This notebook was prepared by Donne Martin. Source and license info is on GitHub.

Solution Notebook

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.

Constraints

  • Can we replace the items once they are placed in the knapsack?
    • Yes, this is the unbounded knapsack problem
  • Can we split an item?
    • No
  • Can we get an input item with weight of 0 or value of 0?
    • No
  • Do we need to return the items that make up the max total value?
    • No, just the total value
  • Can we assume the inputs are valid?
    • No
  • Are the inputs in sorted order by val/weight?
    • Yes
  • Can we assume this fits memory?
    • Yes

Test Cases

  • items or total weight is None -> Exception
  • items or total weight is 0 -> 0
  • General case
total_weight = 8
items
  v | w
  0 | 0
a 1 | 1
b 3 | 2
c 7 | 4

max value = 14 

Algorithm

We'll use bottom up dynamic programming to build a table.

Taking what we learned with the 0/1 knapsack problem, we could solve the problem like the following:

v = value
w = weight

               j              
    -------------------------------------------------
    | v | w || 0 | 1 | 2 | 3 | 4 | 5 |  6 |  7 |  8  |
    -------------------------------------------------
    | 0 | 0 || 0 | 0 | 0 | 0 | 0 | 0 |  0 |  0 |  0  |
  a | 1 | 1 || 0 | 1 | 2 | 3 | 4 | 5 |  6 |  7 |  8  |
i b | 3 | 2 || 0 | 1 | 3 | 4 | 6 | 7 |  9 | 10 | 12  |
  c | 7 | 4 || 0 | 1 | 3 | 4 | 7 | 8 | 10 | 11 | 14  |
    -------------------------------------------------

i = row
j = col

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:

    -------------------------------------------------
    | v | w || 0 | 1 | 2 | 3 | 4 | 5 |  6 |  7 |  8  |
    -------------------------------------------------

    -------------------------------------------------
  a | 1 | 1 || 0 | 1 | 2 | 3 | 4 | 5 |  6 |  7 |  8  |
    -------------------------------------------------

    -------------------------------------------------
  b | 3 | 2 || 0 | 1 | 3 | 4 | 6 | 7 |  9 | 10 | 12  |
    -------------------------------------------------

    -------------------------------------------------
  c | 7 | 4 || 0 | 1 | 3 | 4 | 7 | 8 | 10 | 11 | 14  |
    -------------------------------------------------

if j >= items[i].weight:
    T[j] = max(items[i].value + T[j - items[i].weight],
               T[j])

Complexity:

  • Time: O(n * w), where n is the number of items and w is the total weight
  • Space: O(w), where w is the total weight

Code

Item Class

In [1]:
class Item(object):

    def __init__(self, label, value, weight):
        self.label = label
        self.value = value
        self.weight = weight

    def __repr__(self):
        return self.label + ' v:' + str(self.value) + ' w:' + str(self.weight)

Knapsack Bottom Up

In [2]:
class Knapsack(object):

    def fill_knapsack(self, items, total_weight):
        if items is None or total_weight is None:
            raise TypeError('items or total_weight cannot be None')
        if not items or total_weight == 0:
            return 0
        num_rows = len(items)
        num_cols = total_weight + 1
        T = [0] * (num_cols)
        for i in range(num_rows):
            for j in range(num_cols):
                if j >= items[i].weight:
                    T[j] = max(items[i].value + T[j - items[i].weight],
                               T[j])
        return T[-1]

Unit Test

In [3]:
%%writefile test_knapsack_unbounded.py
import unittest


class TestKnapsack(unittest.TestCase):

    def test_knapsack(self):
        knapsack = Knapsack()
        self.assertRaises(TypeError, knapsack.fill_knapsack, None, None)
        self.assertEqual(knapsack.fill_knapsack(0, 0), 0)
        items = []
        items.append(Item(label='a', value=1, weight=1))
        items.append(Item(label='b', value=3, weight=2))
        items.append(Item(label='c', value=7, weight=4))
        total_weight = 8
        expected_value = 14
        results = knapsack.fill_knapsack(items, total_weight)
        total_weight = 7
        expected_value = 11
        results = knapsack.fill_knapsack(items, total_weight)
        self.assertEqual(results, expected_value)
        print('Success: test_knapsack')

def main():
    test = TestKnapsack()
    test.test_knapsack()


if __name__ == '__main__':
    main()
Overwriting test_knapsack_unbounded.py
In [4]:
%run -i test_knapsack_unbounded.py
Success: test_knapsack