interactive-coding-challenges/recursion_dynamic/knapsack_01/knapsack_solution.ipynb

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   "cell_type": "markdown",
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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)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solution Notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem: Given a knapsack with a total weight capacity and a list of items with weight w(i) and value v(i), determine which items to select to maximize total value.\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",
    "* Can we replace the items once they are placed in the knapsack?\n",
    "    * No, this is the 0/1 knapsack problem\n",
    "* Can we split an item?\n",
    "    * No\n",
    "* Can we get an input item with weight of 0 or value of 0?\n",
    "    * No\n",
    "* Can we assume the inputs are valid?\n",
    "    * No\n",
    "* Are the inputs in sorted order by val/weight?\n",
    "    * Yes, if not we'd need to sort them first\n",
    "* Can we assume this fits memory?\n",
    "    * Yes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Cases\n",
    "\n",
    "* items or total weight is None -> Exception\n",
    "* items or total weight is 0 -> 0\n",
    "* General case\n",
    "\n",
    "<pre>\n",
    "total_weight = 8\n",
    "items\n",
    "  v | w\n",
    "  0 | 0\n",
    "a 2 | 2\n",
    "b 4 | 2\n",
    "c 6 | 4\n",
    "d 9 | 5\n",
    "\n",
    "max value = 13\n",
    "items\n",
    "  v | w\n",
    "b 4 | 2\n",
    "d 9 | 5 \n",
    "</pre>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm\n",
    "\n",
    "We'll use bottom up dynamic programming to build a table.\n",
    "\n",
    "The solution for the top down approach is also provided below.\n",
    "\n",
    "<pre>\n",
    "v = value\n",
    "w = weight\n",
    "\n",
    "               j              \n",
    "    -------------------------------------------------\n",
    "    | v | w || 0 | 1 | 2 | 3 | 4 | 5 | 6  | 7  | 8  |\n",
    "    -------------------------------------------------\n",
    "    | 0 | 0 || 0 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0  |\n",
    "i a | 2 | 2 || 0 | 0 | 2 | 2 | 2 | 2 | 2  | 2  | 2  |\n",
    "  b | 4 | 2 || 0 | 0 | 4 | 4 | 6 | 6 | 6  | 6  | 6  |\n",
    "  c | 6 | 4 || 0 | 0 | 4 | 4 | 6 | 6 | 10 | 10 | 12 |\n",
    "  d | 9 | 5 || 0 | 0 | 4 | 4 | 6 | 9 | 10 | 13 | 13 |\n",
    "    -------------------------------------------------\n",
    "\n",
    "i = row\n",
    "j = col\n",
    "\n",
    "if j >= item[i].weight:\n",
    "    T[i][j] = max(item[i].value + T[i - 1][j - item[i].weight],\n",
    "                  T[i - 1][j])\n",
    "else:\n",
    "    T[i][j] = T[i - 1][j]\n",
    "</pre>\n",
    "\n",
    "Complexity:\n",
    "* Time: O(n * w), where n is the number of items and w is the total weight\n",
    "* Space: O(n * w), where n is the number of items and w is the total weight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Item Class"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Item(object):\n",
    "\n",
    "    def __init__(self, label, value, weight):\n",
    "        self.label = label\n",
    "        self.value = value\n",
    "        self.weight = weight\n",
    "\n",
    "    def __repr__(self):\n",
    "        return self.label + ' v:' + str(self.value) + ' w:' + str(self.weight)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Knapsack Bottom Up"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class Knapsack(object):\n",
    "\n",
    "    def fill_knapsack(self, input_items, total_weight):\n",
    "        if input_items is None or total_weight is None:\n",
    "            raise TypeError('input_items or total_weight cannot be None')\n",
    "        if not input_items or total_weight == 0:\n",
    "            return 0\n",
    "        items = list([Item(label='', value=0, weight=0)] + input_items)\n",
    "        num_rows = len(items)\n",
    "        num_cols = total_weight + 1\n",
    "        T = [[None] * num_cols for _ in range(num_rows)]\n",
    "        for i in range(num_rows):\n",
    "            for j in range(num_cols):\n",
    "                if i == 0 or j == 0:\n",
    "                    T[i][j] = 0\n",
    "                elif j >= items[i].weight:\n",
    "                    T[i][j] = max(items[i].value + T[i - 1][j - items[i].weight],\n",
    "                                  T[i - 1][j])\n",
    "                else:\n",
    "                    T[i][j] = T[i - 1][j]\n",
    "        results = []\n",
    "        i = num_rows - 1\n",
    "        j = num_cols - 1\n",
    "        while T[i][j] != 0:\n",
    "            if T[i - 1][j] ==  T[i][j]:\n",
    "                i -= 1\n",
    "            elif T[i][j - 1] ==  T[i][j]:\n",
    "                j -= 1\n",
    "            else:\n",
    "                results.append(items[i])\n",
    "                i -= 1\n",
    "                j -= items[i].weight\n",
    "        return results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Knapsack Top Down"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class KnapsackTopDown(object):\n",
    "\n",
    "    def fill_knapsack(self, items, total_weight):\n",
    "        if items is None or total_weight is None:\n",
    "            raise TypeError('input_items or total_weight cannot be None')\n",
    "        if not items or not total_weight:\n",
    "            return 0\n",
    "        memo = {}\n",
    "        result = self._fill_knapsack(items, total_weight, memo, index=0)\n",
    "        return result\n",
    "\n",
    "\n",
    "    def _fill_knapsack(self, items, total_weight, memo, index):\n",
    "        if total_weight < 0:\n",
    "            return 0\n",
    "        if not total_weight or index >= len(items):\n",
    "            return items[index - 1].value\n",
    "        if (total_weight, len(items) - index - 1) in memo:\n",
    "            return memo[(total_weight, len(items) - index - 1)] + items[index - 1].value\n",
    "        results = []\n",
    "        for i in range(index, len(items)):\n",
    "            total_weight -= items[i].weight\n",
    "            result = self._fill_knapsack(items, total_weight, memo, index=i + 1)\n",
    "            total_weight += items[i].weight\n",
    "            results.append(result)\n",
    "        results_index = 0\n",
    "        for i in range(index, len(items)):\n",
    "            memo[total_weight, len(items) - i] = max(results[results_index:])\n",
    "            results_index += 1\n",
    "        return max(results) + (items[index - 1].value if index > 0 else 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Knapsack Top Down Alternate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class Result(object):\n",
    "\n",
    "    def __init__(self, total_weight, item):\n",
    "        self.total_weight = total_weight\n",
    "        self.item = item\n",
    "\n",
    "    def __repr__(self):\n",
    "        return 'w:' + str(self.total_weight) + ' i:' + str(self.item)\n",
    "\n",
    "    def __lt__(self, other):\n",
    "        return self.total_weight < other.total_weight\n",
    "\n",
    "\n",
    "def knapsack_top_down_alt(items, total_weight):\n",
    "    if items is None or total_weight is None:\n",
    "        raise TypeError('input_items or total_weight cannot be None')\n",
    "    if not items or not total_weight:\n",
    "        return 0\n",
    "    memo = {}\n",
    "    result = _knapsack_top_down_alt(items, total_weight, memo, index=0)\n",
    "    curr_item = result.item\n",
    "    curr_weight = curr_item.weight\n",
    "    picked_items = [curr_item]\n",
    "    while curr_weight > 0:\n",
    "        total_weight -= curr_item.weight\n",
    "        curr_item = memo[(total_weight, len(items) - len(picked_items))].item\n",
    "    return result\n",
    "\n",
    "\n",
    "def _knapsack_top_down_alt(items, total_weight, memo, index):\n",
    "    if total_weight < 0:\n",
    "        return Result(total_weight=0, item=None)\n",
    "    if not total_weight or index >= len(items):\n",
    "        return Result(total_weight=items[index - 1].value, item=items[index - 1])\n",
    "    if (total_weight, len(items) - index - 1) in memo:\n",
    "        weight=memo[(total_weight, \n",
    "                     len(items) - index - 1)].total_weight + items[index - 1].value\n",
    "        return Result(total_weight=weight,\n",
    "                      item=items[index-1])\n",
    "    results = []\n",
    "    for i in range(index, len(items)):\n",
    "        total_weight -= items[i].weight\n",
    "        result = _knapsack_top_down_alt(items, total_weight, memo, index=i + 1)\n",
    "        total_weight += items[i].weight\n",
    "        results.append(result)\n",
    "    results_index = 0\n",
    "    for i in range(index, len(items)):\n",
    "        memo[(total_weight, len(items) - i)] = max(results[results_index:])\n",
    "        results_index += 1\n",
    "    if index == 0:\n",
    "        result_item = memo[(total_weight, len(items) - 1)].item\n",
    "    else:\n",
    "        result_item = items[index - 1]\n",
    "    weight = max(results).total_weight + (items[index - 1].value if index > 0 else 0)\n",
    "    return Result(total_weight=weight,\n",
    "                  item=result_item)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting test_knapsack.py\n"
     ]
    }
   ],
   "source": [
    "%%writefile test_knapsack.py\n",
    "from nose.tools import assert_equal, assert_raises\n",
    "\n",
    "\n",
    "class TestKnapsack(object):\n",
    "\n",
    "    def test_knapsack_bottom_up(self):\n",
    "        knapsack = Knapsack()\n",
    "        assert_raises(TypeError, knapsack.fill_knapsack, None, None)\n",
    "        assert_equal(knapsack.fill_knapsack(0, 0), 0)\n",
    "        items = []\n",
    "        items.append(Item(label='a', value=2, weight=2))\n",
    "        items.append(Item(label='b', value=4, weight=2))\n",
    "        items.append(Item(label='c', value=6, weight=4))\n",
    "        items.append(Item(label='d', value=9, weight=5))\n",
    "        total_weight = 8\n",
    "        expected_value = 13\n",
    "        results = knapsack.fill_knapsack(items, total_weight)\n",
    "        assert_equal(results[0].label, 'd')\n",
    "        assert_equal(results[1].label, 'b')\n",
    "        total_value = 0\n",
    "        for item in results:\n",
    "            total_value += item.value\n",
    "        assert_equal(total_value, expected_value)\n",
    "        print('Success: test_knapsack_bottom_up')\n",
    "\n",
    "    def test_knapsack_top_down(self):\n",
    "        knapsack = KnapsackTopDown()\n",
    "        assert_raises(TypeError, knapsack.fill_knapsack, None, None)\n",
    "        assert_equal(knapsack.fill_knapsack(0, 0), 0)\n",
    "        items = []\n",
    "        items.append(Item(label='a', value=2, weight=2))\n",
    "        items.append(Item(label='b', value=4, weight=2))\n",
    "        items.append(Item(label='c', value=6, weight=4))\n",
    "        items.append(Item(label='d', value=9, weight=5))\n",
    "        total_weight = 8\n",
    "        expected_value = 13\n",
    "        assert_equal(knapsack.fill_knapsack(items, total_weight), expected_value)\n",
    "        print('Success: test_knapsack_top_down')\n",
    "\n",
    "def main():\n",
    "    test = TestKnapsack()\n",
    "    test.test_knapsack_bottom_up()\n",
    "    test.test_knapsack_top_down()\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    main()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success: test_knapsack_bottom_up\n",
      "Success: test_knapsack_top_down\n"
     ]
    }
   ],
   "source": [
    "%run -i test_knapsack.py"
   ]
  }
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