Clarify the problem constraints and edge cases, then propose a dynamic programming solution with a constant number of states representing the best profit at each day for different transaction phases. Walk through the state transitions and explain how the O(n) time and O(1) space complexity is achieved.
Pro tip: Emphasize that the constant space comes from using a fixed number of variables (not an array) to track states, and mention that the cooldown and fee are naturally incorporated into the state transitions.
Restate the problem to ensure understanding: at most two transactions, fee on sell, one-day cooldown after sell. Ask about edge cases like empty array or single day.
Identify the necessary states: for each transaction, track the best profit when holding a stock (after buy) and when not holding (after sell), considering cooldown. With two transactions, we need states for first buy, first sell, second buy, second sell.
Write recurrence relations for each state: e.g., first buy = max(previous first buy, -price); first sell = max(previous first sell, first buy + price - fee); second buy = max(previous second buy, first sell - price); second sell = max(previous second sell, second buy + price - fee). Incorporate cooldown by ensuring sells happen at least one day after buys.
Initialize variables to represent the states (e.g., -infinity for buys, 0 for sells). Iterate through prices, updating each state in order. Use only a constant number of variables to achieve O(1) space.
Walk through a small example to verify correctness. Discuss time complexity O(n) and space O(1). Mention potential pitfalls like cooldown handling and fee deduction.
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