<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 {
public:
    int memo(vector<int> &coins, int amount, int i, vector<vector<int>> &dp) {
        if(amount==0) return 0;

        if(i==coins.size()-1) {
            if(amount%coins[i]==0) return amount/coins[i];
            return 1e9;
        }

        if(dp[i][amount]!=-1) return dp[i][amount];

        int notTake = memo(coins,amount,i+1,dp);
        int take = INT_MAX;
        if(coins[i]<=amount) take = 1 + memo(coins,amount-coins[i],i,dp);

        return dp[i][amount] = min(notTake,take);
    }

    int tab(vector<int>& coins, int amount) {
        vector<vector<int>> dp(coins.size()+1, vector<int> (amount+1,0)); 
        int n = coins.size();

        for(int i=1;i<=amount;i++) {
            if(i%coins[n-1]==0) dp[n-1][i] = i/coins[n-1];
            else dp[n-1][i] = 1e9;
        }

        for(int i = n-2;i>=0;i--) {
            for(int t = 1;t<=amount;t++) {
                int notTake = dp[i+1][t];
                int take = INT_MAX;
                if(coins[i]<=t && t-coins[i]>=0) take = 1 + dp[i][t-coins[i]];
                dp[i][t] = min(notTake,take);
            }
        }

        return dp[0][amount];
    }

    int spaceOpt(vector<int>& coins, int amount) {
        vector<int> curr(amount+1,0); 
        vector<int> next(amount+1,0); 
        int n = coins.size();

        for(int i=1;i<=amount;i++) {
            if(i%coins[n-1]==0) next[i] = i/coins[n-1];
            else next[i] = 1e9;
        }

        for(int i = n-2;i>=0;i--) {
            for(int t = 1;t<=amount;t++) {
                int notTake = next[t];
                int take = 1e9;
                if(coins[i]<=t && t-coins[i]>=0) take = 1 + curr[t-coins[i]];
                curr[t] = min(notTake,take);
            }
            next = curr;
        }

        return next[amount];
    }

    int coinChange(vector<int>& coins, int amount) {
        vector<vector<int>> dp(coins.size()+1, vector<int> (amount+1,-1));
        
        int ans = spaceOpt(coins,amount);
        if(ans==1e9) return -1;
        return ans;   
    }
};
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