<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:
    def maxProfit(self, prices: List[int], fee: int) -> int:
        # # recursion
        # def recurse(i, hold):
        #     if i==len(prices):
        #         return 0
        #     if hold:
        #         sell = prices[i]-fee + recurse(i+1, False)
        #         nsell = recurse(i+1, True)
        #         return max(sell, nsell)
        #     else:
        #         buy = -prices[i] + recurse(i+1, True)
        #         nbuy = recurse(i+1, False)
        #         return max(buy, nbuy)
        # return recurse(0, False)

        # DP and memoization
        n = len(prices)
        dp = [[-1]*2 for _ in range(n)]
        def recurse(idx, hold):
            if idx==n:
                return 0
            if dp[idx][hold] == -1:
                if hold:
                    sell = prices[idx]-fee + recurse(idx+1, False)
                    nsell = recurse(idx+1, True)
                    dp[idx][hold] = max(sell, nsell)
                else:
                    buy = -prices[idx] + recurse(idx+1, True)
                    nbuy = recurse(idx+1, False)
                    dp[idx][hold] = max(buy, nbuy)
            return dp[idx][hold]
        return recurse(0, False)

        # DP and tabulation
        n = len(prices)
        dp = [[0]*2 for _ in range(n+1)]

        for idx in range(n-1, -1, -1):
            for hold in [True, False]:
                if hold:
                    sell = prices[idx]-fee + dp[idx+1][not hold]
                    nsesll = dp[idx+1][hold]
                    dp[idx][hold] = max(sell, nsell)
                else:
                    buy = -prices[idx] + dp[idx+1][not hold]
                    nbuy = dp[idx+1][hold]
                    dp[idx][hold] = max(buy, nbuy)
        return dp[0][0]

47330