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COMPTE_RENDU.md
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# Compte rendu — TD 3 : complexité algorithmique
Nathan COUCHOT et Lucas POT
## 1. Mesurer le programme
```bash
gcc -O2 banque.c -o banque
time ./banque
```
La version initiale s'exécute en environ **2,26 s**.
| Mesure initiale | Résultat |
| --- | ---: |
| Temps réel (`real`) | 2,26 s |
| Temps CPU utilisateur (`user`) | 2,23 s |
| Temps CPU système (`sys`) | 0,00 s |
| Empreinte | `ff68d260a7a58441` |
L'empreinte attendue est **`ff68d260a7a58441`**.
## 2. Profiler et analyser le programme initial
```bash
gcc -O0 -g -pg banque.c -o banque_pg
./banque_pg
gprof ./banque_pg gmon.out > profil_avant.txt
```
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.
| Fonction | Part du temps propre | Temps propre | Appels |
| --- | ---: | ---: | ---: |
| `trier_par_montant` | 49,43 % | 3,02 s | 1 |
| `rechercher_par_id` | 13,42 % | 0,82 s | 30 000 |
| `nb_paires_opposees` | 11,95 % | 0,73 s | 1 |
| `nb_facons_rendre` | 8,27 % | 0,51 s | 1 appel depuis `main` |
| `longueur` | 6,87 % | 0,42 s | 6 017 |
| `meilleure_periode` | 4,91 % | 0,30 s | 1 |
| `total_par_client` | 1,80 % | 0,11 s | 1 |
| `nb_clients_distincts` | 1,64 % | 0,10 s | 1 |
| `total_par_jour` | 0,82 % | 0,05 s | 1 |
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.
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.
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.
## 3. Complexités algorithmiques et cyclomatiques
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.
| Fonction | Calcule | Actuelle | Objectif |
| --- | --- | --- | --- |
| F1 `trier_par_montant` | tri des transactions | O(n²) | O(n log n) |
| F2 `nb_clients_distincts` | clients différents | O(n²) | O(n + C) |
| F3 `total_par_client` | solde par client | O(n * c) | O(n + C) |
| F4 `total_par_jour` | solde par jour | O(n * J) | O(n + J) |
| F5 `meilleure_periode` | période la plus rentable | O(J³) | O(J) |
| F6 `rechercher_par_id` | recherche par identifiant | O(n) | O(1) par requête |
| F7 `nb_paires_opposees` | paires a + b = 0 | O(n²) | O(n log n) |
| F8 `nb_facons_rendre` | rendu de monnaie | O(S^n) | O(S × k) |
| F9 `categorie` | catégorie d'une transaction | M = 14 | M ≤ 10 |
| F10 `ajouter_ligne` | construction du rapport | O(n) | O(L) au total |
### Complexité cyclomatique des fonctions F1 à F10
On calcule **M = nombre de décisions + 1**.
| Fonction | Décisions | Calcul de M |
| --- | --- | ---: |
| F1 `trier_par_montant` | 2 `for` + 1 `if` + 1 `||` + 1 `&&` | 5 + 1 = **6** |
| F2 `nb_clients_distincts` | 2 `for` + 2 `if` | 4 + 1 = **5** |
| F3 `total_par_client` | 2 `for` + 1 `if` | 3 + 1 = **4** |
| F4 `total_par_jour` | 2 `for` + 1 `if` | 3 + 1 = **4** |
| F5 `meilleure_periode` | 3 `for` + 1 `if` | 4 + 1 = **5** |
| F6 `rechercher_par_id` | 1 `for` + 1 `if` | 2 + 1 = **3** |
| F7 `nb_paires_opposees` | 2 `for` + 1 `if` | 3 + 1 = **4** |
| F8 `nb_facons_rendre` | 2 `if` + 1 `||` | 3 + 1 = **4** |
| F9 `categorie` | 11 `if`/`else if` + 1 `||` + 1 `&&` | 13 + 1 = **14** |
| F10 `ajouter_ligne` | 1 `while` | 1 + 1 = **2** |
banque
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banque_pg
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profil_avant.txt
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Flat profile:
Each sample counts as 0.01 seconds.
% cumulative self self total
time seconds seconds calls s/call s/call name
49.46 3.19 3.19 1 3.19 3.19 trier_par_montant
13.49 4.06 0.87 30000 0.00 0.00 rechercher_par_id
12.40 4.86 0.80 1 0.80 0.80 nb_paires_opposees
7.91 5.37 0.51 1 0.51 0.51 nb_facons_rendre
6.36 5.78 0.41 6017 0.00 0.00 longueur
4.81 6.09 0.31 1 0.31 0.31 meilleure_periode
1.86 6.21 0.12 1 0.12 0.12 total_par_client
1.71 6.32 0.11 1 0.11 0.11 nb_clients_distincts
0.93 6.38 0.06 1 0.06 0.06 total_par_jour
0.93 6.44 0.06 _init
0.16 6.45 0.01 30000 0.00 0.00 categorie
0.00 6.45 0.00 149998 0.00 0.00 alea
0.00 6.45 0.00 6017 0.00 0.00 ajouter_ligne
0.00 6.45 0.00 1 0.00 0.00 empreinte
0.00 6.45 0.00 1 0.00 0.00 generer_transactions
% the percentage of the total running time of the
time program used by this function.
cumulative a running sum of the number of seconds accounted
seconds for by this function and those listed above it.
self the number of seconds accounted for by this
seconds function alone. This is the major sort for this
listing.
calls the number of times this function was invoked, if
this function is profiled, else blank.
self the average number of milliseconds spent in this
ms/call function per call, if this function is profiled,
else blank.
total the average number of milliseconds spent in this
ms/call function and its descendents per call, if this
function is profiled, else blank.
name the name of the function. This is the minor sort
for this listing. The index shows the location of
the function in the gprof listing. If the index is
in parenthesis it shows where it would appear in
the gprof listing if it were to be printed.
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.
Call graph (explanation follows)
granularity: each sample hit covers 4 byte(s) for 0.16% of 6.45 seconds
index % time self children called name
<spontaneous>
[1] 99.1 0.00 6.39 main [1]
3.19 0.00 1/1 trier_par_montant [2]
0.87 0.00 30000/30000 rechercher_par_id [3]
0.80 0.00 1/1 nb_paires_opposees [4]
0.51 0.00 1/1 nb_facons_rendre [5]
0.00 0.41 6017/6017 ajouter_ligne [6]
0.31 0.00 1/1 meilleure_periode [8]
0.12 0.00 1/1 total_par_client [9]
0.11 0.00 1/1 nb_clients_distincts [10]
0.06 0.00 1/1 total_par_jour [11]
0.01 0.00 30000/30000 categorie [13]
0.00 0.00 1/1 generer_transactions [16]
0.00 0.00 1/1 empreinte [15]
-----------------------------------------------
3.19 0.00 1/1 main [1]
[2] 49.5 3.19 0.00 1 trier_par_montant [2]
-----------------------------------------------
0.87 0.00 30000/30000 main [1]
[3] 13.5 0.87 0.00 30000 rechercher_par_id [3]
-----------------------------------------------
0.80 0.00 1/1 main [1]
[4] 12.4 0.80 0.00 1 nb_paires_opposees [4]
-----------------------------------------------
266723738 nb_facons_rendre [5]
0.51 0.00 1/1 main [1]
[5] 7.9 0.51 0.00 1+266723738 nb_facons_rendre [5]
266723738 nb_facons_rendre [5]
-----------------------------------------------
0.00 0.41 6017/6017 main [1]
[6] 6.4 0.00 0.41 6017 ajouter_ligne [6]
0.41 0.00 6017/6017 longueur [7]
-----------------------------------------------
0.41 0.00 6017/6017 ajouter_ligne [6]
[7] 6.4 0.41 0.00 6017 longueur [7]
-----------------------------------------------
0.31 0.00 1/1 main [1]
[8] 4.8 0.31 0.00 1 meilleure_periode [8]
-----------------------------------------------
0.12 0.00 1/1 main [1]
[9] 1.9 0.12 0.00 1 total_par_client [9]
-----------------------------------------------
0.11 0.00 1/1 main [1]
[10] 1.7 0.11 0.00 1 nb_clients_distincts [10]
-----------------------------------------------
0.06 0.00 1/1 main [1]
[11] 0.9 0.06 0.00 1 total_par_jour [11]
-----------------------------------------------
<spontaneous>
[12] 0.9 0.06 0.00 _init [12]
-----------------------------------------------
0.01 0.00 30000/30000 main [1]
[13] 0.2 0.01 0.00 30000 categorie [13]
-----------------------------------------------
0.00 0.00 149998/149998 generer_transactions [16]
[14] 0.0 0.00 0.00 149998 alea [14]
-----------------------------------------------
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
rapport.txt
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