The base case I had no problem with, single pass tracking the running minimum and updating max profit as you go.
Start by clarifying the problem constraints and confirming edge cases, then present the optimal O(n) time, O(1) space solution for the single transaction case using a running minimum. For the follow-ups, build up from the single transaction to unlimited transactions (sum of positive differences) and finally to at most k transactions using dynamic programming, explaining the state transitions and space optimization.
Pro tip: Explicitly discuss the trade-offs between the DP approaches for k transactions (e.g., O(kn) time vs. O(n) time when k >= n/2) and mention that the unlimited case is a special case of k >= n/2. This shows you understand both the problem and practical optimization.
Ask if the array can be empty, if prices are integers, and if multiple transactions are allowed in the follow-ups. Confirm that you cannot sell before buying and that you want to maximize profit.
Explain the one-pass algorithm: track the minimum price seen so far and compute the maximum profit as the difference between the current price and that minimum. This gives O(n) time and O(1) space.
For unlimited transactions, the maximum profit is the sum of all positive differences between consecutive days. This is because you can capture every upward movement.
Use dynamic programming with states: dp[i][j] = maximum profit up to day i with at most j transactions. Optimize to O(kn) time and O(k) space by iterating over transactions and days, or use the O(n) approach when k >= n/2.
Discuss time and space complexities for each solution, and explain why the DP is necessary for general k. Mention that the unlimited case is a special case of k >= n/2 and can be solved in O(n) time.
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