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Arbres binaires ![Binder](https://mybinder.org/badge.svg)](https://mybinder.org/v2/gh/fortierq/binder-mp2i/main?urlpath=git-pull%3Frepo%3Dhttps%253A%252F%252Fgithub.com%252Fmp2i-info%252Fmp2i-info.github.io%26urlpath%3Dlab%252Ftree%252Fmp2i-info.github.io%252F5_arbre%252Ftp_td%252Ftp1_arbre_binaire.ipynb%26branch%3Dmain) \n", - "\n", - "On écrira 2 versions pour chaque fonction : en OCaml et en C.\n", - " \n", - "Pour écrire du code en OCaml et C, on pourra \n", - "- soit utiliser SoS et sélectionner le langage pour chaque cellule\n", - "- soit utiliser 2 notebooks : un en OCaml, l'autre en C\n", - "\n", - "## Définitions" - ] - }, - { - "cell_type": "markdown", - "id": "0ac96207-79c4-44fb-b532-cd789bbc4c15", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Redéfinir deux types d'arbres binaires : en C et en OCaml (si possible sans regarder le cours)." - ] - }, - { - "cell_type": "markdown", - "id": "b7a13fb4-bc48-4e4f-bd25-fa15cb114224", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Définir l'arbre suivant, en C et OCaml. On pourra l'utiliser pour tester les réponses dans les exercices suivants. \n", - "

" - ] - }, - { - "cell_type": "markdown", - "id": "8c6b1af6-d596-4748-9fc0-59f279867707", - "metadata": { - "kernel": "SoS" - }, - "source": [ - "## Nombre de feuilles\n", - "\n", - "On rappelle qu'une feuille est un noeud sans fils (dont les deux sous-arbres sont vides) et qu'un noeud interne est un noeud qui n'est pas une feuille." - ] - }, - { - "cell_type": "markdown", - "id": "3fea9498-bd79-4005-bbaf-951eb51713de", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `n_leaf` renvoyant le nombre de feuilles d'un arbre binaire. Revoir les fonctions sur les arbres écrites en cours, si besoin." - ] - }, - { - "cell_type": "markdown", - "id": "eedc9996", - "metadata": { - "kernel": "SoS" - }, - "source": [ - "Un arbre est **binaire strict** si ses noeuds ont 0 ou 2 fils." - ] - }, - { - "cell_type": "markdown", - "id": "7def0220", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Soit $a$ un arbre binaire strict. Conjecturer une formule reliant le nombre $f(a)$ de feuilles de $a$ avec son nombre $n(a)$ de noeuds internes, puis la prouver par récurrence." - ] - }, - { - "attachments": {}, - "cell_type": "markdown", - "id": "583cad19", - "metadata": {}, - "source": [ - "## Profondeur maximum" - ] - }, - { - "attachments": {}, - "cell_type": "markdown", - "id": "86d05797", - "metadata": {}, - "source": [ - "**Question** : Écrire une fonction `max_depth` renvoyant la profondeur maximum d'un arbre binaire, en complexité linéaire." - ] - }, - { - "cell_type": "markdown", - "id": "7ad40a1e-4613-4815-9687-5f578ad41259", - "metadata": { - "kernel": "SoS", - "tags": [] - }, - "source": [ - "## Diamètre" - ] - }, - { - "cell_type": "markdown", - "id": "838c1982-2d26-48dc-a71d-079f9c30072c", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** :\n", - "1. Rappeler la définition de la hauteur $h(a)$ d'un arbre binaire $a$, sous forme de relation de récurrence. \n", - "2. Écrire une fonction pour calculer la hauteur d'un arbre binaire." - ] - }, - { - "cell_type": "markdown", - "id": "6f1607cb-4cc7-41d1-bd10-6e4b03810717", - "metadata": { - "kernel": "SoS" - }, - "source": [ - "Le **diamètre** $d(a)$ d'un arbre $a$ est la longueur maximum d'un chemin entre 2 noeuds quelconques de cet arbre. Par exemple, le diamètre de l'arbre en exemple est $5$, correspondant au chemin $4 \\rightarrow 3 \\rightarrow 1 \\rightarrow 0 \\rightarrow 5 \\rightarrow 7$." - ] - }, - { - "cell_type": "markdown", - "id": "94bd8172-c538-4844-a13d-cedce4156660", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice**: \n", - "1. Soit $a$ un arbre binaire composé d'une racine, d'un sous-arbre gauche $g$ et d'un sous-arbre $d$. Donner, en la démontrant, une relation permettant de calculer $d(a)$ en utilisant $g$ et $d$. On pourra utiliser la $h(g)$ et $h(d)$. \n", - "2. En déduire une fonction permettant de calculer le diamètre d'un arbre binaire. Quelle est sa complexité ? \n", - "3. Améliorer la fonction précédente pour calculer le diamètre en complexité linéaire." - ] - }, - { - "cell_type": "markdown", - "id": "f49eae9b", - "metadata": { - "kernel": "SoS" - }, - "source": [ - "## Arbre binaire complet" - ] - }, - { - "cell_type": "markdown", - "id": "ee045f1e", - "metadata": { - "kernel": "SoS", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction ayant un entier $h$ en argument et renvoyant un arbre binaire complet de hauteur $h$ avec que des $0$ comme étiquettes." - ] - } - ], - "metadata": { - "interpreter": { - "hash": "638d2bb0c3d5a654c8dabbf1e544957f030902e91031027a7993b713e9c17aba" - }, - "kernelspec": { - "display_name": "SoS", - "language": "sos", - "name": "sos" - }, - "language_info": { - "codemirror_mode": "sos", - "file_extension": ".sos", - "mimetype": "text/x-sos", - "name": "sos", - "nbconvert_exporter": "sos_notebook.converter.SoS_Exporter", - "pygments_lexer": "sos" - }, - "sos": { - "kernels": [ - [ - "OCaml default", - "ocaml-jupyter", - "OCaml", - "", - "" - ], - [ - "SoS", - "sos", - "", - "", - "sos" - ] - ], - "version": "0.22.3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/files/5_arbre/tp_td/tp3_arbre.ipynb b/files/5_arbre/tp_td/tp3_arbre.ipynb deleted file mode 100644 index ce8385a68..000000000 --- a/files/5_arbre/tp_td/tp3_arbre.ipynb +++ /dev/null @@ -1,312 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": { - "kernel": "SoS", - "tags": [] - }, - "source": [ - "#
TP 3 : Arbres
\n", - "\n", - "## Encore des arbres binaires\n", - "\n", - "On rappelle les définitions d'arbres binaires en C et OCaml : " - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "kernel": "OCaml default", - "tags": [] - }, - "outputs": [ - { - "data": { - "text/plain": [ - "type 'a btree = E | N of 'a * 'a btree * 'a btree\n" - ] - }, - "execution_count": 2, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "type 'a btree =\n", - " | E (* arbre vide *)\n", - " | N of 'a * 'a btree * 'a btree;;" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "kernel": "C++17", - "tags": [] - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "Warning in cling::IncrementalParser::CheckABICompatibility():\n", - " Possible C++ standard library mismatch, compiled with __GLIBCXX__ '20200312'\n", - " Extraction of runtime standard library version was: '20210601'\n" - ] - } - ], - "source": [ - "typedef struct btree {\n", - " btree *g;\n", - " btree *d;\n", - " int val;\n", - "} btree;" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "kernel": "C++17" - }, - "outputs": [], - "source": [ - "btree* make_node(int r) {\n", - " btree *t = (btree*)malloc(sizeof(btree));\n", - " t->g = t->d = NULL;\n", - " t->val = r;\n", - " return t;\n", - "}" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17" - }, - "source": [ - "On écrira, si possible, chaque fonction en C et OCaml." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour renvoyer la somme des éléments d'un arbre binaire." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `appartient` déterminant si un élément appartient à un arbre binaire. Quelle est sa complexité? \n", - "**Remarque** : Il sera plus simple de faire un test d'appartenance avec un arbre binaire de recherche." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour savoir si deux arbres binaire sont égaux. Quelle est sa complexité?" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour savoir si un arbre binaire est strict (c'est-à-dire : si tous les noeuds ont $0$ ou $2$ fils)." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17" - }, - "source": [ - "## Arbres quelconques\n", - "\n", - "Dans cet exercice, on répondra d'abord à toutes les questions en OCaml, puis en C, en ayant défini un type qui convient. \n", - "\n", - "Dans le cas général, un arbre peut avoir un nombre quelconque de fils. En OCaml, on peut utiliser une liste pour les fils :" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "kernel": "OCaml default", - "tags": [] - }, - "outputs": [ - { - "data": { - "text/plain": [ - "type 'a tree = N of 'a * 'a tree list\n" - ] - }, - "execution_count": 1, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "type 'a tree = N of 'a * 'a tree list" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "C++17" - }, - "source": [ - "
" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "kernel": "OCaml default" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "val t : int tree =\n", - " N (0,\n", - " [N (1, [N (2, []); N (9, []); N (7, [])]); N (3, []);\n", - " N (5, [N (6, []); N (7, [])])])\n" - ] - }, - "execution_count": 6, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "let t = N(0, \n", - " [N(1, \n", - " [N(2, []);\n", - " N(9, []);\n", - " N(7, [])]);\n", - " N(3, []);\n", - " N(5,\n", - " [N(6, []);\n", - " N(7, [])])\n", - " ])" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "OCaml default", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour renvoyer la hauteur d'un arbre quelconque." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "OCaml default", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour renvoyer le nombre de feuilles d'un arbre quelconque." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "OCaml default" - }, - "source": [ - "Il est possible de représenter un arbre quelconque $t$ en arbre binaire $t'$ où chaque noeud $n$ de $t'$ est de la forme $n = N(r, g, d)$ où $g$ est le premier fils (le plus à gauche) de $n$ dans $t'$, et $d$ est le prochain frère de $n$ dans $t$. Ainsi, l'arbre ci-dessous est transformé en l'arbre binaire suivant :" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "OCaml default" - }, - "source": [ - "
" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "kernel": "OCaml default", - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction pour transformer un arbre quelconque en arbre binaire." - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "SoS", - "language": "sos", - "name": "sos" - }, - "language_info": { - "codemirror_mode": "sos", - "file_extension": ".sos", - "mimetype": "text/x-sos", - "name": "sos", - "nbconvert_exporter": "sos_notebook.converter.SoS_Exporter", - "pygments_lexer": "sos" - }, - "sos": { - "kernels": [ - [ - "C++17", - "xcpp17", - "C++17", - "", - "text/x-c++src" - ], - [ - "OCaml default", - "ocaml-jupyter", - "OCaml", - "", - "text/x-ocaml" - ] - ], - "version": "0.22.3" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/files/5_arbre/tp_td/tp6_segment/draw_segment.ml b/files/5_arbre/tp_td/tp6_segment/draw_segment.ml deleted file mode 100644 index e455c5d12..000000000 --- a/files/5_arbre/tp_td/tp6_segment/draw_segment.ml +++ /dev/null @@ -1,23 +0,0 @@ -type 'a tree = E | N of 'a * 'a tree * 'a tree - -let tree_to_tex file_out t = - let open Printf in - let file_out = sprintf "%s.tex" file_out in - let f = open_out file_out in - fprintf f "\\documentclass[convert={outfile=\\jobname.png}]{standalone}\n -\\usepackage{tikz} -\\usepackage{forest}\n -\\begin{document}\n -\\tikzset{every node/.style={draw, circle}}\n -\\begin{forest}\n"; 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TP 6 : Arbre de segments et ancêtre commun
\n", - "\n", - "**Remarque** : les arbres de segments ne doivent pas être confondus avec les arbres d'intervalles (la page [Wikipedia sur les arbres de segments](https://fr.wikipedia.org/wiki/Arbre_de_segments) traite en fait des arbres d'intervalles)." - ] - }, - { - "cell_type": "markdown", - "id": "58f7ea05-fa3f-4907-8b8b-60e5df53db52", - "metadata": {}, - "source": [ - "## Arbre de segments\n", - "\n", - "Étant donné un tableau `a` et deux indices `i` et `j`, on souhaite pouvoir trouver efficacement le minimum de `a.(i), a.(i + 1), ..., a.(j)` (plus rapidement qu'en regardant chaque élément un par un, ce qui demanderait O($j - i$) opérations). Pour cela, on peut utiliser un **arbre de segments** : \n", - "> Un arbre de segments est un arbre binaire dont les feuilles sont les éléments du tableau (ordonnés de gauche à droite) et chaque noeud $v$ possède une étiquette qui est un triplet `(m, i, j)` tel que le sous-arbre enraciné en `v` contienne les feuilles `t.(i), ... , t.(j)` et `m` est le minimum de ces valeurs. \n", - "\n", - "Par exemple, voici un arbre de segments obtenu à partir du tableau `[|5; 1; 2; 6; 3; 0|]`, où on a représenté seulement les minimums (premiers éléments de chaque triplet) :\n", - "
" - ] - }, - { - "cell_type": "markdown", - "id": "e24f2a62-9a08-4c40-a552-865ef360b9de", - "metadata": {}, - "source": [ - "On utilisera le fichier suivant qui définit un type `tree` et une fonction `tree_to_tex` pour dessiner un arbre (dans un nom de fichier donné en argument) :" - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "id": "dc92fa2b-7370-44df-8a38-d4d29d962912", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "type 'a tree = E | N of 'a * 'a tree * 'a tree\n", - "val tree_to_tex : string -> (int * 'a * 'b) tree -> unit = \n" - ] - } - ], - "source": [ - "#use \"./draw_segment.ml\"" - ] - }, - { - "cell_type": "markdown", - "id": "87478191-d8dc-48e2-8b52-254f92360576", - "metadata": { - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `root` pour renvoyer la racine d'un arbre. On renverra `max_int` si l'arbre est vide." - ] - }, - { - "cell_type": "markdown", - "id": "e991c671-96b5-4388-9736-087b8d2441a2", - "metadata": { - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `build : int array -> (int * int * int) tree` pour construire un arbre de segments récursivement à partir d'un tableau, en faisant en sorte qu'à chaque noeud `N(r, g, d)`, le nombre de noeuds dans `g` et `d` diffère d'au plus $1$." - ] - }, - { - "cell_type": "markdown", - "id": "8d47279d-5330-4516-8729-7c40fe1f750e", - "metadata": { - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Montrer que la hauteur de l'arbre construit précédemment est O($\\log(n)$), si $n$ est le nombre d'éléments du tableau." - ] - }, - { - "cell_type": "markdown", - "id": "635c8db2", - "metadata": { - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `update : int tree -> int -> int` qui met à jour l'arbre de segments en ajoutant un nouvel élément." - ] - }, - { - "cell_type": "markdown", - "id": "16cd6d52-59b8-4f3b-8267-89b3334ff4fb", - "metadata": { - "tags": [ - "exo" - ] - }, - "source": [ - "**Exercice** : Écrire une fonction `mini : int -> int -> int tree -> int` telle que, si `t` est un arbre d'intervalles associé à un tableau `a`, `mini i j t` renvoie le minimum de `a.(i), ..., a.(j)`. Pour cela, on remarquera qu'il y a $4$ cas possibles lorsqu'on est sur un noeud `v = N((r, ri, rj), g, d)` : \n", - "- Si `ri > j || rj < i` : alors le segment `i, ..., j` est disjoint du segment du noeud `v`, on peut renvoyer `max_int` (pas de minimum) \n", - "Sinon, on pose `m = (ri + rj) / 2`. \n", - "- Si `i > m` alors il faut trouver le minimum dans `d` (appel récursif). \n", - "- Si `j < m` alors il faut trouver le minimum dans `g` (appel récursif). \n", - "- Sinon, il faut chercher le segment `i, ..., m` dans `g`, le segment `(m+1), ..., j` dans `d`, et prendre le minimum de ces $2$ valeurs." - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "OCaml default", - "language": "OCaml", - "name": "ocaml-jupyter" - }, - "language_info": { - "codemirror_mode": "text/x-ocaml", - "file_extension": ".ml", - "mimetype": "text/x-ocaml", - "name": "OCaml", - "nbconverter_exporter": null, - "pygments_lexer": "OCaml", - "version": "4.08.1" - } - }, - "nbformat": 4, - 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