#!/usr/bin/env python
# coding: utf-8

# # Brute force

# Sources:
# 
# 1. Guttag, *Introduction to Computation and Programming Using Python*, Revised and Expanded Ed. (2013)
# 2. Cormen et al., *Introduction to Algorithms*, 3rd ed. (2009)
# 3. [Brute force](https://en.wikipedia.org/wiki/Brute_force)

# In *Introduction to Computation and Programming Using Python*, Guttag provides this code (modified) for finding the integer cube root of an integer.

# In[10]:


def int_cube_root(n):
    result = 0
    while result ** 3 < abs(n):
        result += 1
    if result ** 3 != abs(n):
        return '{} is not a perfect cube'.format(n)
    else:
        if n < 0:
            result = -result
        return result


# In[11]:


int_cube_root(7)


# In[12]:


int_cube_root(8)


# "The algorithmic technique used in this program," Guttag writes, "is a variant of **guess and check** called **exhaustive enumeration**. We enumerate all possibilities until we get to the right answer or exhaust the space of possibilities."

# The term **guess and check** is used in algebra to refer to a trial and error approach to solving equations.

# The term **[brute force](https://en.wikipedia.org/wiki/Brute_force)** is perhaps more often used to mean what Guttag means by **exhaustive enumeration**.

# Guttag's subsequent remarks downplay the value of algorithmic optimization:
# 
# "At first blush, this may seem like an incredibly stupid way to solve a problem. Surprisingly, however, exhaustive enumeration algorithms are often the most practical way to solve a problem. They are typically easy to implement and easy to understand. And, in many cases, they run fast enough for all practical purposes. [...] try finding the cube root of 1957816251. The program will seem to finish almost instantaneously."

# In[5]:


get_ipython().run_cell_magic('time', '', '\nint_cube_root(1957816251)\n')


# In[6]:


get_ipython().run_line_magic('timeit', 'int_cube_root(1957816251)')


# "Now," Guttag continues, "try 7406961012236344616."

# In[8]:


get_ipython().run_cell_magic('time', '', '\nint_cube_root(7406961012236344616)\n')


# In[9]:


get_ipython().run_line_magic('timeit', 'int_cube_root(7406961012236344616)')


# "As you can see," Guttag concludes, "even if millions of guesses are required, it’s not usually a problem. Modern computers are amazingly fast. It takes on the order of one nanosecond — one billionth of a second — to execute an instruction. It’s a bit hard to appreciate how fast that is. For perspective, it takes slightly more than a nanosecond for light to travel a single foot (0.3 meters). Another way to think about this is that in the time it takes for the sound of your voice to travel a hundred feet, a modern computer can execute millions of instructions."

# However, Guttag then provides this code (modified) implementing an algorithm to find an approximation to a square root:

# In[24]:


def my_sqrt(n, epsilon):
    step = epsilon ** 2
    guesses = 0
    result = 0.0
    while abs(result ** 2 - n) > epsilon and result * result <= n:
        result += step
        guesses += 1
    if abs(result ** 2 - n) >= epsilon:
        return 'Failed ({} guesses)'.format(guesses)
    return '{} ({} guesses)'.format(result, guesses)


# Here we use exhaustive enumeration to find an approximation that lies within the constant `episilon` of the actual answer. The example Guttag provides, of using it to find an approximation of the square root of 25, demonstrates that, as he puts it, "**exhaustive enumeration is a search technique that works only if the set of values being searched includes the answer**":

# In[25]:


my_sqrt(25, 0.01)


# Of course, if we reduce our `epsilon` value to 1, the number of guesses needed is much fewer:

# In[20]:


my_sqrt(25, 1)


# But this is not possible if we want to find an approximation of the square root of 2, for example, since the square root of 2 is not a rational number.

# In[18]:


my_sqrt(2, 1)


# In[23]:


my_sqrt(2, 0.01)


# "The time has come," Guttag concludes, "to look for a different way to attack the problem. We need to choose a better algorithm rather than fine tune the current one."

# ## See also

# In *Introduction to Algorithms*, Cormen et al. describe their insertion sort implementation as taking an **incremental** approach, processing one element of the collection to be sorted at a time. They contrast this incremental approach with so-called **divide and conquer** approaches.