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

# Solution Notebook¶

## Constraints¶

• Do the coins have to reach exactly n cents?
• Yes
• Can we assume we have an infinite number of coins to make n cents?
• Yes
• Do we need to report the combination(s) of coins that represent the minimum?
• No
• Can we assume the coin denominations are given in sorted order?
• No
• Can we assume this fits memory?
• Yes

## Test Cases¶

• coins: None or n: None -> Exception
• coins: [] or n: 0 -> 0
• coins: [1, 2, 3] or [3, 2, 1], n: 5 -> 2

## Algorithm¶

We'll use top down dynamic programming with memoization.

• Base case: If the total is 0, return 0
• If the total is already in the memo, return it
• For each coin denomination:
• If this coin > total, continue
• Recurse, decreasing total by the coin denomination, keeping track of the min return
• Set memo[total] to the min value + 1
• Return the memo[total]
total: 5
coins: [1,2,3]
memo key: total value: min ways
memo = {
1: 1,
2: 1,
3: 1,
4: 2,
5: 2
}
5
1, 2, 3
/
4
1, 2, 3
/
3
1,    2,    3
/       \     \____
2         1         0
1, 2, 3   1, 2, 3
/   |
1    0
1, 2, 3
/
0


Complexity:

• Time: O(t * n), where t is the total and n is the number of coin denominations
• Space: O(t) for the recursion depth

## Code¶

In [1]:
import sys

class CoinChanger(object):

def make_change(self, coins, total):
if coins is None or total is None:
raise TypeError('coins or total cannot be None')
if not coins or total == 0:
return 0
cache = {}
return self._make_change(coins, total, cache)

def _make_change(self, coins, total, cache):
if total == 0:
return 0
if total in cache:
return cache[total]
min_ways = sys.maxsize
for coin in coins:
if total - coin < 0:
continue
ways = self._make_change(coins, total - coin, cache)
if ways < min_ways:
min_ways = ways
cache[total] = min_ways + 1
return cache[total]


## Unit Test¶

In [2]:
%%writefile test_coin_change_min.py
import unittest

class TestCoinChange(unittest.TestCase):

def test_coin_change(self):
coin_changer = CoinChanger()
self.assertRaises(TypeError, coin_changer.make_change, None, None)
self.assertEqual(coin_changer.make_change([], 0), 0)
self.assertEqual(coin_changer.make_change([1, 2, 3], 5), 2)
self.assertEqual(coin_changer.make_change([3, 2, 1], 5), 2)
self.assertEqual(coin_changer.make_change([3, 2, 1], 8), 3)
print('Success: test_coin_change')

def main():
test = TestCoinChange()
test.test_coin_change()

if __name__ == '__main__':
main()

Overwriting test_coin_change_min.py

In [3]:
%run -i test_coin_change_min.py

Success: test_coin_change