Hard

You are given an integer array prices where prices[i] is the price of a given stock on the ith day, and an integer k.

Find the maximum profit you can achieve. You may complete at most k transactions.

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

Example 1:

Input: k = 2, prices = [2,4,1]

Output: 2

Explanation: Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.

Example 2:

Input: k = 2, prices = [3,2,6,5,0,3]

Output: 7

Explanation: 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.

Constraints:

• 0 <= k <= 100
• 0 <= prices.length <= 1000
• 0 <= prices[i] <= 1000

from typing import List

class Solution:
    def maxProfit(self, k: int, prices: List[int]) -> int:
        n = len(prices)
        dp = [0] * (k + 1)
        maxdp = [float('-inf')] * (k + 1)
        
        for i in range(1, n + 1):
            maxdp[0] = max(maxdp[0], dp[0] - prices[i - 1])
            for j in range(k, 0, -1):
                maxdp[j] = max(maxdp[j], dp[j] - prices[i - 1])
                dp[j] = max(maxdp[j - 1] + prices[i - 1], dp[j])
        
        return dp[k]