Coin Change Ways Problem/Solution

recursion_dynamic/coinchange_ways/coinchange_ways_solution.ipynb

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+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<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>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Solution Notebook"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Problem: Counting Ways of Making Change\n",
+    "\n",
+    "* [Hints](#Hints)\n",
+    "* [Algorithm](#Algorithm)\n",
+    "* [Code](#Code)\n",
+    "* [Unit Test](#Unit-Test)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 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?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 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)\n",
+    "* Space: O(n)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Code"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "def change_ways(n,k,coins):\n",
+    "    arr = [1] + [0] * n\n",
+    "    for coin in coins:\n",
+    "        for i in xrange(coin,n + 1):\n",
+    "            arr[i] += arr[i - coin]\n",
+    "    return 0 if n == 0 else arr[n]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Unit Test\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "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(0,2,[1,2]), 0)\n",
+    "        assert_equal(solution(100,3,[1,2,3]), 884)\n",
+    "        assert_equal(solution(1000,100,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()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Success: test_coinchange_ways\n"
+     ]
+    }
+   ],
+   "source": [
+    "%run -i test_coinchange_ways.py"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
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