Russian Doll Envelopes

To solve the Russian Doll Envelopes problem efficiently, I first analyze the constraints. Since Microsoft often deals with large datasets, I assume we need better than O(N^2) performance. My approach involves two main phases: transformation and optimization. First, I will sort the envelopes. The key insight is that an envelope can only fit inside another if both its width and height are strictly smaller. To simplify this, I sort the envelopes by width in ascending order. However, for envelopes with the same width, I must sort their heights in descending order. This is critical because it ensures that when we process the list, we never accidentally pick two envelopes with the same width for our subsequence, as the larger height appears before the smaller one, breaking any potential increasing sequence. Once sorted, the problem reduces entirely to finding the Longest Increasing Subsequence (LIS) based solely on the height values. The width condition is automatically satisfied by our sorting strategy. For the LIS calculation, instead of using the standard O(N^2) dynamic programming approach, I will implement the patience sorting method using binary search. This allows us to maintain a list of 'tails' for increasing subsequences of various lengths. By iterating through the heights and using binary search to find the insertion point, we can build the longest chain in O(N log N) time. Finally, I will verify the solution against edge cases like zero-width envelopes or duplicate dimensions to ensure robustness.