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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook was prepared by [Donne Martin](https://github.com/donnemartin). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solution Notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem: Generate a list of primes.\n",
    "\n",
    "* [Constraints](#Constraints)\n",
    "* [Test Cases](#Test-Cases)\n",
    "* [Algorithm](#Algorithm)\n",
    "* [Code](#Code)\n",
    "* [Unit Test](#Unit-Test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constraints\n",
    "\n",
    "* Is it correct that 1 is not considered a prime number?\n",
    "    * Yes\n",
    "* Can we assume the inputs are valid?\n",
    "    * No\n",
    "* Can we assume this fits memory?\n",
    "    * Yes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Cases\n",
    "\n",
    "* None -> Exception\n",
    "* Not an int -> Exception\n",
    "* 20 -> [False, False, True, True, False, True, False, True, False, False, False, True, False, True, False, False, False, True, False, True]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "For a number to be prime, it must be 2 or greater and cannot be divisible by another number other than itself (and 1).\n",
    "\n",
    "We'll use the Sieve of Eratosthenes.  All non-prime numbers are divisible by a prime number.\n",
    "\n",
    "* Use an array (or bit array, bit vector) to keep track of each integer up to the max\n",
    "* Start at 2, end at sqrt(max)\n",
    "    * We can use sqrt(max) instead of max because:\n",
    "        * For each value that divides the input number evenly, there is a complement b where a * b = n\n",
    "        * If a > sqrt(n) then b < sqrt(n) because sqrt(n^2) = n\n",
    "    * \"Cross off\" all numbers divisible by 2, 3, 5, 7, ... by setting array[index] to False\n",
    "\n",
    "Complexity:\n",
    "* Time: O(n log log n)\n",
    "* Space: O(n)\n",
    "\n",
    "Wikipedia's animation:\n",
    "\n",
    "![alt text](https://upload.wikimedia.org/wikipedia/commons/b/b9/Sieve_of_Eratosthenes_animation.gif)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "\n",
    "class PrimeGenerator(object):\n",
    "\n",
    "    def generate_primes(self, max_num):\n",
    "        if max_num is None:\n",
    "            raise TypeError('max_num cannot be None')\n",
    "        array = [True] * max_num\n",
    "        array[0] = False\n",
    "        array[1] = False\n",
    "        prime = 2\n",
    "        while prime <= math.sqrt(max_num):\n",
    "            self._cross_off(array, prime)\n",
    "            prime = self._next_prime(array, prime)\n",
    "        return array\n",
    "\n",
    "    def _cross_off(self, array, prime):\n",
    "        for index in range(prime*prime, len(array), prime):\n",
    "            # Start with prime*prime because if we have a k*prime\n",
    "            # where k < prime, this value would have already been\n",
    "            # previously crossed off\n",
    "            array[index] = False\n",
    "\n",
    "    def _next_prime(self, array, prime):\n",
    "        next = prime + 1\n",
    "        while next < len(array) and not array[next]:\n",
    "            next += 1\n",
    "        return next"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting test_generate_primes.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_generate_primes.py\n",
    "from nose.tools import assert_equal, assert_raises\n",
    "\n",
    "\n",
    "class TestMath(object):\n",
    "\n",
    "    def test_generate_primes(self):\n",
    "        prime_generator = PrimeGenerator()\n",
    "        assert_raises(TypeError, prime_generator.generate_primes, None)\n",
    "        assert_raises(TypeError, prime_generator.generate_primes, 98.6)\n",
    "        assert_equal(prime_generator.generate_primes(20), [False, False, True, \n",
    "                                                           True, False, True, \n",
    "                                                           False, True, False, \n",
    "                                                           False, False, True, \n",
    "                                                           False, True, False, \n",
    "                                                           False, False, True, \n",
    "                                                           False, True])\n",
    "        print('Success: generate_primes')\n",
    "\n",
    "\n",
    "def main():\n",
    "    test = TestMath()\n",
    "    test.test_generate_primes()\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    main()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success: generate_primes\n"
     ]
    }
   ],
   "source": [
    "%run -i test_generate_primes.py"
   ]
  }
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