Started fine on the base case but the follow-ups stacked fast.
Start by clearly defining the state and transition for the base stock trading problem, then generalize the DP recurrence to incorporate each new constraint (transaction fees, cooldown) as a modification to the state or transition. For each follow-up, explain how the added constraint changes the state space or transition, and adjust the DP accordingly, ensuring time and space complexity are analyzed.
Pro tip: Always discuss the trade-offs between different DP formulations (e.g., state definitions, space optimization) and proactively mention edge cases like multiple transactions, cooldown after selling, and fee impact on profitability. This shows you think like a production engineer, not just a coder.
Restate the problem: given stock prices, maximize profit with unlimited transactions (or limited k). Confirm constraints like can you buy and sell on the same day, and whether short selling is allowed.
Define DP[i][j] where i is day and j is holding state (0: no stock, 1: holding stock). Write transitions: dp[i][0] = max(dp[i-1][0], dp[i-1][1] + price[i]); dp[i][1] = max(dp[i-1][1], dp[i-1][0] - price[i]).
Modify the sell transition to subtract the fee: dp[i][0] = max(dp[i-1][0], dp[i-1][1] + price[i] - fee). Explain that this discourages frequent trading and may change optimal strategy.
Introduce a cooldown state or adjust transitions: after selling, you cannot buy for one day. Use dp[i][0] = max(dp[i-1][0], dp[i-1][1] + price[i]); dp[i][1] = max(dp[i-1][1], dp[i-2][0] - price[i]) to enforce cooldown.
State time O(n) and space O(n) or O(1) with rolling variables. Discuss how each constraint affects complexity and whether further optimizations are possible.
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