{
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   "source": [
    "# Brute force"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sources:\n",
    "\n",
    "1. Guttag, *Introduction to Computation and Programming Using Python*, Revised and Expanded Ed. (2013)\n",
    "2. Cormen et al., *Introduction to Algorithms*, 3rd ed. (2009)\n",
    "3. [Brute force](https://en.wikipedia.org/wiki/Brute_force)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In *Introduction to Computation and Programming Using Python*, Guttag provides this code (modified) for finding the integer cube root of an integer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def int_cube_root(n):\n",
    "    result = 0\n",
    "    while result ** 3 < abs(n):\n",
    "        result += 1\n",
    "    if result ** 3 != abs(n):\n",
    "        return '{} is not a perfect cube'.format(n)\n",
    "    else:\n",
    "        if n < 0:\n",
    "            result = -result\n",
    "        return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'7 is not a perfect cube'"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_cube_root(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_cube_root(8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"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.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The term **guess and check** is used in algebra to refer to a trial and error approach to solving equations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The term **[brute force](https://en.wikipedia.org/wiki/Brute_force)** is perhaps more often used to mean what Guttag means by **exhaustive enumeration**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Guttag's subsequent remarks downplay the value of algorithmic optimization:\n",
    "\n",
    "\"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.\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 791 µs, sys: 1 µs, total: 792 µs\n",
      "Wall time: 796 µs\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1251"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "int_cube_root(1957816251)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 791 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit int_cube_root(1957816251)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"Now,\" Guttag continues, \"try 7406961012236344616.\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 1.37 s, sys: 7.04 ms, total: 1.38 s\n",
      "Wall time: 1.38 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1949306"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "int_cube_root(7406961012236344616)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 loop, best of 3: 1.34 s per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit int_cube_root(7406961012236344616)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"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.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, Guttag then provides this code (modified) implementing an algorithm to find an approximation to a square root:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def my_sqrt(n, epsilon):\n",
    "    step = epsilon ** 2\n",
    "    guesses = 0\n",
    "    result = 0.0\n",
    "    while abs(result ** 2 - n) > epsilon and result * result <= n:\n",
    "        result += step\n",
    "        guesses += 1\n",
    "    if abs(result ** 2 - n) >= epsilon:\n",
    "        return 'Failed ({} guesses)'.format(guesses)\n",
    "    return '{} ({} guesses)'.format(result, guesses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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**\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'4.999000000001688 (49990 guesses)'"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_sqrt(25, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, if we reduce our `epsilon` value to 1, the number of guesses needed is much fewer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'5.0 (5 guesses)'"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_sqrt(25, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Failed (1 guesses)'"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_sqrt(2, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.410699999999861 (14107 guesses)'"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_sqrt(2, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"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.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## See also"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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."
   ]
  }
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