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3 Commits
| Author | SHA1 | Date | |
|---|---|---|---|
 Nathan Couchot
|
e9c7b77dee | ||
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Nathan Couchot
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8b28803301 | ||
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Nathan Couchot
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2bd16d1b2c |
+109
@@ -0,0 +1,109 @@
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# Compte rendu — TD 3 : complexité algorithmique
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Nathan COUCHOT et Lucas POT
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## 1. Mesurer le programme
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```bash
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gcc -O2 banque.c -o banque
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time ./banque
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```
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La version initiale s'exécute en environ **2,26 s**.
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| Mesure initiale | Résultat |
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| --- | ---: |
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| Temps réel (`real`) | 2,26 s |
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| Temps CPU utilisateur (`user`) | 2,23 s |
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| Temps CPU système (`sys`) | 0,00 s |
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| Empreinte | `ff68d260a7a58441` |
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L'empreinte attendue est **`ff68d260a7a58441`**.
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## 2. Profiler et analyser le programme initial
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```bash
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gcc -O0 -g -pg banque.c -o banque_pg
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./banque_pg
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gprof ./banque_pg gmon.out > profil_avant.txt
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```
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Le temps réel mesuré avec l'instrumentation est **8,35 s**, contre **2,26 s** sans `-pg`. Cette différence vient en partie du coût ajouté par le comptage des appels. Le profil complet est fourni dans profil_avant.txt.
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| Fonction | Part du temps propre | Temps propre | Appels |
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| --- | ---: | ---: | ---: |
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| `trier_par_montant` | 49,43 % | 3,02 s | 1 |
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| `rechercher_par_id` | 13,42 % | 0,82 s | 30 000 |
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| `nb_paires_opposees` | 11,95 % | 0,73 s | 1 |
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| `nb_facons_rendre` | 8,27 % | 0,51 s | 1 appel depuis `main` |
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| `longueur` | 6,87 % | 0,42 s | 6 017 |
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| `meilleure_periode` | 4,91 % | 0,30 s | 1 |
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| `total_par_client` | 1,80 % | 0,11 s | 1 |
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| `nb_clients_distincts` | 1,64 % | 0,10 s | 1 |
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| `total_par_jour` | 0,82 % | 0,05 s | 1 |
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Le tri à bulles est le principal point chaud du profil. `rechercher_par_id` est appelée 30 000 fois et parcourt le tableau à chaque recherche. `ajouter_ligne` appelle `longueur` pour retrouver la fin du rapport ; comme le rapport grandit à chaque ajout, ces recherches répétées reparcourent une quantité croissante de caractères.
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Le graphe d'appels permet aussi de voir le coût récursif de `nb_facons_rendre`. Le profil plat ne montre qu'un appel depuis `main`, mais cet appel entraîne de nombreux appels récursifs. Le temps propre observé ne suffit donc pas à lui seul pour expliquer le coût total.
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Les fonctions les plus coûteuses n'ont pas nécessairement les complexités les plus élevées. Le tri traite 30 000 transactions, tandis que `meilleure_periode` traite 1 000 jours. Le temps mesuré dépend à la fois de la complexité et de la taille effective des données.
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## 3. Complexités algorithmiques et cyclomatiques
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On note **n** le nombre de transactions, **C** le nombre de clients, **J** le nombre de jours, **Q** le nombre de requêtes, **S** la somme à rendre, **k** le nombre de pièces et **L** la longueur finale du rapport. Les paramètres sont considérés variables pour l'analyse, même si le programme utilise des constantes.
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| Fonction | Calcule | Actuelle | Objectif |
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| --- | --- | --- | --- |
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| F1 `trier_par_montant` | tri des transactions | O(n²) | O(n log n) |
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| F2 `nb_clients_distincts` | clients différents | O(n²) | O(n + C) |
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| F3 `total_par_client` | solde par client | O(n * c) | O(n + C) |
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| F4 `total_par_jour` | solde par jour | O(n * J) | O(n + J) |
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| F5 `meilleure_periode` | période la plus rentable | O(J³) | O(J) |
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| F6 `rechercher_par_id` | recherche par identifiant | O(n) | O(1) par requête |
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| F7 `nb_paires_opposees` | paires a + b = 0 | O(n²) | O(n log n) |
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| F8 `nb_facons_rendre` | rendu de monnaie | O(S^n) | O(S × k) |
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| F9 `categorie` | catégorie d'une transaction | M = 14 | M ≤ 10 |
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| F10 `ajouter_ligne` | construction du rapport | O(n) | O(L) au total |
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### Complexité cyclomatique des fonctions F1 à F10
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On calcule **M = nombre de décisions + 1**.
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| Fonction | Décisions | Calcul de M |
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| --- | --- | ---: |
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| F1 `trier_par_montant` | 2 `for` + 1 `if` + 1 `||` + 1 `&&` | 5 + 1 = **6** |
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| F2 `nb_clients_distincts` | 2 `for` + 2 `if` | 4 + 1 = **5** |
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| F3 `total_par_client` | 2 `for` + 1 `if` | 3 + 1 = **4** |
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| F4 `total_par_jour` | 2 `for` + 1 `if` | 3 + 1 = **4** |
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| F5 `meilleure_periode` | 3 `for` + 1 `if` | 4 + 1 = **5** |
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| F6 `rechercher_par_id` | 1 `for` + 1 `if` | 2 + 1 = **3** |
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| F7 `nb_paires_opposees` | 2 `for` + 1 `if` | 3 + 1 = **4** |
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| F8 `nb_facons_rendre` | 2 `if` + 1 `||` | 3 + 1 = **4** |
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| F9 `categorie` | 11 `if`/`else if` + 1 `||` + 1 `&&` | 13 + 1 = **14** |
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| F10 `ajouter_ligne` | 1 `while` | 1 + 1 = **2** |
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## 4. Optimiser et vérifier F1
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Le tri à bulles de `trier_par_montant` effectue O(n²) comparaisons. Il est remplacé par `qsort`, avec un comparateur qui ordonne d'abord les transactions par montant croissant, puis par identifiant croissant en cas d'égalité. Les comparaisons relationnelles évitent le risque de débordement qu'introduirait une soustraction des montants ou des identifiants. Sur la bibliothèque utilisée, le tri a un coût O(n log n), conformément à l'objectif du TD.
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| Mesure | Avant : tri à bulles | Après : `qsort` |
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| --- | ---: | ---: |
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| Temps réel médian | 2,36 s | 0,97 s |
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| Temps utilisateur médian | 2,35 s | 0,97 s |
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| Temps système médian | 0,00 s | 0,00 s |
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| Empreinte | `ff68d260a7a58441` | `ff68d260a7a58441` |
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Sur ces mesures, le temps réel médian baisse d'environ **59 %** (la version optimisée est environ **2,4 fois plus rapide**). Les valeurs affichées par le programme restent identiques et l'empreinte confirme que le rapport n'a pas changé.
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## 4. Optimiser F6 : rechercher une transaction par identifiant
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La recherche initiale parcourait le tableau jusqu'à trouver l'identifiant demandé, soit O(n) par requête et O(Qn) pour Q requêtes. Or les identifiants générés forment une permutation de 0 à n−1. Le programme construit donc une fois un tableau d'index où `index[id]` pointe vers la transaction correspondante. Chaque recherche vérifie que l'identifiant est dans les limites, puis renvoie directement `index[id]` en O(1). La construction coûte O(n), les Q recherches coûtent O(Q), soit O(n + Q) au total. L'index est libéré après les requêtes, avant le tri qui déplace les transactions.
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| Mesure | Avant : recherche linéaire | Après : index |
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| --- | ---: | ---: |
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| Temps réel médian | 0,97 s | 0,80 s |
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| Temps utilisateur médian | 0,97 s | 0,80 s |
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| Temps système médian | 0,00 s | 0,00 s |
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| Empreinte | `ff68d260a7a58441` | `ff68d260a7a58441` |
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Le temps réel médian baisse d'environ **18 %** (environ **1,2 fois plus rapide**). L'empreinte identique confirme que les recherches renvoient les mêmes transactions et que le rapport conserve son contenu.
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@@ -71,18 +71,19 @@ Transaction *generer_transactions(int n)
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/* ------------------------------------------------------------------ */
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/* F1 : trie les transactions par montant croissant, puis par id */
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static int comparer_transactions(const void *a, const void *b)
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{
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const Transaction *x = a;
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const Transaction *y = b;
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if (x->montant != y->montant)
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return (x->montant > y->montant) - (x->montant < y->montant);
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return (x->id > y->id) - (x->id < y->id);
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}
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void trier_par_montant(Transaction *t, int n)
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{
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for (int i = 0; i < n - 1; i++) {
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for (int j = 0; j < n - 1 - i; j++) {
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if (t[j].montant > t[j + 1].montant ||
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(t[j].montant == t[j + 1].montant && t[j].id > t[j + 1].id)) {
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Transaction tmp = t[j];
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t[j] = t[j + 1];
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t[j + 1] = tmp;
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}
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}
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}
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qsort(t, (size_t)n, sizeof(*t), comparer_transactions);
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}
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/* F2 : nombre de clients ayant au moins une transaction */
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@@ -147,12 +148,21 @@ long meilleure_periode(const long *totaux, int nb_jours, int *debut, int *fin)
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}
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/* F6 : retrouve une transaction à partir de son id */
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const Transaction *rechercher_par_id(const Transaction *t, int n, int id)
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const Transaction *rechercher_par_id(const Transaction *const *index, int n, int id)
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{
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if (id < 0 || id >= n)
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return NULL;
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return index[id];
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}
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static const Transaction **construire_index(const Transaction *t, int n)
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{
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const Transaction **index = malloc((size_t)n * sizeof(*index));
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if (index == NULL)
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exit(EXIT_FAILURE);
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for (int i = 0; i < n; i++)
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if (t[i].id == id)
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return &t[i];
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return NULL;
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index[t[i].id] = &t[i];
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return index;
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}
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/* F7 : nombre de paires de transactions de montants opposés (a + b = 0) */
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@@ -258,12 +268,14 @@ int main(void)
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/* recherches par id, dans l'ordre d'origine */
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long somme_requetes = 0;
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unsigned long etat = 12345;
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const Transaction **index = construire_index(t, NB_TRANSACTIONS);
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for (int q = 0; q < NB_REQUETES; q++) {
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etat = etat * 1103515245UL + 12345UL;
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const Transaction *r = rechercher_par_id(t, NB_TRANSACTIONS, (int)((etat >> 8) % NB_TRANSACTIONS));
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const Transaction *r = rechercher_par_id(index, NB_TRANSACTIONS, (int)((etat >> 8) % NB_TRANSACTIONS));
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if (r != NULL)
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somme_requetes += r->montant;
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}
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free(index);
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int nb_clients = nb_clients_distincts(t, NB_TRANSACTIONS);
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@@ -0,0 +1,220 @@
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Flat profile:
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Each sample counts as 0.01 seconds.
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% cumulative self self total
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time seconds seconds calls s/call s/call name
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49.46 3.19 3.19 1 3.19 3.19 trier_par_montant
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13.49 4.06 0.87 30000 0.00 0.00 rechercher_par_id
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12.40 4.86 0.80 1 0.80 0.80 nb_paires_opposees
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7.91 5.37 0.51 1 0.51 0.51 nb_facons_rendre
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6.36 5.78 0.41 6017 0.00 0.00 longueur
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4.81 6.09 0.31 1 0.31 0.31 meilleure_periode
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1.86 6.21 0.12 1 0.12 0.12 total_par_client
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1.71 6.32 0.11 1 0.11 0.11 nb_clients_distincts
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0.93 6.38 0.06 1 0.06 0.06 total_par_jour
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0.93 6.44 0.06 _init
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0.16 6.45 0.01 30000 0.00 0.00 categorie
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0.00 6.45 0.00 149998 0.00 0.00 alea
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0.00 6.45 0.00 6017 0.00 0.00 ajouter_ligne
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0.00 6.45 0.00 1 0.00 0.00 empreinte
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0.00 6.45 0.00 1 0.00 0.00 generer_transactions
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% the percentage of the total running time of the
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time program used by this function.
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cumulative a running sum of the number of seconds accounted
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seconds for by this function and those listed above it.
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self the number of seconds accounted for by this
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seconds function alone. This is the major sort for this
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listing.
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calls the number of times this function was invoked, if
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this function is profiled, else blank.
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self the average number of milliseconds spent in this
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ms/call function per call, if this function is profiled,
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else blank.
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total the average number of milliseconds spent in this
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ms/call function and its descendents per call, if this
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function is profiled, else blank.
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name the name of the function. This is the minor sort
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for this listing. The index shows the location of
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the function in the gprof listing. If the index is
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in parenthesis it shows where it would appear in
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the gprof listing if it were to be printed.
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Copyright (C) 2012-2026 Free Software Foundation, Inc.
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Copying and distribution of this file, with or without modification,
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are permitted in any medium without royalty provided the copyright
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notice and this notice are preserved.
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Call graph (explanation follows)
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granularity: each sample hit covers 4 byte(s) for 0.16% of 6.45 seconds
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index % time self children called name
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<spontaneous>
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[1] 99.1 0.00 6.39 main [1]
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3.19 0.00 1/1 trier_par_montant [2]
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0.87 0.00 30000/30000 rechercher_par_id [3]
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0.80 0.00 1/1 nb_paires_opposees [4]
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0.51 0.00 1/1 nb_facons_rendre [5]
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0.00 0.41 6017/6017 ajouter_ligne [6]
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0.31 0.00 1/1 meilleure_periode [8]
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0.12 0.00 1/1 total_par_client [9]
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0.11 0.00 1/1 nb_clients_distincts [10]
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0.06 0.00 1/1 total_par_jour [11]
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0.01 0.00 30000/30000 categorie [13]
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0.00 0.00 1/1 generer_transactions [16]
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0.00 0.00 1/1 empreinte [15]
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-----------------------------------------------
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3.19 0.00 1/1 main [1]
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[2] 49.5 3.19 0.00 1 trier_par_montant [2]
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-----------------------------------------------
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0.87 0.00 30000/30000 main [1]
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[3] 13.5 0.87 0.00 30000 rechercher_par_id [3]
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-----------------------------------------------
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0.80 0.00 1/1 main [1]
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[4] 12.4 0.80 0.00 1 nb_paires_opposees [4]
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-----------------------------------------------
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266723738 nb_facons_rendre [5]
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0.51 0.00 1/1 main [1]
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||||
[5] 7.9 0.51 0.00 1+266723738 nb_facons_rendre [5]
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||||
266723738 nb_facons_rendre [5]
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-----------------------------------------------
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0.00 0.41 6017/6017 main [1]
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[6] 6.4 0.00 0.41 6017 ajouter_ligne [6]
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0.41 0.00 6017/6017 longueur [7]
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-----------------------------------------------
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0.41 0.00 6017/6017 ajouter_ligne [6]
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||||
[7] 6.4 0.41 0.00 6017 longueur [7]
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-----------------------------------------------
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||||
0.31 0.00 1/1 main [1]
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||||
[8] 4.8 0.31 0.00 1 meilleure_periode [8]
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||||
-----------------------------------------------
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||||
0.12 0.00 1/1 main [1]
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||||
[9] 1.9 0.12 0.00 1 total_par_client [9]
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||||
-----------------------------------------------
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||||
0.11 0.00 1/1 main [1]
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||||
[10] 1.7 0.11 0.00 1 nb_clients_distincts [10]
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||||
-----------------------------------------------
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||||
0.06 0.00 1/1 main [1]
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||||
[11] 0.9 0.06 0.00 1 total_par_jour [11]
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||||
-----------------------------------------------
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||||
<spontaneous>
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||||
[12] 0.9 0.06 0.00 _init [12]
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||||
-----------------------------------------------
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||||
0.01 0.00 30000/30000 main [1]
|
||||
[13] 0.2 0.01 0.00 30000 categorie [13]
|
||||
-----------------------------------------------
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||||
0.00 0.00 149998/149998 generer_transactions [16]
|
||||
[14] 0.0 0.00 0.00 149998 alea [14]
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||||
-----------------------------------------------
|
||||
0.00 0.00 1/1 main [1]
|
||||
[15] 0.0 0.00 0.00 1 empreinte [15]
|
||||
-----------------------------------------------
|
||||
0.00 0.00 1/1 main [1]
|
||||
[16] 0.0 0.00 0.00 1 generer_transactions [16]
|
||||
0.00 0.00 149998/149998 alea [14]
|
||||
-----------------------------------------------
|
||||
|
||||
This table describes the call tree of the program, and was sorted by
|
||||
the total amount of time spent in each function and its children.
|
||||
|
||||
Each entry in this table consists of several lines. The line with the
|
||||
index number at the left hand margin lists the current function.
|
||||
The lines above it list the functions that called this function,
|
||||
and the lines below it list the functions this one called.
|
||||
This line lists:
|
||||
index A unique number given to each element of the table.
|
||||
Index numbers are sorted numerically.
|
||||
The index number is printed next to every function name so
|
||||
it is easier to look up where the function is in the table.
|
||||
|
||||
% time This is the percentage of the `total' time that was spent
|
||||
in this function and its children. Note that due to
|
||||
different viewpoints, functions excluded by options, etc,
|
||||
these numbers will NOT add up to 100%.
|
||||
|
||||
self This is the total amount of time spent in this function.
|
||||
|
||||
children This is the total amount of time propagated into this
|
||||
function by its children.
|
||||
|
||||
called This is the number of times the function was called.
|
||||
If the function called itself recursively, the number
|
||||
only includes non-recursive calls, and is followed by
|
||||
a `+' and the number of recursive calls.
|
||||
|
||||
name The name of the current function. The index number is
|
||||
printed after it. If the function is a member of a
|
||||
cycle, the cycle number is printed between the
|
||||
function's name and the index number.
|
||||
|
||||
|
||||
For the function's parents, the fields have the following meanings:
|
||||
|
||||
self This is the amount of time that was propagated directly
|
||||
from the function into this parent.
|
||||
|
||||
children This is the amount of time that was propagated from
|
||||
the function's children into this parent.
|
||||
|
||||
called This is the number of times this parent called the
|
||||
function `/' the total number of times the function
|
||||
was called. Recursive calls to the function are not
|
||||
included in the number after the `/'.
|
||||
|
||||
name This is the name of the parent. The parent's index
|
||||
number is printed after it. If the parent is a
|
||||
member of a cycle, the cycle number is printed between
|
||||
the name and the index number.
|
||||
|
||||
If the parents of the function cannot be determined, the word
|
||||
`<spontaneous>' is printed in the `name' field, and all the other
|
||||
fields are blank.
|
||||
|
||||
For the function's children, the fields have the following meanings:
|
||||
|
||||
self This is the amount of time that was propagated directly
|
||||
from the child into the function.
|
||||
|
||||
children This is the amount of time that was propagated from the
|
||||
child's children to the function.
|
||||
|
||||
called This is the number of times the function called
|
||||
this child `/' the total number of times the child
|
||||
was called. Recursive calls by the child are not
|
||||
listed in the number after the `/'.
|
||||
|
||||
name This is the name of the child. The child's index
|
||||
number is printed after it. If the child is a
|
||||
member of a cycle, the cycle number is printed
|
||||
between the name and the index number.
|
||||
|
||||
If there are any cycles (circles) in the call graph, there is an
|
||||
entry for the cycle-as-a-whole. This entry shows who called the
|
||||
cycle (as parents) and the members of the cycle (as children.)
|
||||
The `+' recursive calls entry shows the number of function calls that
|
||||
were internal to the cycle, and the calls entry for each member shows,
|
||||
for that member, how many times it was called from other members of
|
||||
the cycle.
|
||||
|
||||
Copyright (C) 2012-2026 Free Software Foundation, Inc.
|
||||
|
||||
Copying and distribution of this file, with or without modification,
|
||||
are permitted in any medium without royalty provided the copyright
|
||||
notice and this notice are preserved.
|
||||
|
||||
Index by function name
|
||||
|
||||
[12] _init [16] generer_transactions [4] nb_paires_opposees
|
||||
[6] ajouter_ligne [7] longueur [3] rechercher_par_id
|
||||
[14] alea [8] meilleure_periode [9] total_par_client
|
||||
[13] categorie [10] nb_clients_distincts [11] total_par_jour
|
||||
[15] empreinte [5] nb_facons_rendre [2] trier_par_montant
|
||||
+6017
File diff suppressed because it is too large
Load Diff
Reference in New Issue
Block a user