Introduction to what computational complexity is, what is it used for and how to improve your code keeping it in mind.

1. Computational
complexity in Python
Jordi Soucheiron (@jordixou) @ pybcn 17/03/2016

2. What is computational complexity?
● It’s a theoretical branch of computer science
● It’s used to determine the cost of algorithms in:
○ Instructions (time)
○ Space
● Algorithms are sorted in classes (like P and NP)

3. The big O

4. Basic rules
● Cut your algorithm in pieces
● The piece with bigger cost wins
● Discard constants
● Examples:
○ O(n!) + O(4n³) + O(1) →O(n!)
○ O(5n!) →O(n!)
○ O(3) →O(1)
○ O(n³) + O(n) →O(n³)

5. Is it a permutation?

6. Is it a permutation?
O(n*n!)

8. Is it a permutation?

9. Is it a permutation?
O(n²)

11. Is it a permutation?

12. Is it a permutation?
O(n log n)

13. Is it a permutation?

14. Is it a permutation?
O(n)

15. Does it really matter?

16. Python lists

17. Python lists
Operation Average Case Amortized Worst Case
Copy O(n) O(n)
Append O(1) O(1)
Insert/Delete item O(n) O(n)
Get/Set Item O(1) O(1)
Iteration O(n) O(n)
Sort O(n log n) O(n log n)
x in s O(n) O(n)
Get length O(1) O(1)

18. Python dictionaries

19. Python dictionaries
Operation Average Case Amortized Worst Case
Copy O(n) O(n)
Get Item O(1) O(n)
Set Item O(1) O(n)
Delete item O(1) O(n)
Iteration O(n) O(n)

20. Python sets
Operation Average Case Worst Case
x in s O(1) O(n)
union s|t O(n) O(n)
intersection s&t O(n) O(n²)
Difference s-t O(n) O(n)
Symmetric Difference s^t O(n) O(n²)

21. Conclusions
● Use the best suited data structure for your use case
● If you need to use an expensive algorithm:
○ Reduce n size first*
○ Split the problem in smaller chunks (n³ > ((n/2)³ + (n/2)³)*
● Sometimes a slower algorithm is just fine:
○ Know your requirements
○ For small inputs it may not be worth it (premature optimization is the root
of all evil)
● Avoid expensive algorithms*
*whenever possible

22. Questions

23. Sources and more information
● https://www.raywenderlich.com/79850/collection-data-structures-swift
● https://www.youtube.com/watch?v=8TDZsaATmqQ
● https://en.wikipedia.org/wiki/Computational_complexity_theory
● https://wiki.python.org/moin/TimeComplexity
● http://www.laurentluce.com/posts/python-list-implementation/
● http://www.laurentluce.com/posts/python-dictionary-implementation/
● https://github.com/jsoucheiron/computational-complexity-talk
● http://xkcd.com/353/