<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 recursiveLogic(vector<int>& prices, int bought, int currentIndex, int fee, vector<vector<int>>& dp) {
        if (currentIndex >= prices.size()) {
            return 0;
        }

        if (dp[currentIndex][bought] != -1) {
            return dp[currentIndex][bought];
        }


        int option1 = 0;
        int option2 = 0;
        if (!bought) {
            // finding index whose next is greater than itself
            while (currentIndex + 1 < prices.size() && prices[currentIndex] >= prices[currentIndex + 1]) {
                currentIndex++;
            }
            // buying
            option1 = -prices[currentIndex] + recursiveLogic(prices, 1, currentIndex + 1, fee, dp);

            // not selling
            option2 = 0 + recursiveLogic(prices, 0, currentIndex + 1, fee, dp);
        }
        else {
            // selling
            option1 = prices[currentIndex] - fee + recursiveLogic(prices, 0, currentIndex + 1, fee, dp);
            
            // not selling
            option2 = 0 + recursiveLogic(prices, 1, currentIndex + 1, fee, dp);
        }

        return dp[currentIndex][bought] = max(option1, option2);
    }
    int maxProfit(vector<int>& prices, int fee) {
        int start = 0;
        while (start + 1 < prices.size() && prices[start] >= prices[start + 1]) {
            start++;
        }

        vector<vector<int>> dp(prices.size() + 1, vector<int>(2, -1));

        int answer = recursiveLogic(prices, 0, start, fee, dp);

        return answer;
    }
};
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