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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.)

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 .

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

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