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   "metadata": {},
   "source": [
    "<small><i>This notebook was prepared by [Donne Martin](http://donnemartin.com). Source and license info is on [GitHub](https://bit.ly/code-notes).</i></small>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem: Find the start of a linked list loop.\n",
    "\n",
    "* [Constraints and Assumptions](#Constraints-and-Assumptions)\n",
    "* [Test Cases](#Test-Cases)\n",
    "* [Algorithm](#Algorithm)\n",
    "* [Code](#Code)\n",
    "* [Unit Test](#Unit-Test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constraints and Assumptions\n",
    "\n",
    "*Problem statements are often intentionally ambiguous.  Identifying constraints and stating assumptions can help to ensure you code the intended solution.*\n",
    "\n",
    "* This is a singly linked list?\n",
    "    * Yes\n",
    "* Can we assume we are always passed a circular linked list?\n",
    "    * No\n",
    "* Can we assume we already have a linked list class that can be used for this problem?\n",
    "    * Yes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Cases\n",
    "\n",
    "* Empty list\n",
    "* Not a circular linked list\n",
    "    * One element\n",
    "    * Two elements\n",
    "    * Three or more elements\n",
    "* General case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "* Use two pointers i, j, initialized to the head\n",
    "* Increment i and j until they meet\n",
    "    * j is incremented twice as fast as i\n",
    "        * If j's next is NULL, we do not have a circular list\n",
    "* When i and j meet, move j to the head\n",
    "* Increment i and j one node at a time until they meet\n",
    "* Where they meet is the start of the loop\n",
    "\n",
    "Complexity:\n",
    "* Time: O(n)\n",
    "* Space: In-place"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%run linked_list.py"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class MyLinkedList(LinkedList):\n",
    "    def find_loop_start(self):\n",
    "        if self.head is None or self.head.next is None:\n",
    "            return\n",
    "        i = self.head\n",
    "        j = self.head\n",
    "        i = i.next\n",
    "        j = j.next.next\n",
    "        \n",
    "        # Increment i and j until they meet\n",
    "        # j is incremented twice as fast as i\n",
    "        while j != i:\n",
    "            i = i.next\n",
    "            if j is None or j.next is None:\n",
    "                return\n",
    "            j = j.next.next\n",
    "            \n",
    "        # When i and j meet, move j to the head\n",
    "        j = self.head\n",
    "        \n",
    "        # Increment i and j one node at a time until \n",
    "        # they meet, which is the start of the loop\n",
    "        while j != i:\n",
    "            i = i.next\n",
    "            j = j.next\n",
    "        return i.data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*It is important to identify and run through general and edge cases from the [Test Cases](#Test-Cases) section by hand.  You generally will not be asked to write a unit test like what is shown below.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test: Empty list\n",
      "Test: Not a circular linked list: One element\n",
      "Test: Not a circular linked list: Two elements\n",
      "Test: Not a circular linked list: Three or more elements\n",
      "Test: General case: Circular linked list\n",
      "Success: test_find_loop_start\n"
     ]
    }
   ],
   "source": [
    "from nose.tools import assert_equal\n",
    "\n",
    "class Test(object):\n",
    "    def test_find_loop_start(self):\n",
    "        print('Test: Empty list')\n",
    "        linked_list = MyLinkedList()\n",
    "        assert_equal(linked_list.find_loop_start(), None)\n",
    "        \n",
    "        print('Test: Not a circular linked list: One element')\n",
    "        head = Node(1)\n",
    "        linked_list = MyLinkedList(head)\n",
    "        assert_equal(linked_list.find_loop_start(), None)\n",
    "        \n",
    "        print('Test: Not a circular linked list: Two elements')\n",
    "        linked_list.append(2)\n",
    "        assert_equal(linked_list.find_loop_start(), None)\n",
    "        \n",
    "        print('Test: Not a circular linked list: Three or more elements')\n",
    "        linked_list.append(3)\n",
    "        assert_equal(linked_list.find_loop_start(), None)\n",
    "        \n",
    "        print('Test: General case: Circular linked list')\n",
    "        node10 = Node(10)\n",
    "        node9 = Node(9, node10)\n",
    "        node8 = Node(8, node9)\n",
    "        node7 = Node(7, node8)\n",
    "        node6 = Node(6, node7)\n",
    "        node5 = Node(5, node6)\n",
    "        node4 = Node(4, node5)\n",
    "        node3 = Node(3, node4)\n",
    "        node2 = Node(2, node3)\n",
    "        node1 = Node(1, node2)\n",
    "        node0 = Node(0, node1)\n",
    "        node10.next = node3\n",
    "        linked_list = MyLinkedList(node0)\n",
    "        assert_equal(linked_list.find_loop_start(), 3)\n",
    "        \n",
    "        print('Success: test_find_loop_start')\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    test = Test()\n",
    "    test.test_find_loop_start()"
   ]
  }
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