Databricks·Software Engineer·Technical Phone Screen·Senior

Senior

May 2026

Summary

Databricks technical screen that went deep into graph theory territory pretty fast. The core problem sounds approachable until you realize the uniformity constraint is basically the whole interview.

Questions Asked (1)

You have n disconnected graph components. Write a function that returns a random set of edges that, when added, merges all components into one connected graph. The catch: the edge set must be sampled uniformly at random from all valid solutions. How do you ensure uniform sampling, and why does a naive approach fail?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I got the basic structure pretty quickly: you need a spanning tree over the components, then pick one endpoint per component per edge.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

First, clarify that the problem is equivalent to sampling a spanning tree of the component graph uniformly at random. Then, explain that a naive approach like randomly connecting components one by one fails because it biases toward certain tree shapes. Finally, describe an algorithm like Wilson's algorithm or a random contraction method to achieve uniform sampling, and justify its correctness.

Pro tip: Mention that the number of valid edge sets is exactly n^{n-2} (Cayley's formula) when each component is treated as a node, and that uniform sampling requires a method like Wilson's algorithm or loop-erased random walks. This shows deep understanding and avoids common pitfalls.

1. Clarify the problem

Restate the problem: we have n disconnected components and need to add edges to connect them into one graph. The added edges must form a spanning tree on the components, and we need to sample uniformly from all such spanning trees.

2. Identify the naive approach and its flaw

A naive approach might repeatedly pick two random components and connect them until all are connected. This fails because it does not sample uniformly; it biases toward trees with certain structures (e.g., star-like trees are less likely than path-like trees).

3. Explain the uniform sampling requirement

Uniform sampling means each possible spanning tree on the n components must have equal probability. The total number of such trees is n^{n-2} by Cayley's formula, so each must have probability 1/n^{n-2}.

4. Present a correct algorithm

Use Wilson's algorithm: start with an empty tree, pick a root, and for each other component, perform a loop-erased random walk from it to the current tree. This yields a uniform spanning tree. Alternatively, use the Aldous-Broder algorithm or random contraction (Karger's algorithm variant).

5. Discuss implementation and complexity

Explain how to implement Wilson's algorithm efficiently, noting that it runs in expected polynomial time. Mention that the edges added are between components, so we need to map component IDs to actual vertices.

Key Points to Mention

The problem reduces to sampling a uniform spanning tree on the component graph.
Naive sequential connection biases the distribution; not all trees are equally likely.
Cayley's formula: there are n^{n-2} labeled trees on n nodes.
Wilson's algorithm uses loop-erased random walks to achieve uniform sampling.
Alternative: Aldous-Broder algorithm or random contraction methods.
Uniform sampling ensures each valid edge set has equal probability, which is crucial for correctness in randomized algorithms.

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