Bertrand Benjamin
e3ffe1e626
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772 lines
16 KiB
Plaintext
772 lines
16 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "6e073068",
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"metadata": {},
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"source": [
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"# Temps d'exécution et complexité\n",
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"\n",
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"Dans ce TP, vous allez chercher à évaluer la rapidité d'exécution de différents programmes.\n",
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"\n",
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"## Recherche d'élément\n",
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"\n",
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"1. Programmer et tester la fonction `chercher` qui prend en argument une liste de nombres et un nombre et qui renvoie si ce nombre est dans la liste"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "3a77b4e6",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Tests"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "50901256",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Fonction\n",
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"def chercher(liste, nombre):\n",
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" pass"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "bf3ab537",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "40a8b167",
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"metadata": {},
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"source": [
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"On voudra tester votre fonction sur des grandes listes de nombre construites aléatoirements avec le module `random`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "6e6e62eb",
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"metadata": {},
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"outputs": [],
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"source": [
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"from random import randint\n",
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"# import de la fonction randint dans le module random"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "6e64bfac",
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"metadata": {},
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"outputs": [],
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"source": [
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"print(randint(1, 10))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "28f0fdac",
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"metadata": {},
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"source": [
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"Le résultat est un nombre aléatoire entre 1 et 10.\n",
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"\n",
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"Les listes seront construites de la manière suivante"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "58200552",
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"metadata": {},
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"outputs": [],
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"source": [
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"liste_aleatoire = [randint(0, 200) for i in range(1000)]\n",
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"print(liste_aleatoire)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fc9b3bf3",
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"metadata": {},
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"source": [
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"2. Vérifier que votre fonction `chercher` fonctionne avec des listes aléatoires à 10, 100, 1000 et 1000 nombres entre 0 et 100."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "ebf2bbc5",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "1b3b40c9",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 100 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "1c5b634f",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 1000 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "e2ad8221",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10000 nombres"
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]
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},
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{
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"cell_type": "markdown",
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"id": "26a61e11",
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"metadata": {},
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"source": [
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"On cherche maintenant à évaluer la rapidité d'exécution de votre fonction. Pour cela on utilisera le module `time` qui permet d'accéder au temps."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "bb9f1569",
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"metadata": {},
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"outputs": [],
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"source": [
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"from time import time\n",
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"# import de la fonction time dans le module time"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "6502ae21",
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"metadata": {},
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"outputs": [],
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"source": [
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"print(time())"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8d97d48f",
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"metadata": {},
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"source": [
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"le résultat est le nombre de secondes écoulée depuis le 1 janvier 1970.\n",
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"\n",
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"3. Pour mesurer le temps d'exécution, on enregistre le temps avant l'exécution de la fonction puis on faire la différente avec le temps après l'exécution.\n",
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"\n",
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"Combien de temps prend la commande `cherche([1, 7, 8, 3, 9, 3], 9)` à s'exécuter?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "eab8639d",
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"metadata": {},
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"outputs": [],
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"source": [
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"t = time()\n",
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"chercher([1, 7, 8, 3, 9, 3], 9)\n",
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"print(time() - t)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a053fc3b",
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"metadata": {},
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"source": [
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"4. Combien de temps met votre fonction pour chercher un nombre dans une liste de 10 nombres? 100 nombres? 1000nombres? 10 000 nombres?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "17f17db3",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "98444ddb",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 100 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "be9d12d2",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 1000 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b9621131",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10000 nombres"
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]
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},
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{
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"cell_type": "markdown",
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"id": "57840abf",
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"metadata": {},
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"source": [
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"5. Le soucis de cette méthode est que l'on est pas à l'abris d'une liste \"facile\". Vous devez faire la même chose en calculant 5 fois le temps pour 5 listes différentes puis faire la moyenne des temps d'exécution. *(je vous encourage à programmer une fonction `moyenne` pour calculer cette moyenne)*"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b67b8fea",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "531e00f3",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "16502cac",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 100 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "e2ea7281",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 1000 nombres"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "370f6d59",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 10000 nombres"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6c13e06b",
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"metadata": {},
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"source": [
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"On va maintenant tracer le temps d'exécution en fonction de la taille de la liste.\n",
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"\n",
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"Pour cela, vous allez utiliser la librairy graphique `matplotlib`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "10e8f28a",
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"metadata": {},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "4318058e",
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"metadata": {},
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"outputs": [],
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"source": [
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"# taille des listes\n",
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"x = [10, 100, 1000, 10000]\n",
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"# Temps\n",
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"t = [1, 2, 3, 4]\n",
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"# le graphique\n",
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"fig, ax = plt.subplots()\n",
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"ax.plot(x, t)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7818c04e",
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"metadata": {},
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"source": [
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"6. Tracer le graphique des temps d'exécution de votre fonction `chercher`. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "a312000a",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "846b0e02",
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"metadata": {},
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"source": [
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"7. Comparer les temps "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0bc4af7b",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "4e3e4ace",
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"metadata": {},
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"source": [
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"## Fonction max\n",
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"\n",
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"On souhaite étudier le temps d'exécution d'une fonction `maximum`.\n",
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"\n",
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"1. Progammer et tester la fonction `maximum` qui prend en argument une liste de liste de nombres et qui renvoie le plus grand élément parmi toutes les listes."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0e5424b5",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Tests\n",
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"assert maximum([[1, 2, 3], [7, 3, 1]]) == 7"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "a0562579",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Fonction"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fee2f5da",
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"metadata": {},
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"source": [
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"2. Mesurer le temps d'exécution de votre fonction sur des listes avec 10, 100, 1000, 10000 et 100000 nombres."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "9004c51c",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "3d16cef9",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "f1914122",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "91d57e06",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b83b39b1",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "4e4324ba",
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"metadata": {},
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"source": [
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"3. Reprendre la question précédente en faisant la moyenne des temps d'exécution sur 5 listes pour chaque taille."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "30d640cd",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "bb771796",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "3a36f920",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "79e54764",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "530204ee",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "ef7c19a2",
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"metadata": {},
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"source": [
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"4. Tracer le graphique de la moyenne de temps d'exécution en fonction de la taille des listes."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "fff676a4",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "deb97308",
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"metadata": {},
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"source": [
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"## Recherche par dichotomie.\n",
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"\n",
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"La recherche par dichotomie permet de grandement améliorer la vitesse de recherche d'un élément dans un liste. **À condition que cette liste soit triée en ordre croissant**.\n",
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"\n",
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"1. Chercher et expliquer la meilleur stratégie pour gagner [au jeu du plus ou moins](https://raw.opytex.org/1NSI/09_Recherche_par_dichotomie_et_complexite/plus_moins/plus_moins.html)"
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]
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},
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{
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"cell_type": "raw",
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"id": "c78246e0",
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"metadata": {},
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "65cf15cd",
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"metadata": {},
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"source": [
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"La résolution du jeux précédent est très similaire à la situation de recherche par dichotomie.\n",
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"\n",
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"2. Pour vous entrainé à la recherche par dichotomie, établir un algorithme qui permet de trouver en le moins de coup possible (en moyenne) un élément dans une liste triée. Pour tester votre stratégie vous pouvez utiliser [ce jeu]()."
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]
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},
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{
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"cell_type": "markdown",
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"id": "a8b722a9",
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"metadata": {},
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "73ccf876",
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"metadata": {},
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"source": [
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"3. On suppose maintenant que la liste `L = [x0, x1, . . . , x]` est triée en ordre croissant, c'est\n",
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"à dire :\n",
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" \n",
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" x0 ⩽ x1 ⩽ · · · ⩽ xn\n",
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" \n",
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"On cherche parmi les éléments de `L` l'élément x.\n",
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" \n",
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"On désigne par `L[i..j]` les éléments de L dont les indices sont compris entre i (inclu) et j\n",
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|
"(inclu). \n",
|
|
"\n",
|
|
"On suppose que parmi les éléments de `L[g..d]`, il y a un élément x. Et on considère un\n",
|
|
"indice m tel que g ⩽ m ⩽ d.\n",
|
|
"\n",
|
|
"Si L[m]<x, dans quelle nouvelle zone est-on sûr que x se trouve ?"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"id": "bf3961fa",
|
|
"metadata": {},
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "645c4300",
|
|
"metadata": {},
|
|
"source": [
|
|
"Si L[m]>x, dans quelle nouvelle zone est-on sûr que x se trouve ?"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"id": "fb42b270",
|
|
"metadata": {},
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "7c6f53e5",
|
|
"metadata": {},
|
|
"source": [
|
|
"Que peut-on affirmer si aucune des deux conditions précédentes n'est valide ?"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"id": "d8cf7292",
|
|
"metadata": {},
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6d046b52",
|
|
"metadata": {},
|
|
"source": [
|
|
"4. Compléter le fonction suivante en respectant la spécification de la docstring"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 1,
|
|
"id": "0afedcb2",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def dichotomie(L, x):\n",
|
|
" \"\"\" Renvoie un indice de l'élément x dans L si x est dedans, et renvoie None sinon\n",
|
|
" \n",
|
|
" On supposera que la liste L est triée en ordre croissant.\n",
|
|
" \"\"\"\n",
|
|
" g = 0\n",
|
|
" d = len(L)-1\n",
|
|
" while g <= l:\n",
|
|
" m = (g + d) // 2\n",
|
|
" if L[m] < x:\n",
|
|
" ...\n",
|
|
" elif L[m] > x:\n",
|
|
" ...\n",
|
|
" else:\n",
|
|
" ...\n",
|
|
" return None"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "5e3bde1e",
|
|
"metadata": {},
|
|
"source": [
|
|
"5. Mesurer le temps d'exécution de votre fonction sur des listes avec 10, 100, 1000, 10000 et 100000 nombres."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "603757b0",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "6959bcfb",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "9a834ec6",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "f3e9a104",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "6093b13a",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "db5d9f35",
|
|
"metadata": {},
|
|
"source": [
|
|
"6. Reprendre la question précédente en faisant la moyenne des temps d'exécution sur 5 listes pour chaque taille."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "906ce312",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "69213d32",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "51e69361",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "d8d068cb",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "b3628a5f",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "a6966564",
|
|
"metadata": {},
|
|
"source": [
|
|
"7. Tracer le graphique de la moyenne de temps d'exécution en fonction de la taille des listes."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "96687215",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 3 (ipykernel)",
|
|
"language": "python",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.10.9"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|