<p>You are given an array <code>prices</code> where <code>prices[i]</code> is the price of a given stock on the <code>i<sup>th</sup></code> day, and an integer <code>fee</code> representing a transaction fee.</p>

<p>Find the maximum profit you can achieve. You may complete as many transactions as you like, but you need to pay the transaction fee for each transaction.</p>

<p><strong>Note:</strong></p>

<ul>
	<li>You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</li>
	<li>The transaction fee is only charged once for each stock purchase and sale.</li>
</ul>

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

<pre>
<strong>Input:</strong> prices = [1,3,2,8,4,9], fee = 2
<strong>Output:</strong> 8
<strong>Explanation:</strong> The maximum profit can be achieved by:
- Buying at prices[0] = 1
- Selling at prices[3] = 8
- Buying at prices[4] = 4
- Selling at prices[5] = 9
The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8.
</pre>

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

<pre>
<strong>Input:</strong> prices = [1,3,7,5,10,3], fee = 3
<strong>Output:</strong> 6
</pre>

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

<ul>
	<li><code>1 &lt;= prices.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= prices[i] &lt; 5 * 10<sup>4</sup></code></li>
	<li><code>0 &lt;= fee &lt; 5 * 10<sup>4</sup></code></li>
</ul>

class Solution {
public:

    int solve(int i, int flag, vector<int> &prices, int fee, vector<int> &dp, vector<int> &sell){
        if(i>=prices.size()) return 0;
        if(flag==-1&&dp[i]!=-1) return dp[i];
        if(flag!=-1&&sell[i]!=-1) return sell[i];
        if(flag==-1){
            int x=-prices[i]+solve(i+1,prices[i],prices,fee,dp,sell);
            int y=solve(i+1,flag,prices,fee,dp,sell);
            dp[i]=max(x,y); 
            return dp[i];
        }
        else{
            int x=prices[i]-fee+solve(i+1,-1,prices,fee,dp,sell);
            int y=solve(i+1,flag,prices,fee,dp,sell);
            sell[i]=max(x,y);
            return sell[i];
        }
    }

    int maxProfit(vector<int>& prices, int fee) {
        int n=prices.size();
        vector<int> dp(n,-1);
        vector<int> sell(n,-1);
        return solve(0,-1,prices,fee,dp,sell);
    }
};
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