<p>You are given an integer 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>k</code>.</p>

<p>Find the maximum profit you can achieve. You may complete at most <code>k</code> transactions: i.e. you may buy at most <code>k</code> times and sell at most <code>k</code> times.</p>

<p><strong>Note:</strong> You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</p>

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

<pre>
<strong>Input:</strong> k = 2, prices = [2,4,1]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.
</pre>

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

<pre>
<strong>Input:</strong> k = 2, prices = [3,2,6,5,0,3]
<strong>Output:</strong> 7
<strong>Explanation:</strong> Buy on day 2 (price = 2) and sell on day 3 (price = 6), profit = 6-2 = 4. Then buy on day 5 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= 100</code></li>
	<li><code>1 &lt;= prices.length &lt;= 1000</code></li>
	<li><code>0 &lt;= prices[i] &lt;= 1000</code></li>
</ul>

class Solution:
    def maxProfit(self, k: int, prices: List[int]) -> int:
        
        @cache
        def trade(day_d, transaction_i):
            
            if day_d <= 0 or transaction_i <= 0:
                # Can not trade ealry than day #0
                # Can not trade with 0 transaction or less
                return 0
            
            
            ## Case_#1
            # not to transact on day d, all transactions are completed before
            not_to_transact = trade(day_d-1, transaction_i)
            
            
            ## Case_#2
            # transact on day_d
            
            # update price gain
            # define price_gain = max( -prices[buy_in] + trade(day_d-1, transaction_i-1) for buy_in in range(day_d) )
            
            if day_d == 1:
                trade.max_gain[transaction_i] = -prices[day_d-1] + trade(day_d-1, transaction_i-1)
            else:
                trade.max_gain[transaction_i] = max(trade.max_gain[transaction_i], -prices[day_d-1] + trade(day_d-1, transaction_i-1) )
                
            
            # transact on day_d, carry out i-th transaction now
            transact = prices[day_d] + trade.max_gain[transaction_i]

            
            return max(not_to_transact, transact)
        
        # ---------------------------------------------------
            
        trade.max_gain = defaultdict(int)    
        return trade(len(prices)-1, k)
48780