<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 helper(vector<int>& prices,int index,bool hold, int fee,vector<vector<int>>&dp) {
        if(index==prices.size())
        return 0;
        if(dp[index][hold]!=-1)
        return dp[index][hold];
     if(!hold){
        if(dp[index][0]!=-1) 
        return dp[index][0];
        int buy=0,wait=0;
        buy=-prices[index]+helper(prices,index+1,true,fee,dp);
        wait=helper(prices,index+1,hold,fee,dp);
        dp[index][0]=max(buy,wait);
        return dp[index][0];

     }
     else{
        if(dp[index][1]!=-1)
        return dp[index][1];
        int sell,wait=0;
        sell=prices[index]-fee+helper(prices,index+1,false,fee,dp);
        wait=helper(prices,index+1,hold,fee,dp);
        dp[index][1]=max(sell,wait);
        return dp[index][1];
     }
    }
          int maxProfit(vector<int>&prices,int fee){
            vector<vector<int>>dp(prices.size(),vector<int>(2,-1));
            int ans=helper(prices,0,false,fee,dp);
            return ans;
          }
        
    
};
86684