<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:
//9,6,5,1 greedy: 9,1,1 dp:6,5
//greedy fails as no unifromity or consistent difference
    int rec(vector<int> &coins, int amount, int i, int n, vector<vector<int>> &dp) {
        if(amount==0) return 0;
        if(amount<0) return INT_MAX;
        if(i==n-1) {
            if(amount%coins[i]==0) return amount/coins[i];
            return INT_MAX;
        }

        if(dp[i][amount]!=-1) return dp[i][amount];
        
        int notTake = rec(coins,amount,i+1,n,dp);
        int take = INT_MAX;
        if(coins[i]<=amount) {
            int d = rec(coins,amount-coins[i],i,n,dp);
            if(d!=INT_MAX) take = 1+d;
        }
        return dp[i][amount]=min(take,notTake);
    }

    int tab(vector<int> &coins, int amount, int n) {
        vector<vector<int>> dp(coins.size()+1, vector<int>(amount+1,0));
        
        for(int i=0;i<=amount;i++) {
            if(i%coins[n-1]==0) dp[n-1][i] = i/coins[n-1];
            else dp[n-1][i] = INT_MAX;
        }

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

        return dp[0][amount];
    }

    int spaceOpt(vector<int> &coins, int amount, int n) {
        vector<int> curr(amount+1,0);
        vector<int> next(amount+1,0);
        
        for(int i=0;i<=amount;i++) {
            if(i%coins[n-1]==0) next[i] = i/coins[n-1];
            else next[i] = INT_MAX;
        }

        for(int i=n-2;i>=0;i--) {
            for(int target = 1;target<=amount;target++) {
                int notTake = next[target];
                int take = INT_MAX;
                if(coins[i]<=target) {
                    int d = curr[target-coins[i]];
                    if(d!=INT_MAX) take = 1+d;
                }
                curr[target]=min(take,notTake);
            }
            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,coins.size());
        if(ans==INT_MAX) return -1;
        return ans;
    }
};
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