Unify Stock-Trading Profit Variants

Problem Statement

You are tasked with designing a unified stock-trading profit calculator. You are given an array prices where prices[i] is the price of a stock on the i-th day, an integer k representing the maximum number of transactions allowed, and a float fee representing a per-transaction cost (incurred at the time of sale).

Implement the following interface:

def max_profit(prices: list[int], k: int, fee: float) -> float:
    pass

Rules:

1. You may engage in at most k transactions (buy then sell).
2. You must sell the stock before you can buy again.
3. You cannot engage in multiple transactions simultaneously.
4. A transaction is defined as a buy followed by a sale; the fee is subtracted from the profit only upon the sale.

Constraints:

• 1 <= prices.length <= 10^4
• 0 <= prices[i] <= 10^5
• 0 <= k <= 10^9
• 0 <= fee <= 10^5

Examples:

Example 1:

• Input: prices = [1, 3, 2, 8, 4, 9], k = 2, fee = 2
• Output: 8.0
• Explanation: Buy at 1, sell at 8 (profit 7 - 2 fee = 5). Buy at 4, sell at 9 (profit 5 - 2 fee = 3). Total = 8.

Example 2:

• Input: prices = [1, 2], k = 10^9, fee = 0
• Output: 1.0
• Explanation: Unlimited transactions allowed, no fee.