I knew this was a DP problem pretty quickly but fumbled the base cases at first.
Start by clarifying the problem and edge cases, then propose a dynamic programming solution that builds up the maximum sum by considering each element either included or excluded. Optimize space by keeping only the last two DP values, and analyze time and space complexity.
Pro tip: Mention that this is a classic DP problem similar to 'House Robber' and that you can achieve O(n) time and O(1) space, showing you recognize patterns and optimize beyond the naive approach.
Confirm that the subsequence can be non-contiguous, elements are integers (possibly negative), and we want maximum sum. Discuss edge cases like empty array, single element, all negatives.
Acknowledge that brute force is exponential, then define DP state: dp[i] = max sum using first i elements. Derive recurrence: dp[i] = max(dp[i-1], dp[i-2] + arr[i-1]).
Observe that dp[i] only depends on dp[i-1] and dp[i-2], so use two variables to reduce space to O(1).
Write clean code, handle edge cases, and walk through a small example to verify correctness.
State time complexity O(n) and space complexity O(1). Discuss potential follow-ups like circular array or returning the subsequence.
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