← J.P. Morgan Interview Insights
I started with the recursive approach and got it working pretty fast, but then they asked me to also write the bottom-up version on the spot and I fumbled the table initialization for a bit.
Start by defining the problem as finding the longest palindromic subsequence (LPS) using dynamic programming. Explain the recurrence: if the first and last characters match, LPS length is 2 + LPS of the inner substring; otherwise, it's the maximum of LPS excluding either end. Then implement top-down with memoization and bottom-up with a table, analyze complexities, and discuss space optimization.
Pro tip: In interviews, always clarify that a subsequence doesn't require contiguous characters, unlike a substring. This shows attention to detail and prevents misunderstandings.
Confirm that the problem asks for the longest palindromic subsequence (not substring) and that characters can be skipped. Define the DP state: dp[i][j] = length of LPS in substring s[i..j].
Write the recurrence: if s[i] == s[j], dp[i][j] = 2 + dp[i+1][j-1] (with base case for i==j being 1); else dp[i][j] = max(dp[i+1][j], dp[i][j-1]).
Implement a recursive function with memoization (e.g., using a 2D array initialized to -1) to avoid recomputing overlapping subproblems. Handle base cases: i > j returns 0, i == j returns 1.
Create a 2D table of size n x n, fill diagonals for substrings of increasing length. Initialize dp[i][i] = 1, then iterate lengths from 2 to n, filling dp[i][j] using the recurrence.
State time complexity O(n^2) and space complexity O(n^2) for both approaches. Explain that space can be reduced to O(n) by observing that each row depends only on the previous row and the current row's left value, so we can use two 1D arrays or even one array with careful updating.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Explain that you can reconstruct the LPS by backtracking through the DP table used to compute the length, or by storing parent pointers during the DP. Then walk through the backtracking process step by step, showing how to build the subsequence from both ends.
Pro tip: Mention that storing parent pointers during DP computation can simplify reconstruction and avoid re-deriving choices, but it increases space complexity. Alternatively, backtracking from the DP table is more space-efficient and demonstrates deeper understanding.
Recall that dp[i][j] stores the length of the longest palindromic subsequence in substring s[i..j]. The recurrence is: if s[i] == s[j], dp[i][j] = dp[i+1][j-1] + 2; else dp[i][j] = max(dp[i+1][j], dp[i][j-1]).
Start with i = 0, j = n-1, and an empty list or deque to store characters. We'll build the subsequence from the outside in.
While i <= j: if s[i] == s[j], add s[i] to the front and back of the result (or store for later), then i++, j--. Else if dp[i+1][j] >= dp[i][j-1], move i++ (skip s[i]); otherwise move j-- (skip s[j]).
If i == j after the loop, add s[i] to the middle of the result. If i > j, no middle character is needed.
Combine the left and right parts (and middle if any) to form the final longest palindromic subsequence string.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Acknowledge the current solution's limitations for scale, then systematically propose optimizations across algorithm, data structures, and system design. Emphasize trade-offs between time, space, and complexity, and validate improvements with testing and metrics.
Pro tip: Quantify the impact of each change (e.g., 'reduces time complexity from O(n^2) to O(n log n)') and mention how you'd validate with benchmarks and edge cases. This shows you think like a production engineer, not just a coder.
Analyze the current approach to find performance bottlenecks, such as high time complexity, redundant computations, or memory issues. Consider profiling or theoretical analysis.
Suggest better algorithms or data structures (e.g., hash maps, heaps, dynamic programming) to reduce complexity. Explain how they address the bottlenecks.
Discuss parallelization, caching, batching, or distributed processing if applicable. Mention how these scale with many test cases.
Compare the proposed changes in terms of time, space, readability, and maintainability. Acknowledge any new complexities introduced.
Describe how you would test the optimized solution with large inputs, measure performance, and ensure correctness. Mention continuous improvement.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.