interactive-coding-challenges/recursion_dynamic/steps/steps_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: You are running up n steps.  If you can take a single, double, or triple step, how many possible ways are there to run up to the nth step?\n",
    "\n",
    "* [Constraints](#Constraints)\n",
    "* [Test Cases](#Test-Cases)\n",
    "* [Algorithm](#Algorithm)\n",
    "* [Code](#Code)\n",
    "* [Unit Test](#Unit-Test)"
   ]
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   "metadata": {},
   "source": [
    "## Constraints\n",
    "\n",
    "* If n == 0, what should the result be?\n",
    "    * Go with 1, but discuss different approaches\n",
    "* Can we assume the inputs are valid?\n",
    "    * No\n",
    "* Can we assume this fits memory?\n",
    "    * Yes"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Cases\n",
    "\n",
    "* None or negative input -> Exception\n",
    "* n == 0 -> 1\n",
    "* n == 1 -> 1\n",
    "* n == 2 -> 2\n",
    "* n == 3 -> 4\n",
    "* n == 4 -> 7\n",
    "* n == 10 -> 274"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "To get to step n, we will need to have gone:\n",
    "\n",
    "* One step from n-1\n",
    "* Two steps from n-2\n",
    "* Three steps from n-3\n",
    "\n",
    "If we go the one step route above, we'll be at n-1 before taking the last step.  To get to step n-1, we will need to have gone:\n",
    "\n",
    "* One step from n-1-1\n",
    "* Two steps from n-1-2\n",
    "* Three steps from n-1-2\n",
    "\n",
    "Continue this process until we reach the start.\n",
    "\n",
    "Base case:\n",
    "\n",
    "* If n < 0: return 0\n",
    "* If n == 0: return 1\n",
    "\n",
    "Note, if we had chosen n == 0 to return 0 instead, we would need to add additional base cases.  Otherwise we'd be adding multiple 0's once we hit the base cases and not get any result > 0.\n",
    "\n",
    "Recursive case:\n",
    "\n",
    "We'll memoize the solution to improve performance.\n",
    "\n",
    "* Use the memo if we've already processed the current step.\n",
    "* Update the memo by adding the recursive calls to step(n-1), step(n-2), step(n-3)\n",
    "\n",
    "Complexity:\n",
    "* Time: O(n), if using memoization\n",
    "* Space: O(n), where n is the recursion depth\n",
    "\n",
    "Note: The number of ways will quickly overflow the bounds of an integer."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "class Steps(object):\n",
    "\n",
    "    def count_ways(self, num_steps):\n",
    "        if num_steps is None or num_steps < 0:\n",
    "            raise TypeError('num_steps cannot be None or negative')\n",
    "        cache = {}\n",
    "        return self._count_ways(num_steps, cache)\n",
    "\n",
    "    def _count_ways(self, num_steps, cache):\n",
    "        if num_steps < 0:\n",
    "            return 0\n",
    "        if num_steps == 0:\n",
    "            return 1\n",
    "        if num_steps in cache:\n",
    "            return cache[num_steps]\n",
    "        cache[num_steps] = (self._count_ways(num_steps-1, cache) +\n",
    "                            self._count_ways(num_steps-2, cache) +\n",
    "                            self._count_ways(num_steps-3, cache))\n",
    "        return cache[num_steps]"
   ]
  },
  {
   "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_steps.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_steps.py\n",
    "from nose.tools import assert_equal, assert_raises\n",
    "\n",
    "\n",
    "class TestSteps(object):\n",
    "\n",
    "    def test_steps(self):\n",
    "        steps = Steps()\n",
    "        assert_raises(TypeError, steps.count_ways, None)\n",
    "        assert_raises(TypeError, steps.count_ways, -1)\n",
    "        assert_equal(steps.count_ways(0), 1)\n",
    "        assert_equal(steps.count_ways(1), 1)\n",
    "        assert_equal(steps.count_ways(2), 2)\n",
    "        assert_equal(steps.count_ways(3), 4)\n",
    "        assert_equal(steps.count_ways(4), 7)\n",
    "        assert_equal(steps.count_ways(10), 274)\n",
    "        print('Success: test_steps')\n",
    "\n",
    "\n",
    "def main():\n",
    "    test = TestSteps()\n",
    "    test.test_steps()\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_steps\n"
     ]
    }
   ],
   "source": [
    "%run -i test_steps.py"
   ]
  }
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