# Author: Huahua
# Time complexity: O(n)
# Space complexity: O(n)
class Solution:
  def maxScoreSightseeingPair(self, values: List[int]) -> int:    
    # score = (values[i] + i) + (values[j] - j)
    @cache
    def dp(j: int) -> Tuple[int, int]:
      """Returns:
        1) Max score of values[0..j]
        2) Largest values[i] + i (i <= j)."""
      if j < 0: return 0, 0
      s, v = dp(j - 1)
      return max(s, v + values[j] - j), max(v, values[j] + j)
    
    return dp(len(values) - 1)[0]

# 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]:
      """Returns:
        1) max profit of prices[0:i].
        2) min price of prices[0:i]."""
      if i < 0: return 0, 10**9
      max_profit, min_price = dp(i - 1)
      return max(max_profit, prices[i] - min_price), min(min_price, prices[i])

    return dp(len(prices) - 1)[0]

# 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]:
      """Returns:
      1) Max profit after i-th day. Holding no stocks. Can buy.
      2) Max balance after i-th day. Holding a stock. Can sell.
      """
      if i < 0: return 0, -10**9
      profit, balance = dp(i - 1)
      
      return (max(profit, balance + prices[i]), # do nothing, sell
              max(balance, profit - prices[i]))  # do nothing, buy
    
    return dp(len(prices) - 1)[0]