# big_O 0.7

Empirical estimation of time complexity from execution time

big_O is a Python module to estimate the time complexity of Python code from its execution time. You can use it to analyze how your functions scale with inputs of increasing size.

big_O executes a Python function for input of increasing size N, and measures its execution time. From the measurements, big_O fits a set of time complexity classes and returns the best fitting class. This is an empirical way to compute the asymptotic class of a function in “Big-O”. notation. (Strictly speaking, we’re empirically computing the Big Theta class.)

## Usage

For concreteness, let’s say we would like to compute the asymptotic behavior of a simple function that finds the maximum element in a list of positive integers:

```>>> def find_max(x):
...     """Find the maximum element in a list of positive integers."""
...     max_ = 0
...     for el in x:
...         if el > max_:
...             max_ = el
...     return max_
...
```

To do this, we call big_o.big_o passing as argument the function and a data generator that provides lists of random integers of length N:

```>>> import big_o
>>> positive_int_generator = lambda n: big_o.datagen.integers(n, 0, 10000)
>>> best, others = big_o.big_o(find_max, positive_int_generator, n_repeats=100)
>>> print best
Linear: time = -0.0028 + 6.1E-06*n
```

big_o inferred that the asymptotic behavior of the find_max fuction is linear, and returns an object containing the fitted coefficients for the complexity class. The second return argument, others, contains a dictionary of all fitted classes with the residuals from the fit as keys:

```>>> for class_, residuals in others.items():
...     print class_, '   (res: %.2G)' % residuals
...
Linear: time = 0.0015 + 5.8E-06*n    (res: 8.9E-05)
Polynomial: time = -12 * x^0.95    (res: 0.0093)
Logarithmic: time = -0.44 + 0.073*log(n)    (res: 0.14)
Linearithmic: time = 0.016 + 5E-07*n*log(n)    (res: 0.00064)
Exponential: time = -4.1 * 4.5E-05^n    (res: 14)
Constant: time = 0.29    (res: 0.35)
Cubic: time = 0.15 + 5.1E-16*n^3    (res: 0.062)
Quadratic: time = 0.1 + 5.4E-11*n^2    (res: 0.026)
```

## Submodules

• big_o.datagen: this sub-module contains common data generators, including an identity generator that simply returns N (datagen.n_), and a data generator that returns a list of random integers of length N (datagen.integers).
• big_o.complexities: this sub-module defines the complexity classes to be fit to the execution times. Unless you want to define new classes, you don’t need to worry about it.

## Standard library examples

Sorting a list in Python is O(n*log(n)) (a.k.a. ‘linearithmic’):

```>>> big_o.big_o(sorted, lambda n: big_o.datagen.integers(n, -100, 100))
(<big_o.complexities.Linearithmic object at 0x031DA9D0>, ...)
```

## numpy examples

Creating an array:

• numpy.zeros is O(n), since it needs to initialize every element to 0:

```>>> import numpy as np
>>> big_o.big_o(np.zeros, big_o.datagen.n_, max_n=1000000, n_repeats=5) # doctest: +ELLIPSIS
(<class 'big_o.big_o.Linear'>, ...)
```
• numpy.empty instead just allocates the memory, and is thus O(1):

```>>> big_o.big_o(np.empty, big_o.datagen.n_, max_n=1000000, n_repeats=5) # doctest: +ELLIPSIS
(<class 'big_o.big_o.Constant'> ...)
```