interactive-coding-challenges/recursion_dynamic/magic_index/magic_index_solution.ipynb

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   "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)."
   ]
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   "source": [
    "# Solution Notebook"
   ]
  },
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   "metadata": {},
   "source": [
    "## Problem: Find the magic index in an array, where array[i] = i.\n",
    "\n",
    "* [Constraints](#Constraints)\n",
    "* [Test Cases](#Test-Cases)\n",
    "* [Algorithm](#Algorithm)\n",
    "* [Code](#Code)\n",
    "* [Unit Test](#Unit-Test)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constraints\n",
    "\n",
    "* Is the array sorted?\n",
    "    * Yes\n",
    "* Are the elements in the array distinct?\n",
    "    * No\n",
    "* Does a magic index always exist?\n",
    "    * No\n",
    "* If there is no magic index, do we just return -1?\n",
    "    * Yes\n",
    "* Are negative values allowed in the array?\n",
    "    * Yes\n",
    "* If there are multiple magic values, what do we return?\n",
    "    * Return the left-most one\n",
    "* Can we assume this fits memory?\n",
    "    * Yes"
   ]
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   "cell_type": "markdown",
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   "source": [
    "## Test Cases\n",
    "\n",
    "* None input -> -1\n",
    "* Empty array -> -1\n",
    "\n",
    "<pre>\n",
    "a[i]  -4 -2  2  6  6  6  6 10\n",
    "  i    0  1  2  3  4  5  6  7\n",
    "</pre>\n",
    "\n",
    "Result: 2\n",
    "\n",
    "<pre>\n",
    "a[i]  -4 -2  1  6  6  6  6 10\n",
    "  i    0  1  2  3  4  5  6  7\n",
    "</pre>\n",
    "\n",
    "Result: 6\n",
    "\n",
    "<pre>\n",
    "a[i]  -4 -2  1  6  6  6  7 10\n",
    "  i    0  1  2  3  4  5  6  7\n",
    "</pre>\n",
    "\n",
    "Result: -1"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "We'll use a binary search to split the search space in half on each iteration.  To obtain more efficiency, we can do a little better than a naive left and half split.\n",
    "\n",
    "In the example below, we see that i == 5 cannot be the magic index, otherwise a[5] would have to equal 5 (note a[4] == 6).\n",
    "\n",
    "<pre>\n",
    "a[i]  -4 -2  2  6  6  6  6 10\n",
    "  i    0  1  1  3  4  5  6  7\n",
    "                  mid\n",
    "</pre>\n",
    "\n",
    "Similarly, in the example below we can further trim the left search space.\n",
    "\n",
    "<pre>\n",
    "a[i]  -4 -2  2  2  2  6  6 10\n",
    "  i    0  1  2  3  4  5  6  7\n",
    "                  mid\n",
    "</pre>\n",
    "\n",
    "\n",
    "* Calculate mid\n",
    "* If mid == array[mid], return mid\n",
    "* Recurse on the left side of the array\n",
    "    * start: 0\n",
    "    * end: min(mid-1, array[mid]\n",
    "* Recurse on the right side of the array\n",
    "    * start: max(mid+1, array[mid]\n",
    "    * end: end\n",
    "\n",
    "Complexity:\n",
    "* Time: O(log(n))\n",
    "* Space: O(log(n))"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "\n",
    "\n",
    "class MagicIndex(object):\n",
    "\n",
    "    def find_magic_index(self, array):\n",
    "        if array is None or not array:\n",
    "            return -1\n",
    "        return self._find_magic_index(array, 0, len(array) - 1)\n",
    "\n",
    "    def _find_magic_index(self, array, start, end):\n",
    "        if end < start or start < 0 or end >= len(array):\n",
    "            return -1\n",
    "        mid = (start + end) // 2\n",
    "        if mid == array[mid]:\n",
    "            return mid\n",
    "        left_end = min(mid - 1, array[mid])\n",
    "        left_result = self._find_magic_index(array, start, end=left_end)\n",
    "        if left_result != -1:\n",
    "            return left_result\n",
    "        right_start = max(mid + 1, array[mid])\n",
    "        right_result = self._find_magic_index(array, start=right_start, end=end)\n",
    "        if right_result != -1:\n",
    "            return right_result\n",
    "        return -1"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting test_find_magic_index.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_find_magic_index.py\n",
    "from nose.tools import assert_equal\n",
    "\n",
    "\n",
    "class TestFindMagicIndex(object):\n",
    "\n",
    "    def test_find_magic_index(self):\n",
    "        magic_index = MagicIndex()\n",
    "        assert_equal(magic_index.find_magic_index(None), -1)\n",
    "        assert_equal(magic_index.find_magic_index([]), -1)\n",
    "        array = [-4, -2, 2, 6, 6, 6, 6, 10]\n",
    "        assert_equal(magic_index.find_magic_index(array), 2)\n",
    "        array = [-4, -2, 1, 6, 6, 6, 6, 10]\n",
    "        assert_equal(magic_index.find_magic_index(array), 6)\n",
    "        array = [-4, -2, 1, 6, 6, 6, 7, 10]\n",
    "        assert_equal(magic_index.find_magic_index(array), -1)\n",
    "        print('Success: test_find_magic')\n",
    "\n",
    "\n",
    "def main():\n",
    "    test = TestFindMagicIndex()\n",
    "    test.test_find_magic_index()\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    main()"
   ]
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success: test_find_magic\n"
     ]
    }
   ],
   "source": [
    "%run -i test_find_magic_index.py"
   ]
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