322. coin change

You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money.

Return the fewest number of coins that you need to make up that amount. If that amount of money cannot be made up by any combination of the coins, return -1.

You may assume that you have an infinite number of each kind of coin.

solution

Use dynamic programming to keep track of the minimum number of coins required to make each amount from 0 to amount.

Let $d_i$ be the value of the dp array at i. Let $C$ be the set of coin values, and $c_i$ be the value of the coin at index $i$ in the coins array. Then, our recursive relation is as follows:

$$M_i=\{\begin{array}{l}1\text{ if }i\in C,\\ \min (d_{i-c_i})\forall c_i\in C\text{ otherwise}\end{array}$$

def coinChange(self, coins: List[int], amount: int) -> int:
	dp = [0]*(amount+1)
	
	for i in range(1, amount+1):
		m = float("inf")
		for coin in coins:
			if i-coin >= 0 and dp[i-coin] >= 0:
				m = min(m, dp[i-coin]+1)
			dp[i] = m
	
	return dp[-1] if dp[-1] != float("inf") else -1