best-time-to-buy-and-sell-stock-v

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

You are allowed to make at most k transactions, where each transaction can be either of the following:

• 
  

  Normal transaction: Buy on day i, then sell on a later day j where i < j. You profit prices[j] - prices[i].

  
  


• 
  

  Short selling transaction: Sell on day i, then buy back on a later day j where i < j. You profit prices[i] - prices[j].

  
  

Note that you must complete each transaction before starting another. Additionally, you can't buy or sell on the same day you are selling or buying back as part of a previous transaction.

Return the maximum total profit you can earn by making at most k transactions.

Example 1:

Input: prices = [1,7,9,8,2], k = 2

Output: 14

Explanation:

• A normal transaction: buy the stock on day 0 for $1 then sell it on day 2 for $9.


• A short selling transaction: sell the stock on day 3 for $8 then buy back on day 4 for $2.

Example 2:

Input: prices = [12,16,19,19,8,1,19,13,9], k = 3

Output: 36

Explanation:

• A normal transaction: buy the stock on day 0 for $12 then sell it on day 2 for $19.


• A short selling transaction: sell the stock on day 3 for $19 then buy back on day 4 for $8.


• A normal transaction: buy the stock on day 5 for $1 then sell it on day 6 for $19.

Constraints:

• 2 <= prices.length <= 103


• 1 <= prices[i] <= 109


• 1 <= k <= prices.length / 2