diff --git a/slides/cours_19.md b/slides/cours_19.md
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+---
+title: "Arbres AVL"
+date: "2025-03-28"
+---
+
+# Rappel: Algorithme d'insertion
+
+* Insérer le noeud comme d'habitude.
+* Mettre à jour les facteurs d'équilibre jusqu'à la racine (ou au premier
+ noeud déséquilibré).
+* Rééquilibrer le noeud si nécessaire.
+
+# Rappel: les cas de déséquilibre
+
+::: columns
+
+:::: column
+
+## Cas 1a
+
+* `u`, `v`, `w` même hauteur.
+* déséquilibre en `B` après insertion dans `u`
+
+
+
+::::
+
+:::: column
+
+## Cas 1a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v` `w` à la même hauteur.
+* `v` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+# Rappel: Les cas de déséquilibre
+
+
+::: columns
+
+:::: column
+
+## Cas 2a
+
+* `h(v1)=h(v2), h(u)=h(w)`.
+* déséquilibre en `C` après insertion dans `v2`
+
+
+
+::::
+
+:::: column
+
+## Cas 2a
+
+* Comment rééquilibrer?
+
+. . .
+
+* ramène `u`, `v2`, `w` à la même hauteur (`v1` pas tout à fait).
+* `v2` Ã droite de `B` (gauche de `C`)
+* `B` Ã droite de `A` (gauche de `C`)
+* `v1` Ã droite de `A` (gauche de `B`)
+
+
+
+::::
+
+:::
+
+# Rappel: Rééquilibrage
+
+## Rotation simple
+
+
+
+# Rappel: La rotation gauche-droite
+
+## Le cas 2a/b
+
+
+
+# Un petit exercice
+
+* Insérer les nœuds suivants dans un arbre AVL
+
+```
+25 | 60 | 35 | 10 | 5 | 20 | 65 | 45 | 70 | 40
+ | 50 | 55 | 30 | 15
+```
+
+```
+
+
+
+
+
+
+
+
+
+
+```
+
+# La suppression dans un arbre AVL
+
+\Huge La suppression dans un arbre AVL
+
+# Suppression dans un arbre AVL
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: supprimer 10
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((10));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11((12))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: supprimer 10
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL
+
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: supprimer 8
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((8))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((9))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation de 12
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((9))-->id1((4));
+ id0-->id2((12));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9(( ))
+ id2-->id10((14))
+ id10-->id11(( ))
+ id10-->id12((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+ style id9 fill:#fff,stroke:#fff
+ style id11 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: rotation de 12
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((9))-->id1((4));
+ id0-->id2((14));
+ id1-->id3((2));
+ id1-->id4((6));
+ id3-->id5((1));
+ id3-->id6(( ))
+ id4-->id7(( ))
+ id4-->id8((7))
+ id2-->id9((12))
+ id2-->id10((16))
+ style id6 fill:#fff,stroke:#fff
+ style id7 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+1. On supprime comme d'habitude.
+2. On rééquilibre si besoin à l'endroit de la suppression.
+
+* Facile non?
+
+. . .
+
+* Plus dur....
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL 2.0
+
+::: columns
+
+:::: column
+
+## Algorithme par problème: suppression de 30
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((30));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4((40));
+ id3-->id5(( ));
+ id3-->id6((20))
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id5 fill:#fff,stroke:#fff
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation GD autour de 40
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((40));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4(( ));
+ id3-->id5(( ));
+ id3-->id6((20))
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id4 fill:#fff,stroke:#fff
+ style id5 fill:#fff,stroke:#fff
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Suppression dans un arbre AVL 2.0
+
+::: columns
+
+:::: column
+
+## Argl! 50 est déséquilibré!
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((50))-->id1((20));
+ id0-->id2((100));
+ id1-->id3((10));
+ id1-->id4((40));
+ id2-->id7((80))
+ id2-->id8((200))
+ id7-->id9((70))
+ id7-->id10((90))
+ id9-->id11((60))
+ id9-->id12(( ))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id12 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::: column
+
+. . .
+
+## Algorithme par problème: rotation DG autour de 50
+
+```{.mermaid format=pdf width=400 loc=figs/}
+graph TD;
+ id0((80))-->id1((50));
+ id0-->id2((100));
+ id1-->id3((20));
+ id1-->id4((70));
+ id3-->id5((10));
+ id3-->id6((40));
+ id4-->id9((60))
+ id4-->id10(( ))
+ id2-->id7((90))
+ id2-->id8((200))
+ id8-->id13(( ))
+ id8-->id14((300))
+ style id10 fill:#fff,stroke:#fff
+ style id13 fill:#fff,stroke:#fff
+```
+
+::::
+
+:::
+
+# Résumé de la suppression
+
+1. On supprime comme pour un arbre binaire de recherche.
+2. Si un nœud est déséquilibré, on le rééquilibre.
+ * Cette opération peut déséquilibrer un autre nœud.
+3. On continue à rééquilibrer tant qu'il y a des nœuds à équilibrer.
+
+
+# Les arbres quaternaires
+
+\Huge Les arbres quaternaires
+
+
+# Les arbres quaternaires
+
+## Définition
+
+Arbre dont chaque nœud a 4 enfants ou aucun.
+
+
+
+# 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
+
+
+
+::::
+
+:::: {.column width=70%}
+
+## Sous la forme d'un arbre quaternaire?
+
+. . .
+
+
+
+**Économie?**
+
+. . .
+
+Image 64 pixels, arbre 25 nœuds.
+
+::::
+
+:::
+
+
+# Structure de données
+
+::: columns
+
+:::: {.column width=50%}
+
+## Pseudo-code?
+
+. . .
+
+```python
+struct node
+ info
+ node sup_gauche, sup_droit,
+ inf_gauche, inf_droit
+```
+
+
+
+::::
+
+:::: {.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)
+ retourne
+ 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 Ã 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 Ã 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?
+
+. . .
+
+
+
+# Fonctions utiles (2/4)
+
+* 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
+
+
+
+* Quelle feuille pour 31 (`li=2`, `co=3`)?
+* 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 (3/4)
+
+::: 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)
+
+
+
+* 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 (4/4)
+
+\footnotesize
+
+## Pseudo-code
+
+```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_Ã _arbre(matrice)
+ arbre = creer_arbre(profondeur)
+ pour li de 0 Ã nb_lignes(matrice)
+ pour co de 0 Ã 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;
+}
+```
+
+
+# Remplir la matrice
+
+## A partir de l'arbre (pseudo-code, 3min, matrix)?
+
+. . .
+
+```C
+matrice arbre_Ã _matrice(arbre)
+ matrice = creer_matrice(nb_lignes(arbre), nb_colonnes(arbre))
+ pour li de 0 Ã nb_lignes(matrice)
+ pour co de 0 Ã nb_colonnes(matrice)
+ noeud = position(li, co, arbre)
+ matrice[co][li] = noeud.info
+ retourne matrice
+```
+
+. . .
+
+## 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;
+ }
+ }
+```
+
+
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+<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->sg2 -->
+<g id="edge5" class="edge">
+<title>sd1->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"/>
+<polygon fill="#000000" stroke="#000000" points="167.0084,-102.4596 156.4711,-101.357 164.0208,-108.7901 167.0084,-102.4596"/>
+</g>
+<!-- sd2 -->
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+<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->sd2 -->
+<g id="edge6" class="edge">
+<title>sd1->sd2</title>
+<path fill="none" stroke="#000000" d="M295.6918,-149.1278C278.6445,-137.763 253.5981,-121.0654 234.4656,-108.3104"/>
+<polygon fill="#000000" stroke="#000000" points="236.4031,-105.3956 226.1411,-102.7607 232.5201,-111.2199 236.4031,-105.3956"/>
+</g>
+<!-- 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->ig2 -->
+<g id="edge7" class="edge">
+<title>sd1->ig2</title>
+<path fill="none" stroke="#000000" d="M306.2854,-144.5708C302.0403,-136.0807 296.8464,-125.6929 292.1337,-116.2674"/>
+<polygon fill="#000000" stroke="#000000" points="295.237,-114.6477 287.6343,-107.2687 288.976,-117.7782 295.237,-114.6477"/>
+</g>
+<!-- id2 -->
+<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>
+</g>
+<!-- sd1->id2 -->
+<g id="edge8" class="edge">
+<title>sd1->id2</title>
+<path fill="none" stroke="#000000" d="M323.7146,-144.5708C327.9597,-136.0807 333.1536,-125.6929 337.8663,-116.2674"/>
+<polygon fill="#000000" stroke="#000000" points="341.024,-117.7782 342.3657,-107.2687 334.763,-114.6477 341.024,-117.7782"/>
+</g>
+<!-- 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>
+</g>
+<!-- ig1->sg3 -->
+<g id="edge9" class="edge">
+<title>ig1->sg3</title>
+<path fill="none" stroke="#000000" d="M479.7307,-146.7307C469.803,-136.803 456.6847,-123.6847 445.5637,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="447.7933,-109.8436 438.2473,-105.2473 442.8436,-114.7933 447.7933,-109.8436"/>
+</g>
+<!-- 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>
+</g>
+<!-- ig1->sd3 -->
+<g id="edge10" class="edge">
+<title>ig1->sd3</title>
+<path fill="none" stroke="#000000" d="M495,-143.8314C495,-136.131 495,-126.9743 495,-118.4166"/>
+<polygon fill="#000000" stroke="#000000" points="498.5001,-118.4132 495,-108.4133 491.5001,-118.4133 498.5001,-118.4132"/>
+</g>
+<!-- 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>
+</g>
+<!-- ig1->ig3 -->
+<g id="edge11" class="edge">
+<title>ig1->ig3</title>
+<path fill="none" stroke="#000000" d="M510.2693,-146.7307C520.197,-136.803 533.3153,-123.6847 544.4363,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="547.1564,-114.7933 551.7527,-105.2473 542.2067,-109.8436 547.1564,-114.7933"/>
+</g>
+<!-- id3 -->
+<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>
+</g>
+<!-- ig1->id3 -->
+<g id="edge12" class="edge">
+<title>ig1->id3</title>
+<path fill="none" stroke="#000000" d="M516.8705,-151.0647C541.2222,-138.8889 580.762,-119.119 608.2285,-105.3858"/>
+<polygon fill="#000000" stroke="#000000" points="610.0165,-108.4049 617.3955,-100.8022 606.886,-102.1439 610.0165,-108.4049"/>
+</g>
+<!-- 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>
+</g>
+<!-- id1->sg4 -->
+<g id="edge13" class="edge">
+<title>id1->sg4</title>
+<path fill="none" stroke="#000000" d="M767.7307,-146.7307C757.803,-136.803 744.6847,-123.6847 733.5637,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="735.7933,-109.8436 726.2473,-105.2473 730.8436,-114.7933 735.7933,-109.8436"/>
+</g>
+<!-- sd4 -->
+<g id="node15" class="node">
+<title>sd4</title>
+<ellipse fill="none" stroke="#000000" cx="783" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="783" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000">0</text>
+</g>
+<!-- id1->sd4 -->
+<g id="edge14" class="edge">
+<title>id1->sd4</title>
+<path fill="none" stroke="#000000" d="M783,-143.8314C783,-136.131 783,-126.9743 783,-118.4166"/>
+<polygon fill="#000000" stroke="#000000" points="786.5001,-118.4132 783,-108.4133 779.5001,-118.4133 786.5001,-118.4132"/>
+</g>
+<!-- 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>
+</g>
+<!-- id1->ig4 -->
+<g id="edge15" class="edge">
+<title>id1->ig4</title>
+<path fill="none" stroke="#000000" d="M798.2693,-146.7307C808.197,-136.803 821.3153,-123.6847 832.4363,-112.5637"/>
+<polygon fill="#000000" stroke="#000000" points="835.1564,-114.7933 839.7527,-105.2473 830.2067,-109.8436 835.1564,-114.7933"/>
+</g>
+<!-- id4 -->
+<g id="node17" class="node">
+<title>id4</title>
+<ellipse fill="none" stroke="#000000" cx="927" cy="-90" rx="27" ry="18"/>
+<text text-anchor="middle" x="927" y="-85.8" font-family="Times,serif" font-size="14.00" fill="#000000"> </text>
+</g>
+<!-- id1->id4 -->
+<g id="edge16" class="edge">
+<title>id1->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>
+<!-- sg5 -->
+<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>
+</g>
+<!-- sg2->sg5 -->
+<g id="edge17" class="edge">
+<title>sg2->sg5</title>
+<path fill="none" stroke="#000000" d="M115.6918,-77.1278C98.6445,-65.763 73.5981,-49.0654 54.4656,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="56.4031,-33.3956 46.1411,-30.7607 52.5201,-39.2199 56.4031,-33.3956"/>
+</g>
+<!-- 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>
+</g>
+<!-- sg2->sd5 -->
+<g id="edge18" class="edge">
+<title>sg2->sd5</title>
+<path fill="none" stroke="#000000" d="M126.2854,-72.5708C122.0403,-64.0807 116.8464,-53.6929 112.1337,-44.2674"/>
+<polygon fill="#000000" stroke="#000000" points="115.237,-42.6477 107.6343,-35.2687 108.976,-45.7782 115.237,-42.6477"/>
+</g>
+<!-- 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>
+</g>
+<!-- sg2->ig5 -->
+<g id="edge19" class="edge">
+<title>sg2->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"/>
+</g>
+<!-- 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>
+</g>
+<!-- sg2->id5 -->
+<g id="edge20" class="edge">
+<title>sg2->id5</title>
+<path fill="none" stroke="#000000" d="M154.3082,-77.1278C171.3555,-65.763 196.4019,-49.0654 215.5344,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="217.4799,-39.2199 223.8589,-30.7607 213.5969,-33.3956 217.4799,-39.2199"/>
+</g>
+<!-- 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->sg6 -->
+<g id="edge21" class="edge">
+<title>id4->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"/>
+</g>
+<!-- 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->sd6 -->
+<g id="edge22" class="edge">
+<title>id4->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->ig6 -->
+<g id="edge23" class="edge">
+<title>id4->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>
+<!-- id6 -->
+<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->id6 -->
+<g id="edge24" class="edge">
+<title>id4->id6</title>
+<path fill="none" stroke="#000000" d="M946.3082,-77.1278C963.3555,-65.763 988.4019,-49.0654 1007.5344,-36.3104"/>
+<polygon fill="#000000" stroke="#000000" points="1009.4799,-39.2199 1015.8589,-30.7607 1005.5969,-33.3956 1009.4799,-39.2199"/>
+</g>
+</g>
+</svg>
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diff --git a/slides/figs/quad_img_simple.svg b/slides/figs/quad_img_simple.svg
new file mode 100644
index 0000000000000000000000000000000000000000..f9fea0d611cafcfc47bc0bdc6f3dd00f607c7f38
--- /dev/null
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@@ -0,0 +1,257 @@
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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"/>
+<!-- 0 -->
+<g id="node1" class="node">
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+<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->sg1 -->
+<g id="edge1" class="edge">
+<title>0->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->sd1 -->
+<g id="edge2" class="edge">
+<title>0->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>
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+<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>
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+<!-- 0->id1 -->
+<g id="edge4" class="edge">
+<title>0->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"/>
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+<!-- 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->sg2 -->
+<g id="edge5" class="edge">
+<title>sg1->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"/>
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+<!-- sd2 -->
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+<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 -->
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+<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"/>
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+<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>
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+<title>sg1->ig2</title>
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+<polygon fill="#000000" stroke="#000000" points="174.5001,-46.4132 171,-36.4133 167.5001,-46.4133 174.5001,-46.4132"/>
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+<text text-anchor="middle" x="243" y="-13.8" font-family="Times,serif" font-size="14.00" fill="#000000">7</text>
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+</g>
+<!-- sd1->sg3 -->
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+<path fill="none" stroke="#000000" d="M437.1295,-79.0647C412.7778,-66.8889 373.238,-47.119 345.7715,-33.3858"/>
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