big_O

big_O is a Python module to estimate the time complexity of Python code from its execution time. It can be used to analyze how 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 = -0.00035 + 2.7E-06*n (sec)

big_o inferred that the asymptotic behavior of the find_max function 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('{!s:<60s} (res: {:.2G})'.format(class, residuals)) ... Exponential: time = -5 * 4.6E-05^n (sec) (res: 15) Linear: time = -0.00035 + 2.7E-06*n (sec) (res: 6.3E-05) Quadratic: time = 0.046 + 2.4E-11*n^2 (sec) (res: 0.0056) Linearithmic: time = 0.0061 + 2.3E-07*n*log(n) (sec) (res: 0.00016) Cubic: time = 0.067 + 2.3E-16*n^3 (sec) (res: 0.013) Logarithmic: time = -0.2 + 0.033*log(n) (sec) (res: 0.03) Constant: time = 0.13 (sec) (res: 0.071) Polynomial: time = -13 * x^0.98 (sec) (res: 0.0056)

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, 10000, 50000)) (<big_o.complexities.Linearithmic object at 0x031DA9D0>, ...)

Inserting elements at the beginning of a list is O(n):

>>> def insert_0(lst): ... lst.insert(0, 0) ... >>> print(big_o.big_o(insert_0, big_o.datagen.range_n, n_measures=100)[0]) Linear: time = -4.2E-06 + 7.9E-10*n (sec)

Inserting elements at the beginning of a queue is O(1):

>>> from collections import deque >>> def insert_0_queue(queue): ... queue.insert(0, 0) ... >>> def queue_generator(n): ... return deque(range(n)) ... >>> print(big_o.big_o(insert_0_queue, queue_generator, n_measures=100)[0]) Constant: time = 2.2E-06 (sec)

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=100000, n_repeats=100) (<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=100000, n_repeats=100) (<class 'big_o.big_o.Constant'> ...)

Additional examples

We can compare the estimated time complexities of different Fibonacci number implementations. The naive implementation is exponential O(2^n). Since this implementation is very inefficient we'll reduce the maximum tested n:

>>> def fib_naive(n): ... if n < 0: ... return -1 ... if n < 2: ... return n ... return fib_naive(n-1) + fib_naive(n-2) ... >>> print(big_o.big_o(fib_naive, big_o.datagen.n, n_repeats=20, min_n=2, max_n=25)[0]) Exponential: time = -11 * 0.47^n (sec)

A more efficient implementation to find Fibonacci numbers involves using dynamic programming and is linear O(n):

>>> def fib_dp(n): ... if n < 0: ... return -1 ... if n < 2: ... return n ... a = 0 ... b = 1 ... for i in range(2, n+1): ... a, b = b, a+b ... return b ... >>> print(big_o.big_o(fib_dp, big_o.datagen.n, n_repeats=100, min_n=200, max_n=1000)[0]) Linear: time = -1.8E-06 + 7.3E-06*n (sec)

License

big_O is released under BSD-3. See LICENSE.txt .