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], n: 5 -> 5

## Algorithm¶

We'll use a bottom-up dynamic programming approach.

The rows (i) represent the coin values.
The columns (j) represent the totals.

-------------------------
| 0 | 1 | 2 | 3 | 4 | 5 |
-------------------------
0 | 1 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 2 | 2 | 3 | 3 |
3 | 1 | 1 | 2 | 3 | 4 | 5 |
-------------------------

Number of ways to get total n with coin[n] equals:
* Number of ways to get total n with coin[n - 1] plus
* Number of ways to get total n - coin[n]

if j == 0:
T[i][j] = 1
if row == 0:
T[i][j] = 0
if coins[i] >= j
T[i][j] = T[i - 1][j] + T[i][j - coins[i]]
else:
T[i][j] = T[i - 1][j]

The answer will be in the bottom right corner of the matrix.


Complexity:

• Time: O(i * j)
• Space: O(i * j)

## Code¶

In [1]:
class CoinChanger(object):

def make_change(self, coins, total):
if coins is None or total is None:
return None
if not coins or total == 0:
return 0
coins = [0] + coins
num_rows = len(coins)
num_cols = total + 1
T = [[None] * num_cols for _ in range(num_rows)]
for i in range(num_rows):
for j in range(num_cols):
if i == 0:
T[i][j] = 0
continue
if j == 0:
T[i][j] = 1
continue
if coins[i] <= j:
T[i][j] = T[i - 1][j] + T[i][j - coins[i]]
else:
T[i][j] = T[i - 1][j]
return T[num_rows - 1][num_cols - 1]


## Unit Test¶

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

class Challenge(unittest.TestCase):

def test_coin_change(self):
coin_changer = CoinChanger()
self.assertEqual(coin_changer.make_change([1, 2], 0), 0)
self.assertEqual(coin_changer.make_change([1, 2, 3], 5), 5)
self.assertEqual(coin_changer.make_change([1, 5, 25, 50], 10), 3)
print('Success: test_coin_change')

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

if __name__ == '__main__':
main()

Overwriting test_coin_change.py

In [3]:
%run -i test_coin_change.py

Success: test_coin_change