{
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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 the max total value you can carry.\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",
" * Yes, this is the unbounded 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",
"* Do we need to return the items that make up the max total value?\n",
" * No, just the total value\n",
"* Can we assume the inputs are valid?\n",
" * No\n",
"* Are the inputs in sorted order by val/weight?\n",
" * Yes\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",
"\n",
"total_weight = 8\n",
"items\n",
" v | w\n",
" 0 | 0\n",
"a 1 | 1\n",
"b 3 | 2\n",
"c 7 | 4\n",
"\n",
"max value = 14 \n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algorithm\n",
"\n",
"We'll use bottom up dynamic programming to build a table. \n",
"\n",
"Taking what we learned with the 0/1 knapsack problem, we could solve the problem like the following:\n",
"\n",
"\n",
"\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",
" a | 1 | 1 || 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n",
"i b | 3 | 2 || 0 | 1 | 3 | 4 | 6 | 7 | 9 | 10 | 12 |\n",
" c | 7 | 4 || 0 | 1 | 3 | 4 | 7 | 8 | 10 | 11 | 14 |\n",
" -------------------------------------------------\n",
"\n",
"i = row\n",
"j = col\n",
"\n",
"\n",
"\n",
"However, unlike the 0/1 knapsack variant, we don't actually need to keep space of O(n * w), where n is the number of items and w is the total weight. We just need a single array that we update after we process each item:\n",
"\n",
"\n",
"\n",
" -------------------------------------------------\n",
" | v | w || 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n",
" -------------------------------------------------\n",
"\n",
" -------------------------------------------------\n",
" a | 1 | 1 || 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n",
" -------------------------------------------------\n",
"\n",
" -------------------------------------------------\n",
" b | 3 | 2 || 0 | 1 | 3 | 4 | 6 | 7 | 9 | 10 | 12 |\n",
" -------------------------------------------------\n",
"\n",
" -------------------------------------------------\n",
" c | 7 | 4 || 0 | 1 | 3 | 4 | 7 | 8 | 10 | 11 | 14 |\n",
" -------------------------------------------------\n",
"\n",
"if j >= items[i].weight:\n",
" T[j] = max(items[i].value + T[j - items[i].weight],\n",
" T[j])\n",
"\n",
"\n",
"\n",
"Complexity:\n",
"* Time: O(n * w), where n is the number of items and w is the total weight\n",
"* Space: O(w), where 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, items, total_weight):\n",
" if items is None or total_weight is None:\n",
" raise TypeError('items or total_weight cannot be None')\n",
" if not items or total_weight == 0:\n",
" return 0\n",
" num_rows = len(items)\n",
" num_cols = total_weight + 1\n",
" T = [0] * (num_cols)\n",
" for i in range(num_rows):\n",
" for j in range(num_cols):\n",
" if j >= items[i].weight:\n",
" T[j] = max(items[i].value + T[j - items[i].weight],\n",
" T[j])\n",
" return T[-1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Unit Test"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Overwriting test_knapsack_unbounded.py\n"
]
}
],
"source": [
"%%writefile test_knapsack_unbounded.py\n",
"from nose.tools import assert_equal, assert_raises\n",
"\n",
"\n",
"class TestKnapsack(object):\n",
"\n",
" def test_knapsack(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=1, weight=1))\n",
" items.append(Item(label='b', value=3, weight=2))\n",
" items.append(Item(label='c', value=7, weight=4))\n",
" total_weight = 8\n",
" expected_value = 14\n",
" results = knapsack.fill_knapsack(items, total_weight)\n",
" total_weight = 7\n",
" expected_value = 11\n",
" results = knapsack.fill_knapsack(items, total_weight)\n",
" assert_equal(results, expected_value)\n",
" print('Success: test_knapsack')\n",
"\n",
"def main():\n",
" test = TestKnapsack()\n",
" test.test_knapsack()\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_knapsack\n"
]
}
],
"source": [
"%run -i test_knapsack_unbounded.py"
]
}
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