# Author: Huahua
# Time complexity: O(n)
# Space complexity: O(n)
class Solution:
  def maxProfit(self, prices: List[int]) -> int:
    
    @cache
    def dp(i: int) -> Tuple[int, int, int]:
      """
      Returns
      1) sold: Max profit w/o holdings. Can't buy. Can't sell.
      2) holding: Max profit w/ holdings. Can't buy, Can sell.
      3) cooldown: Max profit w/o holdings. Can buy. Can't sell."""
      if i < 0: return -10**9, -10**9, 0
      sold, holding, cooldown = dp(i - 1)
      
      return (holding + prices[i], # sold
              max(holding, cooldown - prices[i]), # do nothing, or buy.
              max(cooldown, sold)) # cooldown
    
    return max(dp(len(prices) - 1))

# Author: Huahua
# Time complexity: O(n)
# Space complexity: O(n)
class Solution:
  def maxProfit(self, prices: List[int], fee: int) -> int:
    
    @cache
    def dp(i: int) -> Tuple[int, int]:
      """
      Returns:
      1) sold: Max profit w/o holdings.
      2) holding: Max profit w/ holdings."""
      if i < 0: return 0, -10**9
      sold, holding = dp(i - 1)
      return (max(sold, holding + prices[i] - fee), # do nothing, sell
              max(holding, sold - prices[i]))       # do nothing, buy
    
    return dp(len(prices) - 1)[0]