Finished with time to spare and got a trivial follow-up that was basically one line added to my solution.
Start by restating the problem in your own words and clarifying constraints and edge cases. Then, work through small examples to identify patterns, derive a recurrence or closed-form formula, and validate it. Finally, discuss algorithmic implementation, complexity, and potential optimizations.
Pro tip: Verbalize your thought process and explicitly connect the combinatorics to algorithmic choices (e.g., dynamic programming vs. closed-form). This demonstrates both mathematical maturity and engineering pragmatism.
Ask clarifying questions to ensure you understand the problem's constraints, input/output format, and edge cases. Restate the problem in your own words to confirm.
Compute answers for small inputs manually or with simple code to identify patterns. Look for sequences, symmetries, or recurrences.
Based on patterns, derive a mathematical formula or recurrence relation. Consider combinatorial principles like inclusion-exclusion, generating functions, or dynamic programming.
Test the formula against small cases and edge cases. Discuss time/space complexity and possible optimizations (e.g., memoization, closed-form simplification).
Outline an algorithm to compute the answer efficiently. Mention potential pitfalls (e.g., overflow, modular arithmetic) and how to handle them.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The base problem was fine, solved it quickly.
First, clarify the tree's structural property and the core problem to solve. Then, design an algorithm to solve the problem, ensuring it leverages the property. Finally, construct a tree that satisfies the property, possibly using the algorithm's insights or a separate constructive method.
Pro tip: Always discuss time and space complexity trade-offs, and mention edge cases like empty trees or single nodes. Show that you can not only solve the problem but also generate valid test cases.
Ask clarifying questions to fully understand the structural property and the core problem. Confirm input/output formats and constraints.
Develop an algorithm to solve the core problem, explaining how it exploits the tree's property. Consider different approaches (e.g., DFS, DP, greedy) and choose the most efficient.
Analyze the time and space complexity of your algorithm. Discuss potential optimizations and trade-offs.
Devise a method to construct a tree that satisfies the given property. This could be done by generating a random tree and checking the property, or by a direct construction based on the property.
Walk through an example to demonstrate your algorithm and construction. Test edge cases and verify correctness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.