The complexity constraint is what made this annoying.
Clarify the problem constraints and edge cases, then explain the DP state and recurrence before coding. Implement the bottom-up DP with O(K·N) time and O(N) space, and test with a small example.
Pro tip: Emphasize that you can optimize space by using two 1D arrays (for buy and sell states) and that the O(K·N) time is optimal for this problem. Also, mention that you can reduce K to min(K, N//2) to avoid unnecessary work.
Ask clarifying questions: Can you buy and sell on the same day? Is K large? What if K >= N/2? Define the DP state clearly: dp[k][i] = max profit with at most k transactions up to day i.
Derive the recurrence: dp[k][i] = max(dp[k][i-1], max_{j<i}(prices[i] - prices[j] + dp[k-1][j])). Explain how to optimize the inner max by maintaining a running maximum.
Show how to reduce space from O(K·N) to O(N) by using two 1D arrays (prev and curr) or by updating in place with careful ordering. Mention that we can also use two arrays for buy and sell states.
Write clean Python code with comments. Test with edge cases: empty array, K=0, K >= N/2, increasing/decreasing prices. Walk through a small example to verify correctness.
State time complexity O(K·N) and space O(N). Discuss trade-offs: if K is large, we can use the greedy approach for unlimited transactions (O(N) time).
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