Big-O-notatie begrijpen

Datastructuren en algoritmen in Python

Miriam Antona

Software Engineer

Big-O-notatie

  • Meet de worstcasescomplexiteit van een algoritme
    • Tijdcomplexiteit: tijd tot volledige uitvoering
    • Geheugencomplexiteit: extra geheugengebruik
  • Geen seconden/bytes
    • Resultaat verschilt per hardware
  • Wiskundige notatie: $O(1)$, $O(n)$, $O(n^2)$...
Datastructuren en algoritmen in Python

Big-O-notatie

Een grafische weergave van verschillende algoritmen in Big-O-notatie.

Datastructuren en algoritmen in Python

$O(1)$

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

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

constant(colors)

blue
Datastructuren en algoritmen in Python

$O(1)$

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

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

constant(colors)

blue
Datastructuren en algoritmen in Python

$O(n)$

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

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

linear(colors)

green

yellow

blue

pink
Datastructuren en algoritmen in Python

$O(n)$

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

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

linear(colors)
  • n=4: 4 bewerkingen
Datastructuren en algoritmen in Python

$O(n)$

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 bewerkingen
  • n=7: 7 bewerkingen
  • n=100: 100 bewerkingen
  • ...
  • $O(n)$-complexiteit
Datastructuren en algoritmen in Python

$O(n^2)$

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 bewerkingen
  • n=100: (100 x 100) 10.000 bewerkingen
  • kwadratisch patroon
  • $O(n^2)$-complexiteit
green green
green yellow
green blue
yellow green
yellow yellow
yellow blue
blue green
blue yellow
blue blue
Datastructuren en algoritmen in Python

$O(n^3)$

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 bewerkingen
  • n=10: (10 x 10 x 10) 1.000 bewerkingen
  • kubisch patroon
  • $O(n^3)$-complexiteit
Datastructuren en algoritmen in Python

Big-O-notatie berekenen

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)
Datastructuren en algoritmen in Python

Big-O-notatie berekenen

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
Datastructuren en algoritmen in Python

Big-O-notatie berekenen

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
Datastructuren en algoritmen in Python

Big-O-notatie berekenen

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)
Datastructuren en algoritmen in Python

Big-O-notatie vereenvoudigen

  1. Verwijder constanten
    • $O(4 + 2n + 2m)$ -> $O(n + m)$
  2. Andere variabelen voor andere input
    • $O(n + m)$
  3. Verwijder kleinere termen
    • $O(n + n^2)$
Datastructuren en algoritmen in Python

Big-O-notatie vereenvoudigen

  1. Verwijder constanten
    • $O(4 + 2n + 2m)$ -> $O(n + m)$
  2. Andere variabelen voor andere input
    • $O(n + m)$
  3. Verwijder kleinere termen
    • $O(n + n^2)$ -> $O(n^2)$
Datastructuren en algoritmen in Python

Laten we oefenen!

Datastructuren en algoritmen in Python

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