NURO·Software Engineer·Technical Phone Screen·Senior

Senior

Apr 2026

Summary

Nuro coding round, one algorithmic problem about clustering points by proximity. The follow-up on optimizing for large sparse inputs was where things got interesting.

Questions Asked (1)

Given N points in 2D space and a distance threshold k, group all points such that any two points within distance k of each other (transitively) belong to the same cluster. How would you approach this, and how would you optimize it when N is very large but the points are spread out sparsely?

Algorithms & Data StructuresSystem DesignTechnical Trade-offs

Author's notes

The union-find part I got pretty quickly, just iterate over all pairs, check distance, union if below k.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem: this is connected components in a graph where edges exist between points within distance k. For small N, use union-find with O(N^2) pairwise checks; for large sparse N, use spatial partitioning (e.g., grid or k-d tree) to avoid unnecessary comparisons, then apply union-find on candidate pairs.

Pro tip: Mention that the choice of spatial index depends on data distribution and dimensionality; for sparse 2D, a uniform grid with cell size k is often simpler and faster than a k-d tree. Also, discuss how to handle edge cases like duplicate points and points exactly at distance k.

1. Clarify requirements and constraints

Confirm the distance metric (Euclidean?), whether k is inclusive, and the expected scale of N. Ask about memory and time constraints, and whether points are static or dynamic.

2. Model as graph connectivity

Explain that the problem reduces to finding connected components in a graph where vertices are points and edges connect points within distance k. This allows using union-find (disjoint set) for efficient merging.

3. Naive approach and its complexity

Describe the O(N^2) pairwise comparison approach: for each pair, check distance and union if within k. This is simple but infeasible for large N.

4. Optimize with spatial partitioning

For large sparse data, use a spatial index like a grid with cell size k (or k-d tree) to only compare points in neighboring cells. This reduces comparisons to near-linear time for sparse data.

5. Implement and analyze trade-offs

Outline the algorithm: build spatial index, for each point query neighbors within k, union them. Discuss time/space complexity and trade-offs between grid, k-d tree, and other methods (e.g., sweep line).

Key Points to Mention

Union-Find (Disjoint Set Union) with path compression and union by rank for near-constant time operations.
Spatial indexing: uniform grid with cell size k ensures that points within distance k are in the same or adjacent cells.
Time complexity: naive O(N^2), optimized O(N + M) where M is number of pairs within k, which is near-linear for sparse data.
Handling sparse data: grid works well when points are uniformly distributed; for clustered data, consider adaptive grids or k-d trees.
Edge cases: points exactly at distance k (inclusive), duplicate points, and points on cell boundaries.
Alternative approaches: DBSCAN clustering (which uses similar concepts) or using a sweep line with balanced BST for 1D, but 2D requires more complex structures.

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