First, clarify the problem constraints and edge cases, then propose a dynamic programming solution that sorts envelopes by width ascending and height descending, followed by finding the longest increasing subsequence (LIS) on heights. Optimize the LIS step using binary search to achieve O(n log n) time complexity.
Pro tip: Mention that sorting height in descending order for equal widths prevents invalid chains where two envelopes have the same width. Also, discuss how this problem is a variation of the classic Russian Doll Envelopes problem, showing pattern recognition.
Ask about input size, whether envelopes can be rotated, and if dimensions are integers. Confirm that both width and height must be strictly smaller.
Sort by width ascending; for equal widths, sort by height descending. This ensures that when we process envelopes, we only consider strictly increasing heights for valid chains.
Extract the heights in sorted order and find the longest strictly increasing subsequence. This gives the maximum chain length.
Use binary search (patience sorting) to compute LIS in O(n log n) time, explaining that a naive DP would be O(n^2).
State that sorting takes O(n log n) and LIS takes O(n log n), so overall O(n log n) time and O(n) space.
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