{
"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: 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)"
]
},
{
"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"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Test Cases\n",
"\n",
"* None input -> -1\n",
"* Empty array -> -1\n",
"\n",
"\n",
"a[i] -4 -2 2 6 6 6 6 10\n",
" i 0 1 2 3 4 5 6 7\n",
"\n",
"\n",
"Result: 2\n",
"\n",
"\n",
"a[i] -4 -2 1 6 6 6 6 10\n",
" i 0 1 2 3 4 5 6 7\n",
"\n",
"\n",
"Result: 6\n",
"\n",
"\n",
"a[i] -4 -2 1 6 6 6 7 10\n",
" i 0 1 2 3 4 5 6 7\n",
"\n",
"\n",
"Result: -1"
]
},
{
"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",
"\n",
"a[i] -4 -2 2 6 6 6 6 10\n",
" i 0 1 1 3 4 5 6 7\n",
" mid\n",
"\n",
"\n",
"Similarly, in the example below we can further trim the left search space.\n",
"\n",
"\n",
"a[i] -4 -2 2 2 2 6 6 10\n",
" i 0 1 2 3 4 5 6 7\n",
" mid\n",
"\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))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Code"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
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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"
]
},
{
"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()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Success: test_find_magic\n"
]
}
],
"source": [
"%run -i test_find_magic_index.py"
]
}
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