Clarify the problem constraints and then present a dynamic programming solution that builds up the maximum sum for each house by considering whether to include it or skip it. Explain the recurrence relation and optimize space to O(1) by keeping only the last two values.
Pro tip: Mention that this is a classic DP problem and that the optimal substructure allows for a linear time solution; also note that the same pattern applies to similar problems like 'House Robber' on LeetCode, showing pattern recognition.
Ask clarifying questions: Are all numbers non-negative? Can the array be empty? Is the input mutable? Confirm that the goal is to maximize the sum without picking adjacent elements.
Define dp[i] as the maximum sum considering houses 0..i. The recurrence is dp[i] = max(dp[i-1], dp[i-2] + nums[i]), with base cases dp[0] = nums[0] and dp[1] = max(nums[0], nums[1]).
Observe that dp[i] only depends on the previous two values, so we can reduce space complexity to O(1) by keeping two variables (prev2 and prev1) and updating them iteratively.
Write clean code implementing the O(1) space solution. Walk through a small example (e.g., [2,7,9,3,1]) to verify correctness and handle edge cases like empty array or single element.
State that the time complexity is O(n) and space complexity is O(1). Mention that this is optimal since we must examine each house at least once.
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