=50%}
## En C?
. . .
```C
struct _node {
int info;
struct _node *sup_left;
struct _node *sup_right;
struct _node *inf_left;
struct _node *inf_right;
};
```
* Pourquoi le `*` est important?
. . .
* Type récursif => taille inconnue à la compilation.
::::
:::
# Une fonctionnalité simple
\footnotesize
## La fonction `est_feuille(noeud)`
* Problème avec cette implémentation?
```python
bool est_feuille(noeud)
retournee
est_vide(sup_gauche(noeud)) &&
est_vide(sup_droit(noeud)) &&
est_vide(inf_gauche(noeud)) &&
est_vide(inf_droit(noeud))
```
. . .
* Inutile d'avoir 4 conditions (soit 4 enfants soit aucun!)
* Facile d'en oublier un!
* Comment changer la structure pour que ça soit moins terrible?
. . .
```python
struct node
info
node enfant[4]
```
# Structure de données
## En C?
. . .
```C
typedef struct _node {
int info;
struct _node *child[4];
} node;
```
## Fonction `is_leaf(node *tree)`?
. . .
```C
bool is_leaf(node *tree) {
return (NULL == tree->child[0]); // only first matters
}
```
# Problème à résoudre
* Construire un arbre quaternaire à partir d'une image:
* Créer l'arbre (allouer la mémoire pour tous les nœuds),
* Le remplir avec les valeurs des pixels.
* Compression de l'image:
* Si les pixels sont les mêmes dans le quadrant on supprime le sous-arbre (sans perte)
* Si les pixels dévient pas trop on supprime le quadrant (avec perte)
# Création de l'arbre
## Comment créer un arbre de profondeur `prof` (3min)?
. . .
```python
arbre creer_arbre(prof)
n = nouveau_noeud() # alloue la mémoire
si prof > 0
pour i = 0 à 3
n.enfant[i] = creer_arbre(prof-1)
retourne n
```
## En `C` (3 min, matrix)?
. . .
```C
node *qt_create(int depth) {
node *n = calloc(1, sizeof(node));
if (depth > 0) {
for (int i = 0; i < 4; ++i) {
n->child[i] = qt_create(depth-1);
}
}
return n;
}
```
# Le nombre de nœuds?
## Comment implémenter la fonction (pseudo-code, 5min, matrix)?
. . .
```C
entier nombre_nœuds(arbre)
si est_feuille(arbre)
retourne 1
sinon
somme = 1
pour i de 0 à 3
somme += nombre_nœuds(arbre.enfant[i])
retourne somme
```
# Le nombre de nœuds?
## Comment implémenter la fonction en C (3min, matrix)?
. . .
```C
int size(node *qt) {
if (is_leaf(qt)) {
return 1;
} else {
int sum = 1;
for (int i = 0; i < 4; ++i) {
sum += size(qt->child[i]);
}
return sum;
}
}
```
# La profondeur en C?
## Implémentation (5min, matrix)
. . .
\footnotesize
```C
int max(int x, int y) {
return (x >= y ? x : y);
}
int max_depth(int depths[4]) {
int m = depths[0];
for (int i = 1; i < 4; ++i) {
m = max(m, depths[i]);
}
return m;
}
int depth(node *qt) {
int depths[] = {0, 0, 0, 0};
if (is_leaf(qt)) {
return 0;
} else {
for (int i = 0; i < 4; ++i) {
depths[i] = depth(qt->child[i]);
}
return 1 + max_depth(depths);
}
}
```
# Fonctions utiles (1/4)
## Comment remplir un arbre depuis une matrice?
```
SG=0 | SD=1
21 | 12 | 4 | 4
9 | 7 | 4 | 4
-----------------
1 | 1 | 0 | 31
1 | 1 | 3 | 27
IG=2 | ID=3
```
## Quel arbre cela représente?
. . .
![L'arbre correspondant](figs/quad_img_simple.svg)
# Fonctions utiles (3/5)
* On veut transformer une ligne/colonne en feuille.
* Comment?
::: columns
:::: {.column width=40%}
## Soit `ligne=2`, `colonne=3`
```
SG=0 | SD=1
21 | 12 | 4 | 4
9 | 7 | 4 | 4
-----------------
1 | 1 | 0 | 31
1 | 1 | 3 | 27
IG=2 | ID=3
```
::::
:::: {.column width=70%}
## Trouver un algorithme
![Déterminer un algorithme.](figs/quad_img_simple.svg)
* Quelle feuille?
* Plus important: quel chemin?
. . .
* `co -> G/D`, `li -> S/I`,
* `2 * (li / 2) + co / 2 -> 2 * 1 + 1 = 3`
* `2 * ((li % 2) / 1) + (co % 2) / 1 -> 2 * 0 + 1 = 1`
* Comment généraliser?
::::
:::
# Fonctions utiles (4/5)
::: columns
:::: {.column width=40%}
## Soit `ligne=2`, `colonne=3`
```
SG=0 | SD=1
21 | 12 | 4 | 4
9 | 7 | 4 | 4
-----------------
1 | 1 | 0 | 31
1 | 1 | 3 | 27
IG=2 | ID=3
```
::::
:::: {.column width=70%}
## Trouver un algorithme (prendre plusieurs exemples, 15min, matrix)
![Déterminer un algorithme.](figs/quad_img_simple.svg)
* Comment généraliser?
. . .
```C
noeud position(li, co, arbre)
d = profondeur(arbre);
tant_que (d > 1)
index = 2 * ((li % 2^d) / 2^(d-1)) +
(col % 2^d) / 2^(d-1)
arbre = arbre.enfant[index]
d -= 1
retourne arbre
```
::::
:::
# Fonctions utiles (5/5)
\footnotesize
## Pseudocode
```C
noeud position(li, co, arbre)
d = profondeur(arbre);
tant_que (d > 1)
index = 2 * ((li % 2^d) / 2^(d-1)) +
(col % 2^d) / 2^(d-1)
arbre = arbre.enfant[index]
d -= 1
retourne arbre
```
## Écrire le code `C` correspondant (5min, matrix)
```C
```
# Remplir l'arbre
## A partir d'une matrice (pseudo-code, 5min, matrix)?
. . .
```C
arbre matrice_à_arbre(matrice)
arbre = creer_arbre(profondeur)
pour li de 0 à nb_lignes(matrice)
pour co de 0 à nb_colonnes(matrice)
noeud = position(li, co, arbre)
noeud.info = matrice[co][li]
retourne arbre
```
. . .
## A partir d'une matrice (C, 5min, matrix)?
. . .
\footnotesize
```C
node *matrix_to_qt(int nb_li, int nb_co, int matrix[nb_li][nb_co], int depth)
{
node *qt = qt_create(depth);
for (int li = 0; li < nd_li; ++li) {
for (int co = 0; co < nd_co; ++co) {
node *current = position(li, co, qt);
current->info = matrix[li][co];
}
}
return qt;
}
```
<!--
Deja fait plus haut
# La profondeur?
## Comment implémenter la fonction profondeur?
* Quelle signature?
. . .
```C
entier profondeur(arbre)
```
* Quel pseudo-code (3min, matrix)?
. . .
```C
entier profondeur(arbre)
profondeurs = [0, 0, 0, 0];
si est_feuille(arbre)
retourne 0
sinon
pour i 0 à 3
profondeurs[i] = 1 + profondeur(arbre.enfant[i])
retourne max(p)
```
# La profondeur en C?
## Implémentation (5min, matrix)
. . .
```C
int max(int x, int y) {
return (x >= y ? x : y);
}
int max_depth(int depths[4]) {
int m = depths[0];
for (int i = 1; i < 4; ++i) {
m = max(m, depths[i]);
}
return m;
}
int depth(node *qt) {
int depths[] = {0, 0, 0, 0};
if (is_leaf(qt)) {
return 0;
} else {
depths[i] = 1 + depth(qt->child[i]);
return max_depth(depths);
}
}
``` -->
# Remplir la matrice
## A partir de l'arbre (pseudo-code, 3min, matrix)?
. . .
```C
matrice arbre_à_matrice(arbre)
pour li de 0 à nb_lignes(matrice)
pour co de 0 à nb_colonnes(matrice)
noeud = position(li, co, arbre)
matrice[co][li] = noeud.info
retourne arbre
```
. . .
## A partir de l'arbre (C, 3min, matrix)?
. . .
\footnotesize
```C
void qt_to_matrix(node *qt, int nb_li, int nb_co, int matrix[nb_li][nb_co])
for (int li = 0; li < nd_li; ++li) {
for (int co = 0; co < nd_co; ++co) {
node *current = position(li, co, qt);
matrix[li][co] = current->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 à nb_colonnes(matrice) / 2
pour j de 0 à 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 à 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 à 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]);
}
}
}
```
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<title>ig1</title>
<ellipse fill="none" stroke="#000000" cx="279" cy="-162" rx="27" ry="18"/>
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<!-- 0&#45;&gt;ig1 -->
<g id="edge3" class="edge">
<title>0-&gt;ig1</title>
<path fill="none" stroke="#000000" d="M251.7146,-216.5708C255.9597,-208.0807 261.1536,-197.6929 265.8663,-188.2674"/>
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<!-- id1 -->
<g id="node5" class="node">
<title>id1</title>
<ellipse fill="none" stroke="#000000" cx="351" cy="-162" rx="27" ry="18"/>
<text text-anchor="middle" x="351" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000">7</text>
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<!-- 0&#45;&gt;id1 -->
<g id="edge4" class="edge">
<title>0-&gt;id1</title>
<path fill="none" stroke="#000000" d="M262.3082,-221.1278C279.3555,-209.763 304.4019,-193.0654 323.5344,-180.3104"/>
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<g id="node6" class="node">
<title>sg2</title>
<ellipse fill="none" stroke="#000000" cx="171" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="171" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
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<!-- ig1&#45;&gt;sg2 -->
<g id="edge5" class="edge">
<title>ig1-&gt;sg2</title>
<path fill="none" stroke="#000000" d="M259.6918,-149.1278C242.6445,-137.763 217.5981,-121.0654 198.4656,-108.3104"/>
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<title>sd2</title>
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<text text-anchor="middle" x="243" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">2</text>
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<g id="edge6" class="edge">
<title>ig1-&gt;sd2</title>
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<g id="node8" class="node">
<title>ig2</title>
<ellipse fill="none" stroke="#000000" cx="315" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="315" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<g id="edge7" class="edge">
<title>ig1-&gt;ig2</title>
<path fill="none" stroke="#000000" d="M287.7146,-144.5708C291.9597,-136.0807 297.1536,-125.6929 301.8663,-116.2674"/>
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<g id="node9" class="node">
<title>id2</title>
<ellipse fill="none" stroke="#000000" cx="387" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="387" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
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<!-- ig1&#45;&gt;id2 -->
<g id="edge8" class="edge">
<title>ig1-&gt;id2</title>
<path fill="none" stroke="#000000" d="M298.3082,-149.1278C315.3555,-137.763 340.4019,-121.0654 359.5344,-108.3104"/>
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<g id="node10" class="node">
<title>sg3</title>
<ellipse fill="none" stroke="#000000" cx="27" cy="-18" rx="27" ry="18"/>
<text text-anchor="middle" x="27" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">12</text>
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<!-- sg2&#45;&gt;sg3 -->
<g id="edge9" class="edge">
<title>sg2-&gt;sg3</title>
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<g id="node11" class="node">
<title>sd3</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">1</text>
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<!-- sg2&#45;&gt;sd3 -->
<g id="edge10" class="edge">
<title>sg2-&gt;sd3</title>
<path fill="none" stroke="#000000" d="M155.7307,-74.7307C145.803,-64.803 132.6847,-51.6847 121.5637,-40.5637"/>
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<g id="node12" class="node">
<title>ig3</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">2</text>
</g>
<!-- sg2&#45;&gt;ig3 -->
<g id="edge11" class="edge">
<title>sg2-&gt;ig3</title>
<path fill="none" stroke="#000000" d="M171,-71.8314C171,-64.131 171,-54.9743 171,-46.4166"/>
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<g id="node13" class="node">
<title>id3</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">3</text>
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<!-- sg2&#45;&gt;id3 -->
<g id="edge12" class="edge">
<title>sg2-&gt;id3</title>
<path fill="none" stroke="#000000" d="M186.2693,-74.7307C196.197,-64.803 209.3153,-51.6847 220.4363,-40.5637"/>
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<g id="node14" class="node">
<title>sg4</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">8</text>
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<!-- id2&#45;&gt;sg4 -->
<g id="edge13" class="edge">
<title>id2-&gt;sg4</title>
<path fill="none" stroke="#000000" d="M371.7307,-74.7307C361.803,-64.803 348.6847,-51.6847 337.5637,-40.5637"/>
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<g id="node15" class="node">
<title>sd4</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">4</text>
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<!-- id2&#45;&gt;sd4 -->
<g id="edge14" class="edge">
<title>id2-&gt;sd4</title>
<path fill="none" stroke="#000000" d="M387,-71.8314C387,-64.131 387,-54.9743 387,-46.4166"/>
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<g id="node16" class="node">
<title>ig4</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">5</text>
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<!-- id2&#45;&gt;ig4 -->
<g id="edge15" class="edge">
<title>id2-&gt;ig4</title>
<path fill="none" stroke="#000000" d="M402.2693,-74.7307C412.197,-64.803 425.3153,-51.6847 436.4363,-40.5637"/>
<polygon fill="#000000" stroke="#000000" points="439.1564,-42.7933 443.7527,-33.2473 434.2067,-37.8436 439.1564,-42.7933"/>
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<g id="node17" class="node">
<title>id4</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">9</text>
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<!-- id2&#45;&gt;id4 -->
<g id="edge16" class="edge">
<title>id2-&gt;id4</title>
<path fill="none" stroke="#000000" d="M408.8705,-79.0647C433.2222,-66.8889 472.762,-47.119 500.2285,-33.3858"/>
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<g id="node2" class="node">
<title>sg1</title>
<ellipse fill="none" stroke="#000000" cx="243" cy="-162" rx="27" ry="18"/>
<text text-anchor="middle" x="243" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
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<!-- 0&#45;&gt;sg1 -->
<g id="edge1" class="edge">
<title>0-&gt;sg1</title>
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<title>sd1</title>
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<!-- 0&#45;&gt;sd1 -->
<g id="edge2" class="edge">
<title>0-&gt;sd1</title>
<path fill="none" stroke="#000000" d="M371.7307,-218.7307C361.803,-208.803 348.6847,-195.6847 337.5637,-184.5637"/>
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<g id="node4" class="node">
<title>ig1</title>
<ellipse fill="none" stroke="#000000" cx="495" cy="-162" rx="27" ry="18"/>
<text text-anchor="middle" x="495" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
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<!-- 0&#45;&gt;ig1 -->
<g id="edge3" class="edge">
<title>0-&gt;ig1</title>
<path fill="none" stroke="#000000" d="M406.3082,-221.1278C423.3555,-209.763 448.4019,-193.0654 467.5344,-180.3104"/>
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<g id="node5" class="node">
<title>id1</title>
<ellipse fill="none" stroke="#000000" cx="783" cy="-162" rx="27" ry="18"/>
<text text-anchor="middle" x="783" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
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<!-- 0&#45;&gt;id1 -->
<g id="edge4" class="edge">
<title>0-&gt;id1</title>
<path fill="none" stroke="#000000" d="M413.2893,-229.2201C482.1457,-216.7008 667.8702,-182.9327 746.7603,-168.589"/>
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<!-- sg2 -->
<g id="node6" class="node">
<title>sg2</title>
<ellipse fill="none" stroke="#000000" cx="135" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="135" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
</g>
<!-- sd1&#45;&gt;sg2 -->
<g id="edge5" class="edge">
<title>sd1-&gt;sg2</title>
<path fill="none" stroke="#000000" d="M293.5289,-150.643C288.8008,-148.3177 283.7773,-145.9788 279,-144 232.255,-124.6376 217.745,-127.3624 171,-108 169.2085,-107.2579 167.3824,-106.4653 165.5502,-105.6416"/>
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<!-- sd2 -->
<g id="node7" class="node">
<title>sd2</title>
<ellipse fill="none" stroke="#000000" cx="207" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="207" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
</g>
<!-- sd1&#45;&gt;sd2 -->
<g id="edge6" class="edge">
<title>sd1-&gt;sd2</title>
<path fill="none" stroke="#000000" d="M295.6918,-149.1278C278.6445,-137.763 253.5981,-121.0654 234.4656,-108.3104"/>
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<!-- ig2 -->
<g id="node8" class="node">
<title>ig2</title>
<ellipse fill="none" stroke="#000000" cx="279" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="279" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
</g>
<!-- sd1&#45;&gt;ig2 -->
<g id="edge7" class="edge">
<title>sd1-&gt;ig2</title>
<path fill="none" stroke="#000000" d="M306.2854,-144.5708C302.0403,-136.0807 296.8464,-125.6929 292.1337,-116.2674"/>
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<g id="node9" class="node">
<title>id2</title>
<ellipse fill="none" stroke="#000000" cx="351" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="351" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- sd1&#45;&gt;id2 -->
<g id="edge8" class="edge">
<title>sd1-&gt;id2</title>
<path fill="none" stroke="#000000" d="M323.7146,-144.5708C327.9597,-136.0807 333.1536,-125.6929 337.8663,-116.2674"/>
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<!-- sg3 -->
<g id="node10" class="node">
<title>sg3</title>
<ellipse fill="none" stroke="#000000" cx="423" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="423" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- ig1&#45;&gt;sg3 -->
<g id="edge9" class="edge">
<title>ig1-&gt;sg3</title>
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<!-- sd3 -->
<g id="node11" class="node">
<title>sd3</title>
<ellipse fill="none" stroke="#000000" cx="495" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="495" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- ig1&#45;&gt;sd3 -->
<g id="edge10" class="edge">
<title>ig1-&gt;sd3</title>
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<!-- ig3 -->
<g id="node12" class="node">
<title>ig3</title>
<ellipse fill="none" stroke="#000000" cx="567" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="567" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
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<!-- ig1&#45;&gt;ig3 -->
<g id="edge11" class="edge">
<title>ig1-&gt;ig3</title>
<path fill="none" stroke="#000000" d="M510.2693,-146.7307C520.197,-136.803 533.3153,-123.6847 544.4363,-112.5637"/>
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<g id="node13" class="node">
<title>id3</title>
<ellipse fill="none" stroke="#000000" cx="639" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="639" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- ig1&#45;&gt;id3 -->
<g id="edge12" class="edge">
<title>ig1-&gt;id3</title>
<path fill="none" stroke="#000000" d="M516.8705,-151.0647C541.2222,-138.8889 580.762,-119.119 608.2285,-105.3858"/>
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<!-- sg4 -->
<g id="node14" class="node">
<title>sg4</title>
<ellipse fill="none" stroke="#000000" cx="711" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="711" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- id1&#45;&gt;sg4 -->
<g id="edge13" class="edge">
<title>id1-&gt;sg4</title>
<path fill="none" stroke="#000000" d="M767.7307,-146.7307C757.803,-136.803 744.6847,-123.6847 733.5637,-112.5637"/>
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<!-- sd4 -->
<g id="node15" class="node">
<title>sd4</title>
<ellipse fill="none" stroke="#000000" cx="783" cy="-90" rx="27" ry="18"/>
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<!-- id1&#45;&gt;sd4 -->
<g id="edge14" class="edge">
<title>id1-&gt;sd4</title>
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<!-- ig4 -->
<g id="node16" class="node">
<title>ig4</title>
<ellipse fill="none" stroke="#000000" cx="855" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="855" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
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<!-- id1&#45;&gt;ig4 -->
<g id="edge15" class="edge">
<title>id1-&gt;ig4</title>
<path fill="none" stroke="#000000" d="M798.2693,-146.7307C808.197,-136.803 821.3153,-123.6847 832.4363,-112.5637"/>
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<g id="node17" class="node">
<title>id4</title>
<ellipse fill="none" stroke="#000000" cx="927" cy="-90" rx="27" ry="18"/>
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<!-- id1&#45;&gt;id4 -->
<g id="edge16" class="edge">
<title>id1-&gt;id4</title>
<path fill="none" stroke="#000000" d="M804.8705,-151.0647C829.2222,-138.8889 868.762,-119.119 896.2285,-105.3858"/>
<polygon fill="#000000" stroke="#000000" points="898.0165,-108.4049 905.3955,-100.8022 894.886,-102.1439 898.0165,-108.4049"/>
</g>
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<g id="node18" class="node">
<title>sg5</title>
<ellipse fill="none" stroke="#000000" cx="27" cy="-18" rx="27" ry="18"/>
<text text-anchor="middle" x="27" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
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<!-- sg2&#45;&gt;sg5 -->
<g id="edge17" class="edge">
<title>sg2-&gt;sg5</title>
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<!-- sd5 -->
<g id="node19" class="node">
<title>sd5</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">1</text>
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<!-- sg2&#45;&gt;sd5 -->
<g id="edge18" class="edge">
<title>sg2-&gt;sd5</title>
<path fill="none" stroke="#000000" d="M126.2854,-72.5708C122.0403,-64.0807 116.8464,-53.6929 112.1337,-44.2674"/>
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<!-- ig5 -->
<g id="node20" class="node">
<title>ig5</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">1</text>
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<!-- sg2&#45;&gt;ig5 -->
<g id="edge19" class="edge">
<title>sg2-&gt;ig5</title>
<path fill="none" stroke="#000000" d="M143.7146,-72.5708C147.9597,-64.0807 153.1536,-53.6929 157.8663,-44.2674"/>
<polygon fill="#000000" stroke="#000000" points="161.024,-45.7782 162.3657,-35.2687 154.763,-42.6477 161.024,-45.7782"/>
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<!-- id5 -->
<g id="node21" class="node">
<title>id5</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">0</text>
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<!-- sg2&#45;&gt;id5 -->
<g id="edge20" class="edge">
<title>sg2-&gt;id5</title>
<path fill="none" stroke="#000000" d="M154.3082,-77.1278C171.3555,-65.763 196.4019,-49.0654 215.5344,-36.3104"/>
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<!-- sg6 -->
<g id="node22" class="node">
<title>sg6</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>
<!-- id4&#45;&gt;sg6 -->
<g id="edge21" class="edge">
<title>id4-&gt;sg6</title>
<path fill="none" stroke="#000000" d="M907.6918,-77.1278C890.6445,-65.763 865.5981,-49.0654 846.4656,-36.3104"/>
<polygon fill="#000000" stroke="#000000" points="848.4031,-33.3956 838.1411,-30.7607 844.5201,-39.2199 848.4031,-33.3956"/>
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<!-- sd6 -->
<g id="node23" class="node">
<title>sd6</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">1</text>
</g>
<!-- id4&#45;&gt;sd6 -->
<g id="edge22" class="edge">
<title>id4-&gt;sd6</title>
<path fill="none" stroke="#000000" d="M918.2854,-72.5708C914.0403,-64.0807 908.8464,-53.6929 904.1337,-44.2674"/>
<polygon fill="#000000" stroke="#000000" points="907.237,-42.6477 899.6343,-35.2687 900.976,-45.7782 907.237,-42.6477"/>
</g>
<!-- ig6 -->
<g id="node24" class="node">
<title>ig6</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">1</text>
</g>
<!-- id4&#45;&gt;ig6 -->
<g id="edge23" class="edge">
<title>id4-&gt;ig6</title>
<path fill="none" stroke="#000000" d="M935.7146,-72.5708C939.9597,-64.0807 945.1536,-53.6929 949.8663,-44.2674"/>
<polygon fill="#000000" stroke="#000000" points="953.024,-45.7782 954.3657,-35.2687 946.763,-42.6477 953.024,-45.7782"/>
</g>
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<g id="node25" class="node">
<title>id6</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">0</text>
</g>
<!-- id4&#45;&gt;id6 -->
<g id="edge24" class="edge">
<title>id4-&gt;id6</title>
<path fill="none" stroke="#000000" d="M946.3082,-77.1278C963.3555,-65.763 988.4019,-49.0654 1007.5344,-36.3104"/>
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<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 184)">
<title>G</title>
<polygon fill="#ffffff" stroke="transparent" points="-4,4 -4,-184 1138,-184 1138,4 -4,4"/>
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<g id="node1" class="node">
<title>0</title>
<ellipse fill="none" stroke="#000000" cx="567" cy="-162" rx="27" ry="18"/>
<text text-anchor="middle" x="567" y="-157.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
</g>
<!-- sg1 -->
<g id="node2" class="node">
<title>sg1</title>
<ellipse fill="none" stroke="#000000" cx="171" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="171" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">SG</text>
</g>
<!-- 0&#45;&gt;sg1 -->
<g id="edge1" class="edge">
<title>0-&gt;sg1</title>
<path fill="none" stroke="#000000" d="M540.7107,-157.2201C471.8543,-144.7008 286.1298,-110.9327 207.2397,-96.589"/>
<polygon fill="#000000" stroke="#000000" points="207.7449,-93.1236 197.28,-94.7782 206.4926,-100.0107 207.7449,-93.1236"/>
</g>
<!-- sd1 -->
<g id="node3" class="node">
<title>sd1</title>
<ellipse fill="none" stroke="#000000" cx="459" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="459" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">SD</text>
</g>
<!-- 0&#45;&gt;sd1 -->
<g id="edge2" class="edge">
<title>0-&gt;sd1</title>
<path fill="none" stroke="#000000" d="M547.6918,-149.1278C530.6445,-137.763 505.5981,-121.0654 486.4656,-108.3104"/>
<polygon fill="#000000" stroke="#000000" points="488.4031,-105.3956 478.1411,-102.7607 484.5201,-111.2199 488.4031,-105.3956"/>
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<!-- ig1 -->
<g id="node4" class="node">
<title>ig1</title>
<ellipse fill="none" stroke="#000000" cx="675" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="675" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">IG</text>
</g>
<!-- 0&#45;&gt;ig1 -->
<g id="edge3" class="edge">
<title>0-&gt;ig1</title>
<path fill="none" stroke="#000000" d="M586.3082,-149.1278C603.3555,-137.763 628.4019,-121.0654 647.5344,-108.3104"/>
<polygon fill="#000000" stroke="#000000" points="649.4799,-111.2199 655.8589,-102.7607 645.5969,-105.3956 649.4799,-111.2199"/>
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<!-- id1 -->
<g id="node5" class="node">
<title>id1</title>
<ellipse fill="none" stroke="#000000" cx="963" cy="-90" rx="27" ry="18"/>
<text text-anchor="middle" x="963" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">ID</text>
</g>
<!-- 0&#45;&gt;id1 -->
<g id="edge4" class="edge">
<title>0-&gt;id1</title>
<path fill="none" stroke="#000000" d="M593.2893,-157.2201C662.1457,-144.7008 847.8702,-110.9327 926.7603,-96.589"/>
<polygon fill="#000000" stroke="#000000" points="927.5074,-100.0107 936.72,-94.7782 926.2551,-93.1236 927.5074,-100.0107"/>
</g>
<!-- sg2 -->
<g id="node6" class="node">
<title>sg2</title>
<ellipse fill="none" stroke="#000000" cx="27" cy="-18" rx="27" ry="18"/>
<text text-anchor="middle" x="27" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">21</text>
</g>
<!-- sg1&#45;&gt;sg2 -->
<g id="edge5" class="edge">
<title>sg1-&gt;sg2</title>
<path fill="none" stroke="#000000" d="M149.1295,-79.0647C124.7778,-66.8889 85.238,-47.119 57.7715,-33.3858"/>
<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&#45;&gt;sd2 -->
<g id="edge6" class="edge">
<title>sg1-&gt;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&#45;&gt;ig2 -->
<g id="edge7" class="edge">
<title>sg1-&gt;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&#45;&gt;id2 -->
<g id="edge8" class="edge">
<title>sg1-&gt;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&#45;&gt;sg3 -->
<g id="edge9" class="edge">
<title>sd1-&gt;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">4</text>
</g>
<!-- sd1&#45;&gt;sd3 -->
<g id="edge10" class="edge">
<title>sd1-&gt;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&#45;&gt;ig3 -->
<g id="edge11" class="edge">
<title>sd1-&gt;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&#45;&gt;id3 -->
<g id="edge12" class="edge">
<title>sd1-&gt;id3</title>
<path fill="none" stroke="#000000" d="M474.2693,-74.7307C484.197,-64.803 497.3153,-51.6847 508.4363,-40.5637"/>
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<!-- 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&#45;&gt;sg4 -->
<g id="edge13" class="edge">
<title>ig1-&gt;sg4</title>
<path fill="none" stroke="#000000" d="M659.7307,-74.7307C649.803,-64.803 636.6847,-51.6847 625.5637,-40.5637"/>
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</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&#45;&gt;sd4 -->
<g id="edge14" class="edge">
<title>ig1-&gt;sd4</title>
<path fill="none" stroke="#000000" d="M675,-71.8314C675,-64.131 675,-54.9743 675,-46.4166"/>
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<!-- 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">1</text>
</g>
<!-- ig1&#45;&gt;ig4 -->
<g id="edge15" class="edge">
<title>ig1-&gt;ig4</title>
<path fill="none" stroke="#000000" d="M690.2693,-74.7307C700.197,-64.803 713.3153,-51.6847 724.4363,-40.5637"/>
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<!-- 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>
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<!-- ig1&#45;&gt;id4 -->
<g id="edge16" class="edge">
<title>ig1-&gt;id4</title>
<path fill="none" stroke="#000000" d="M696.8705,-79.0647C721.2222,-66.8889 760.762,-47.119 788.2285,-33.3858"/>
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<!-- 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&#45;&gt;sg5 -->
<g id="edge17" class="edge">
<title>id1-&gt;sg5</title>
<path fill="none" stroke="#000000" d="M947.7307,-74.7307C937.803,-64.803 924.6847,-51.6847 913.5637,-40.5637"/>
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<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>
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<!-- id1&#45;&gt;sd5 -->
<g id="edge18" class="edge">
<title>id1-&gt;sd5</title>
<path fill="none" stroke="#000000" d="M963,-71.8314C963,-64.131 963,-54.9743 963,-46.4166"/>
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<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>
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<!-- id1&#45;&gt;ig5 -->
<g id="edge19" class="edge">
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<path fill="none" stroke="#000000" d="M978.2693,-74.7307C988.197,-64.803 1001.3153,-51.6847 1012.4363,-40.5637"/>
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<!-- 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>
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<g id="edge20" class="edge">
<title>id1-&gt;id5</title>
<path fill="none" stroke="#000000" d="M984.8705,-79.0647C1009.2222,-66.8889 1048.762,-47.119 1076.2285,-33.3858"/>
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<ellipse fill="none" stroke="#000000" cx="154" cy="-106.8" rx="27" ry="18"/>
<text text-anchor="middle" x="154" y="-102.6" font-family="Times,serif" font-size="14.00" fill="#000000">info</text>
</g>
<!-- sg1 -->
<g id="node2" class="node">
<title>sg1</title>
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<g id="edge1" class="edge">
<title>0-&gt;sg1</title>
<path fill="none" stroke="#000000" d="M128.139,-101.2097C99.5091,-94.5402 55.8191,-82.7801 43.9116,-70.8 37.256,-64.1038 33.2179,-54.7877 30.7686,-45.8424"/>
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<g id="node3" class="node">
<title>sd1</title>
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<text text-anchor="middle" x="116" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
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<g id="edge2" class="edge">
<title>0-&gt;sd1</title>
<path fill="none" stroke="#000000" d="M135.1259,-93.5052C128.1258,-87.4583 120.9598,-79.6658 117.1136,-70.8 113.866,-63.314 112.7418,-54.6514 112.6573,-46.5351"/>
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<g id="node4" class="node">
<title>ig1</title>
<ellipse fill="none" stroke="#000000" cx="192" cy="-18" rx="27" ry="18"/>
<text text-anchor="middle" x="192" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
</g>
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<g id="edge3" class="edge">
<title>0-&gt;ig1</title>
<path fill="none" stroke="#000000" d="M161.8951,-89.258C164.5048,-83.401 167.4038,-76.8308 170,-70.8 173.6125,-62.4084 177.4698,-53.2291 180.9294,-44.9121"/>
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<g id="node5" class="node">
<title>id1</title>
<ellipse fill="none" stroke="#000000" cx="277" cy="-18" rx="27" ry="18"/>
<text text-anchor="middle" x="277" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
</g>
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<g id="edge4" class="edge">
<title>0-&gt;id1</title>
<path fill="none" stroke="#000000" d="M179.5839,-100.9465C198.6633,-95.5835 224.5126,-86.1061 243,-70.8 251.566,-63.708 258.7036,-53.8853 264.1911,-44.6338"/>
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\ No newline at end of file
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