interactive-coding-challenges/recursion_dynamic/coinchange_ways/coinchange_ways_solution.ipynb

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    "<small><i>This notebook was prepared by [mrb00l34n](http://github.com/mrb00l34n). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges).</i></small>"
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   "source": [
    "# Solution Notebook"
   ]
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    "## Problem: Counting Ways of Making Change\n",
    "\n",
    "* [Hints](#Hints)\n",
    "* [Algorithm](#Algorithm)\n",
    "* [Code](#Code)\n",
    "* [Unit Test](#Unit-Test)"
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    "## Hints\n",
    "\n",
    "* Can you think of a way to build up to a solution?\n",
    "* If there are 2 ways of making 3, and you are now given a coin of value v, how many ways can you make 3 + v?\n",
    "* Can you think of a way to divide the problem into smaller subproblems?"
   ]
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    "## Algorithm\n",
    "\n",
    "One possible solution using dynamic programming:\n",
    "* Create an array, s.t arr[i] = # of ways to make change for i\n",
    "* Initialize arr[0] = 1, arr[i>0] = 0\n",
    "* For each coin, and for each index from coin to n, increment arr[i] by arr[i - coin]\n",
    "\n",
    "How does this work?\n",
    "* As we iterate through each coin, we are adding the ways of making arr[i - coin] to arr[i]\n",
    "* If we have 2 ways of making 4, and are now iterating on a coin of value 3, there should be 2 ways of making 7.\n",
    "* We are essentially adding the coin we are iterating on to the # of ways of making arr[i].\n",
    "\n",
    "Complexity:\n",
    "* Time: O(mn); let the number of coins be m. We iterate from arr[coin] -> arr[n], or ~ n operations on each coin, hence n*m. \n",
    "* Space: O(n)"
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   "source": [
    "## Code"
   ]
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   "source": [
   "def change_ways(n, coins):\n",
   "    arr = [1] + [0] * n\n",
   "    for coin in coins:\n",
   "        for i in range(coin, n + 1):\n",
   "            arr[i] += arr[i - coin]\n",
   "    return 0 if n == 0 else arr[n]"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit Test\n",
    "\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
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    "collapsed": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting test_coinchange_ways.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_coinchange_ways.py\n",
    "from nose.tools import assert_equal\n",
    "\n",
    "\n",
    "class Challenge(object):\n",
    "\n",
    "    def test_coinchange_ways(self,solution):\n",
    "        assert_equal(solution(0, [1, 2]), 0)\n",
    "        assert_equal(solution(100, [1, 2, 3]), 884)\n",
    "        assert_equal(solution(1000, range(1, 101)), 15658181104580771094597751280645)\n",
    "        print('Success: test_coinchange_ways')\n",
    "\n",
    "\n",
    "def main():\n",
    "    test = Challenge()\n",
    "    test.test_coinchange_ways(change_ways)\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    main()"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success: test_coinchange_ways\n"
     ]
    }
   ],
   "source": [
    "%run -i test_coinchange_ways.py"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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