3573. Best Time to Buy and Sell Stock V

3573. Best Time to Buy and Sell Stock V

Difficulty: Medium

Topics: Array, Dynamic Programming, Biweekly Contest 158

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: We can make $14 of profit through 2 transactions:
  • 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: We can make $36 of profit through 3 transactions:
  • 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

Hint:

1. Use dynamic programming.
2. Keep the following states: idx, transactionsDone, transactionType, isTransactionRunning.
3. Transactions transition from completed -> running and from running -> completed.

Solution:

We need to maximize profit with at most k transactions, where each transaction can be either:

• Normal: Buy then sell (profit = prices[j] - prices[i])
• Short: Sell then buy (profit = prices[i] - prices[j])

Key constraints:

• No overlapping transactions
• Must complete one transaction before starting another
• Can't perform another trade on the same day as a previous transaction's completion
• k ≤ prices.length / 2

Dynamic Programming Approach

We can use DP with state variables:

• i: current day index
• t: number of transactions completed
• state: 0 (not holding any position), 1 (holding a long position after buy), 2 (holding a short position after sell)

State Transitions

1. State 0 (no position):

   • Do nothing: dp[i][t][0] → dp[i+1][t][0]
   • Start long (buy): dp[i][t][0] - prices[i] → dp[i+1][t][1]
   • Start short (sell): dp[i][t][0] + prices[i] → dp[i+1][t][2]

2. State 1 (holding long):

   • Do nothing: dp[i][t][1] → dp[i+1][t][1]
   • Sell (complete normal transaction): dp[i][t][1] + prices[i] → dp[i+1][t+1][0]

3. State 2 (holding short):

   • Do nothing: dp[i][t][2] → dp[i+1][t][2]
   • Buy back (complete short transaction): dp[i][t][2] - prices[i] → dp[i+1][t+1][0]

Let's implement this solution in PHP: 3573. Best Time to Buy and Sell Stock V

<?php
/**
 * @param Integer[] $prices
 * @param Integer $k
 * @return Integer
 */
function maximumProfit($prices, $k) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo maximumProfit([[1,7,9,8,2], 2) . "\n";                 // Output: 14
echo maximumProfit([12,16,19,19,8,1,19,13,9], 3) . "\n";    // Output: 36
?>

Explanation:

1. Initialization: We start with 0 profit, 0 transactions, and no position.
2. State Transitions: At each day, we consider all possible actions from each state.
3. State 0 (no position): We can start a new transaction (long or short) or do nothing.
4. State 1 (holding long): We can either continue holding or sell to complete the transaction.
5. State 2 (holding short): We can either continue holding or buy back to complete the transaction.
6. Transaction Limit: We only increment the transaction count when completing a transaction, and we can't exceed k.
7. Final Answer: The maximum profit after n days with any number of transactions ≤ k and no open positions.

Complexity

• Time: O(n × k)
• Space: O(n × k) or optimized to O(k)

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