{
 "cells": [
  {
   "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: Given an array of n integers, find an int not in the input.  Use a minimal amount of memory.\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",
    "* Are we working with non-negative ints?\n",
    "    * Yes\n",
    "* What is the range of the integers?\n",
    "    * Discuss the approach for 4 billion integers\n",
    "    * Implement for 32 integers\n",
    "* Can we assume the inputs are valid?\n",
    "    * No"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Cases\n",
    "\n",
    "* None -> Exception\n",
    "* [] -> Exception\n",
    "* General case\n",
    "    * There is an int excluded from the input -> int\n",
    "    * There isn't an int excluded from the input -> None"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "The problem states to use a minimal amount of memory.  We'll use a bit vector to keep track of the inputs.\n",
    "\n",
    "Say we are given 4 billion integers, which is 2^32 integers.  The number of non-negative integers would be 2^31.  With a bit vector, we'll need 4 billion bits to map each integer to a bit.  Say we had only 1 GB of memory or 2^32 bytes.  This would leave us with 8 billion bits.\n",
    "\n",
    "To simplify this exercise, we'll work with an input of up to 32 ints that we'll map to a bit vector of 32 bits.\n",
    "\n",
    "<pre>\n",
    "\n",
    "input = [0, 1, 2, 3, 4...28, 29, 31]\n",
    "\n",
    "bytes          [         1          ]  [          2         ] [          3          ] [          4          ]\n",
    "index       =  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31\n",
    "bit_vector  =  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1\n",
    "\n",
    "result = 30\n",
    "\n",
    "* Loop through each item in the input, setting bit_vector[item] = True.\n",
    "* Loop through the bit_vector, return the first index where bit_vector[item] == False.\n",
    "\n",
    "</pre>\n",
    "\n",
    "Complexity:\n",
    "* Time: O(b), where b is the number of bits\n",
    "* Space: O(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from bitstring import BitArray  # Run pip install bitstring\n",
    "\n",
    "\n",
    "class Bits(object):\n",
    "\n",
    "    def new_int(self, array, max_size):\n",
    "        if not array:\n",
    "            raise TypeError('array cannot be None or empty')\n",
    "        bit_vector = BitArray(max_size)\n",
    "        for item in array:\n",
    "            bit_vector[item] = True\n",
    "        for index, item in enumerate(bit_vector):\n",
    "            if not item:\n",
    "                return index\n",
    "        return None"
   ]
  },
  {
   "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_new_int.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_new_int.py\n",
    "from nose.tools import assert_equal, assert_raises\n",
    "\n",
    "\n",
    "class TestBits(object):\n",
    "\n",
    "    def test_new_int(self):\n",
    "        bits = Bits()\n",
    "        max_size = 32\n",
    "        assert_raises(TypeError, bits.new_int, None, max_size)\n",
    "        assert_raises(TypeError, bits.new_int, [], max_size)\n",
    "        data = [item for item in range(30)]\n",
    "        data.append(31)\n",
    "        assert_equal(bits.new_int(data, max_size), 30)\n",
    "        data = [item for item in range(32)]\n",
    "        assert_equal(bits.new_int(data, max_size), None)\n",
    "        print('Success: test_find_int_excluded_from_input')\n",
    "\n",
    "\n",
    "def main():\n",
    "    test = TestBits()\n",
    "    test.test_new_int()\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    main()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success: test_find_int_excluded_from_input\n"
     ]
    }
   ],
   "source": [
    "%run -i test_new_int.py"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
