Databricks·Software Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

Apr 2026

Summary

One round at Databricks for a software engineering role, and the interviewer threw something I'd genuinely never seen before. Not your typical graph problem.

Questions Asked (1)

Given a list of node groups (e.g. [[1],[2,3],[4,5,6]]), connect all groups into a single connected graph using the minimum number of edges (k groups need k-1 edges). Each edge must be formed by picking one node from each of two different groups. The output must be uniformly random across all valid spanning structures.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The minimum edges part clicked for me pretty fast, basically just a spanning tree over groups.

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AI HintsAI Generated

Suggested Approach

First, clarify that the problem reduces to building a random spanning tree on the group-level complete graph, then selecting a random node from each group for each edge. Use a randomized algorithm like random Kruskal or random Prüfer code to ensure uniform distribution over all possible spanning trees, and handle node selection independently.

Pro tip: Mention that uniformity requires careful handling: if you simply pick random edges, you might bias toward certain trees. Using a known uniform spanning tree algorithm (e.g., random Kruskal with random edge weights) guarantees uniformity. Also, note that the node choices are independent of the tree structure, so you can sample them separately.

1. Clarify the problem and constraints

Confirm that the goal is to connect k groups with exactly k-1 edges, each edge between two different groups, and that the output must be uniformly random among all valid spanning structures (i.e., all possible trees on the groups, with all possible node selections).

2. Reduce to spanning tree on groups

Observe that the group-level connectivity forms a spanning tree on k nodes. The number of possible trees is k^(k-2) by Cayley's formula, and each tree can be realized with product of group sizes choices for node endpoints.

3. Choose a uniform spanning tree algorithm

Select an algorithm that generates a uniformly random spanning tree on the complete graph of k groups, such as random Kruskal (assign random weights to all possible group-pair edges and run Kruskal) or random Prüfer code.

4. Sample nodes for each edge

For each edge in the spanning tree, independently pick a random node from each of the two groups involved. This ensures uniform selection among all possible node pairs for that edge.

5. Verify uniformity and complexity

Argue that the combination of a uniform spanning tree and independent uniform node choices yields a uniform distribution over all valid spanning structures. Discuss time complexity: O(k^2) for random Kruskal or O(k) for Prüfer, plus O(k) for node selection.

Key Points to Mention

Cayley's formula: number of spanning trees on k labeled nodes is k^(k-2).
Uniform spanning tree algorithms: random Kruskal (assign i.i.d. random weights to edges) and random Prüfer code.
Independence of node selection: once the tree structure is fixed, each edge's node choices are independent and uniform.
Total number of valid structures: k^(k-2) * product over edges of (size of group A * size of group B).
Potential pitfalls: naive random edge selection may not yield uniform distribution; need to ensure each tree is equally likely.
Complexity considerations: random Kruskal on complete graph of k groups takes O(k^2 log k) if sorting, but can be optimized; Prüfer is O(k).

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.