<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>amount</code> representing a total amount of money.</p>

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

<p>You may assume that you have an infinite number of each kind of coin.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> coins = [1,2,5], amount = 11
<strong>Output:</strong> 3
<strong>Explanation:</strong> 11 = 5 + 5 + 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> coins = [2], amount = 3
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> coins = [1], amount = 0
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= coins.length &lt;= 12</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>0 &lt;= amount &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    def coinChange(self, coins: List[int], amount: int) -> int:

        # top-down approach
        # min_coins = 0

        # @cache
        # def coinChangeHelper(curr_amount):

        #     if curr_amount == amount:
        #         return 0
            
        #     if curr_amount > amount:
        #         return -1
                
        #     min_coins = float('inf')

        #     for coin in coins:
        #         c1 = coinChangeHelper(curr_amount + coin)
        #         if c1 != -1:
        #             min_coins = min(c1 + 1, min_coins)
            
        #     return min_coins

        # res = coinChangeHelper(0)
        # if res < float('inf'):
        #     return res
        
        # return -1


        # bottom up solution
        dp = defaultdict(int)
        dp[0] = 0
        for cost in range(1, amount + 1):
            min_cost = float('inf')
            for coin in coins:
                if cost - coin in dp:
                    min_cost = min(1 + dp[cost - coin], min_cost)
            dp[cost] = min_cost

        print(dp)
        return dp[amount] if dp[amount] < float('inf') else -1
        






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