I knew there was a DP angle here but spent too long trying to build a full table before realizing the greedy approach is cleaner.
Start by clarifying the definition of an alternating subsequence (e.g., differences alternate in sign) and whether the subsequence must be contiguous. Then, discuss a dynamic programming approach that tracks the longest alternating subsequence ending at each index with either a positive or negative difference, optimizing to O(n) time and O(1) space by observing that the length increases only at local extrema.
Pro tip: Mention that the problem reduces to counting the number of sign changes in the sequence of differences between consecutive elements after removing duplicates, which can be done in a single pass. This shows deep insight and can lead to a very clean solution.
Ask whether the subsequence must be contiguous and confirm the definition of alternating (e.g., differences alternate between positive and negative). Also, check if equal elements are allowed and how they affect alternation.
Explain a brute force approach (exponential) and then a dynamic programming approach that uses two arrays (up and down) to track the longest alternating subsequence ending at each index with the last difference positive or negative.
Show that the DP can be optimized by only keeping track of the last two values and the current lengths, or by counting sign changes in the differences between consecutive elements after removing duplicates.
Discuss edge cases such as empty array, single element, all equal elements, and strictly increasing/decreasing arrays. Explain how the algorithm handles them.
State the time and space complexity (O(n) time, O(1) space) and walk through a small example to verify correctness.
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