Canonical Coin Change

Greedy Algorithms Coin Change Algorithm

Problem: Canonical Coin Change

Given a list of coin denominations and a target amount, implement a greedy algorithm to determine the minimum number of coins required to make the target amount. In this problem, you can assume the coin denominations form a canonical coin system (for example, US coins: [25, 10, 5, 1]). If the target amount cannot be made using the given coins, your function should return -1.

Details:

  • You are provided an array of coin denominations (not necessarily sorted) and an integer amount.
  • Your task is to use a greedy approach: at each step, pick the largest coin denomination that does not exceed the remaining amount.
  • If you cannot exactly reach the target amount with the given coins, return -1.
  • Consider edge cases: when amount is 0 or when no coins can be used.

Example:

For coin denominations [1, 5, 10, 25] and target amount 63, the greedy algorithm would work as follows:

  1. Start with 63. The largest coin ≤ 63 is 25. Use it: 63 - 25 = 38.
  2. Again, the largest coin ≤ 38 is 25. Use it: 38 - 25 = 13.
  3. Largest coin ≤ 13 is 10. Use it: 13 - 10 = 3.
  4. Largest coin ≤ 3 is 1. Use it three times: 3 - 3*1 = 0.

Total coins used = 2 (25-cent) + 1 (10-cent) + 3 (1-cent) = 6.

Note:

While a greedy strategy is optimal for canonical coin systems (like US coins), it may not work for arbitrary coin denominations. For this problem, you can assume that the coin set will allow a greedy solution to be optimal.

Happy coding on this November day!