Skip to main content

Empirical estimation of time complexity from execution time

Project description

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.)


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)


  • 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)


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

Copyright (c) 2011-2018, Pietro Berkes. All rights reserved.

Project details

Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

Source Distribution

big_O-0.10.2.tar.gz (10.3 kB view hashes)

Uploaded source

Supported by

AWS AWS Cloud computing Datadog Datadog Monitoring Facebook / Instagram Facebook / Instagram PSF Sponsor Fastly Fastly CDN Google Google Object Storage and Download Analytics Huawei Huawei PSF Sponsor Microsoft Microsoft PSF Sponsor NVIDIA NVIDIA PSF Sponsor Pingdom Pingdom Monitoring Salesforce Salesforce PSF Sponsor Sentry Sentry Error logging StatusPage StatusPage Status page