← Databricks Interview Insights
The two-pass structure tripped me up at first.
Clarify the aggregate function and tree properties, then outline a two-phase algorithm: first compute a topological order (e.g., via BFS) to process nodes bottom-up, then apply tree DP to combine children's results into each node's aggregate. Discuss complexity and edge cases, and be ready to code the solution.
Pro tip: Mention that BFS alone doesn't guarantee bottom-up order; you need to reverse the BFS order or use post-order DFS. Showing awareness of this subtlety demonstrates deep understanding.
Ask about the aggregate function (sum, min, max, count, etc.), tree size, and whether the tree is rooted or unrooted. Confirm that 'subtree' means all descendants of a node.
Decide on a traversal to get a bottom-up processing order. BFS gives level order; reverse it to process leaves first. Alternatively, use iterative post-order DFS to avoid recursion limits.
For each node, define the aggregate value based on its own value and the aggregates of its children. Specify how to combine children's results (e.g., sum, max) and handle base cases (leaves).
Code the solution using the chosen order, ensuring each node is processed after its children. Analyze time and space complexity, typically O(n) time and O(n) space.
Test on small trees, skewed trees, and single-node trees. Discuss handling of large inputs, recursion depth, and potential optimizations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by restating the base 1D tree DP and then introduce the extra dimension as an additional state parameter, explaining how it changes the DP definition and transitions. Derive the recurrence for a generic extra dimension (e.g., budget, parity, or ancestor usage), and analyze the time and space complexity in terms of the new state size. Emphasize the trade-offs and potential optimizations like state compression or pruning.
Pro tip: Mention that the extra dimension often turns the DP into a knapsack-like problem on trees, and that the complexity is typically O(N * K^2) where K is the dimension size, but can be optimized to O(N * K) with careful merging. Also, note that the choice of extra dimension affects whether the DP can be computed bottom-up or requires rerooting.
Briefly describe the original tree DP (e.g., maximum independent set) and define the extra state dimension (e.g., remaining budget, parity of path length, number of used ancestors). Explain what the DP state represents.
Specify dp[u][state][extra] and write the recurrence for combining children, showing how the extra dimension is updated (e.g., adding costs, toggling parity, incrementing usage). Highlight that transitions often involve convolution or knapsack merging.
Derive time and space complexity: O(N * S * K) or O(N * K^2) depending on merging. Discuss optimizations like limiting K to subtree size, using prefix sums, or applying small-to-large merging.
Address trade-offs between different extra dimensions (e.g., budget vs. parity) and how they affect implementation. Mention edge cases like K=0, negative values, or cycles if the tree is not rooted.
Summarize when this 3D DP is preferable over other approaches (e.g., greedy, flow) and how it scales for large N and K. Optionally, mention real-world applications like resource allocation in distributed systems.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.