Computational complexity

The computational complexity of an algorithm is the amount of resources (time and memory) required to run it.

Arithmetic complexity of computations

Big-O notation

Big-O notation gives an upper bound of the complexity in the worst case, helping to quantify performance as the input becomes arbitrarily large.

Where n is the size of the input; complexities ordered from smallest to largest:

Notation Name
$O(1)$ Constant time
$O(\log(n))$ Logarithmic time
$O(n)$ Linear time
$O(n \log(n))$ Linearithmic time
$O(n^2)$ Quadric time
$O(n^3)$ Cubic time
$O(b^n)$, b > 1 Exponential time
$O(n!)$ Factorial time

Popular algorithms:

• $O(\log(n))$ - binary search
• $O(n)$ - linear search
• $O(n \log(n))$ - merge sort
• $O(n^2)$ - selection sort, bubble sort
• $O(n!)$ - finding all permutations of a string

Big-Omega notation

Big-Omega notation gives a lower bound of the complexity in the best case.

Popular algorithms:

• $\Omega(1)$ - linear search, binary search
• $\Omega(n)$ - bubble sort
• $\Omega(n \log(n))$ - merge sort
• $\Omega(n^2)$ - selection sort

Big-Theta notation

Big-Theta notation gives a asymptotic bound (upper and lower bounds) of the complexity in the average case.

Popular algorithms:

• $\Theta(n \log(n))$ - merge sort
• $\Theta(n^2)$ - selection sort