Comprendre la notation Big O

Structures de données et algorithmes en Python

Miriam Antona

Software Engineer

Notation Big O

Mesure la complexité au pire cas d’un algorithme
- Complexité en temps : temps d’exécution total
- Complexité en espace : mémoire supplémentaire
N’utilise pas secondes/ octets
- Résultats différents selon le matériel
Expressions mathématiques : $O(1)$, $O(n)$, $O(n^2)$...

Notation Big O

Une représentation graphique de différents types d’algorithmes en notation Big O.

$O(1)$

colors = ['green', 'yellow', 'blue', 'pink']

def constant(colors):
    print(colors[2])

constant(colors)

blue

$O(1)$

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple', 'red']

def constant(colors):
    print(colors[2])  # O(1)

constant(colors)

blue

$O(n)$

colors = ['green', 'yellow', 'blue', 'pink']

def linear(colors):
  for color in colors:
    print(color)    

linear(colors)

green

yellow

blue

pink

$O(n)$

colors = ['green', 'yellow', 'blue', 'pink'] # n=4

def linear(colors):
  for color in colors:
    print(color)    # O(4)

linear(colors)

n=4 : 4 opérations

$O(n)$

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple'] # n=7

def linear(colors):
  for color in colors:
    print(color)    # O(7)

linear(colors)

n=4 : 4 opérations
n=7 : 7 opérations
n=100 : 100 opérations
...
Complexité $O(n)$

$O(n^2)$

colors = ['green', 'yellow', 'blue']

def quadratic(colors):  
  for first in colors:
      for second in colors:
          print(first, second)

quadratic(colors)

n=3 : (3 x 3) 9 opérations
n=100 : (100 x 100) 10 000 opérations
motif quadratique
Complexité $O(n^2)$

green green
green yellow
green blue
yellow green
yellow yellow
yellow blue
blue green
blue yellow
blue blue

$O(n^3)$

colors = ['green', 'yellow', 'blue']

def cubic(colors):  
  for color1 in colors:
      for color2 in colors:
          for color3 in colors:
              print(color1, color2, color3)

cubic(colors)

n=3 : (3 x 3 x 3) 27 opérations
n=10 : (10 x 10 x 10) 1 000 opérations
motif cubique
Complexité $O(n^3)$

Calculer la notation Big O

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple']
other_colors = ['orange', 'brown']

def complex_algorithm(colors):
  color_count = 0

  for color in colors:
      print(color)
      color_count += 1

  for other_color in other_colors:
      print(other_color)
      color_count += 1

  print(color_count)

complex_algorithm(colors)

Calculer la notation Big O

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple']  # O(1)
other_colors = ['orange', 'brown']  # O(1)


def complex_algorithm(colors):
  color_count = 0          # O(1)


  for color in colors:
    print(color)           # O(n)
    color_count += 1       # O(n)


  for other_color in other_colors:
    print(other_color)     # O(m)
    color_count += 1       # O(m)


  print(color_count)       # O(1)


complex_algorithm(colors)  # O(4

Calculer la notation Big O

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple']  # O(1)
other_colors = ['orange', 'brown']  # O(1)

def complex_algorithm(colors):
  color_count = 0          # O(1)

  for color in colors:
    print(color)           # O(n)
    color_count += 1       # O(n)

  for other_color in other_colors:
    print(other_color)     # O(m)
    color_count += 1       # O(m)

  print(color_count)       # O(1)

complex_algorithm(colors)  # O(4 + 2n

Calculer la notation Big O

colors = ['green', 'yellow', 'blue', 'pink', 'black', 'white', 'purple']  # O(1)
other_colors = ['orange', 'brown']  # O(1)

def complex_algorithm(colors):
  color_count = 0          # O(1)

  for color in colors:
    print(color)           # O(n)
    color_count += 1       # O(n)

  for other_color in other_colors:
    print(other_color)     # O(m)
    color_count += 1       # O(m)

  print(color_count)       # O(1)

complex_algorithm(colors)  # O(4 + 2n + 2m)

Simplifier la notation Big O

Supprimer les constantes
- $O(4 + 2n + 2m)$ -> $O(n + m)$
Variables distinctes pour des entrées distinctes
- $O(n + m)$
Supprimer les termes plus petits
- $O(n + n^2)$

Simplifier la notation Big O

Supprimer les constantes
- $O(4 + 2n + 2m)$ -> $O(n + m)$
Variables distinctes pour des entrées distinctes
- $O(n + m)$
Supprimer les termes plus petits
- $O(n + n^2)$ -> $O(n^2)$

Passons à la pratique !

Structures de données et algorithmes en Python