Meta·Software Engineer·Technical Phone Screen·Senior

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Apr 2026

Summary

Meta Research Scientist interview with a pretty meaty algorithms question about sparse data structures. One question but it had a lot of layers to it, and the follow-up on matrices caught me a bit flat-footed.

Questions Asked (1)

Given two sparse vectors (and as a follow-up, two sparse matrices), design a data representation and an algorithm to compute their dot product or matrix product efficiently. Walk through your choice of representation, time and space complexity in terms of non-zero entries, and the trade-offs involved.

Algorithms & Data StructuresSystem DesignTechnical Trade-offs

Author's notes

I went with list of (index, value) pairs for the vector case and talked through a two-pointer merge approach.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem and constraints, then propose a representation like a list of (index, value) pairs for sparse vectors and a dictionary of keys (DOK) or compressed sparse row (CSR) for matrices. Explain the dot product algorithm using two pointers for sorted vectors, and for matrices, discuss iterating over non-zero entries and accumulating results. Analyze time and space complexity in terms of non-zero entries (nnz) and discuss trade-offs between different representations.

Pro tip: Emphasize that the choice of representation depends on the operations needed: for dot product, sorted lists are efficient; for matrix multiplication, CSR is often better due to row-wise access. Also, mention that in practice, libraries like SciPy use optimized formats, but understanding the underlying principles is key.

1. Clarify requirements and constraints

Ask about the expected size of vectors/matrices, sparsity level, and whether the data is static or dynamic. Confirm if the dot product is between two vectors or a matrix and a vector, and for matrices, whether it's matrix-matrix multiplication.

2. Choose data representation

For vectors, propose a list of (index, value) pairs sorted by index. For matrices, consider CSR (compressed sparse row) or DOK (dictionary of keys). Explain why these are efficient for sparse data.

3. Design algorithm for dot product

For vectors, use two pointers to iterate through both lists simultaneously, multiplying values when indices match. For matrix-vector product, iterate over non-zero entries of the matrix and accumulate into the result vector.

4. Extend to matrix multiplication

For matrix-matrix product, iterate over non-zero entries of the first matrix, and for each, iterate over non-zero entries of the corresponding row of the second matrix (if using CSR), accumulating into the result matrix.

5. Analyze complexity and trade-offs

Discuss time complexity: O(nnz1 + nnz2) for vector dot product, O(nnz1 * avg row nnz2) for matrix multiplication. Space complexity: O(nnz). Compare with dense representations and mention trade-offs like ease of updates vs. efficient multiplication.

Key Points to Mention

Sparse vector representation: list of (index, value) pairs, sorted by index for efficient intersection.
Sparse matrix representations: CSR (compressed sparse row), CSC (compressed sparse column), DOK (dictionary of keys), and their trade-offs.
Two-pointer technique for vector dot product: O(nnz1 + nnz2) time.
Matrix multiplication algorithm: iterate over non-zero entries and accumulate, avoiding dense operations.
Time and space complexity in terms of non-zero entries (nnz), not dimensions.
Trade-offs: CSR is efficient for row-wise operations but costly to modify; DOK is efficient for incremental construction but slower for arithmetic.

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