I went straight to COO for the dot product because it felt natural, two lists of (index, value) pairs and you merge them.
Start by clarifying the sparse vector representation (e.g., dictionary or sorted index-value pairs) and the expected output format for matrix multiplication. Then implement the dot product using an efficient merge or hash-based intersection, and extend the same logic to sparse matrix multiplication by iterating over non-zero entries and accumulating results in a sparse output structure. Discuss trade-offs between different representations and algorithms, and analyze time/space complexity.
Pro tip: Emphasize that sparse operations should scale with the number of non-zeros (nnz), not the full dimensions, and proactively mention how you'd handle edge cases like empty vectors or mismatched dimensions. This shows production-level awareness and avoids naive dense implementations.
Ask about the input format for sparse vectors (e.g., dictionary, list of (index, value) pairs, CSR) and the expected output for matrix multiplication (sparse or dense). Confirm constraints like vector length, sparsity level, and whether indices are sorted.
Choose an approach: if indices are sorted, use a two-pointer merge to compute the dot product in O(nnz1 + nnz2) time; otherwise, use a hash map for O(nnz1 + nnz2) expected time. Explain why this is better than iterating over all dimensions.
For C = A * B, iterate over non-zero entries of A and B, and accumulate contributions to C. Use a dictionary or CSR-like structure to store only non-zero results, ensuring the algorithm runs in time proportional to the number of non-zero multiplications.
Discuss time and space complexity in terms of nnz. Compare hash-based vs. sorted-merge approaches, and mention when a dense implementation might be preferable (e.g., very low sparsity).
Consider empty vectors, zero values, mismatched dimensions, and duplicate indices. Walk through a small example to verify correctness and discuss potential optimizations like parallelization or blocking for large matrices.
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This is where the interview actually lived.
Start by defining COO, CSR, and CSC in terms of their storage layouts and access patterns. Then, for each operation, analyze the required access patterns and match them to the format that minimizes overhead and maximizes efficiency. Conclude with a concise recommendation for each operation.
Pro tip: Emphasize that the best format depends on the specific operation and access pattern, and mention that many libraries (e.g., SciPy) provide conversions between formats, so the choice often involves a trade-off between conversion cost and operation efficiency.
Briefly describe COO (coordinate list), CSR (compressed sparse row), and CSC (compressed sparse column), focusing on their storage structures and typical use cases.
Consider the operation: dot product of two sparse vectors. Determine which format allows efficient element-wise multiplication and summation, likely COO due to its simplicity for merging sorted indices.
For y = A * x, where A is sparse and x is dense, CSR is efficient because it enables row-wise access to compute each output element as a dot product of a row with x.
For C = A * B, where both are sparse, CSR is often preferred for A and CSC for B to efficiently compute the product by iterating over rows of A and columns of B, though other strategies exist.
Concisely state the most appropriate format for each operation and justify based on access patterns and computational complexity.
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Straightforward once you've thought about the formats.
Clarify which sparse operations are being asked about (e.g., sparse-dense matrix multiplication, sparse-sparse addition, transpose, or conversion) and state assumptions about the sparse format (CSR/CSC/COO). Then derive time and memory complexity in terms of nnz, explaining how the format's storage and access patterns affect the bounds.
Pro tip: Mention that for many sparse operations, the constant factors and memory access patterns (e.g., irregular gathers in SpMM) often dominate in practice, so asymptotic nnz bounds alone don't tell the full performance story—especially on GPUs.
Ask which specific sparse operations are in scope (e.g., SpMV, SpMM, SpGEMM, transpose, element-wise add) and which storage format is assumed (CSR, CSC, COO). State that complexity depends on both.
Explain that sparse formats store only non-zeros plus index metadata, so memory is O(nnz) for COO (row, col, value) and O(nnz + n) for CSR/CSC (values, indices, row/column pointers).
For SpMV: O(nnz). For SpMM (sparse A m×k times dense B k×n): O(nnz * n). For SpGEMM: O(nnz(A) * nnz(B) / k) in the worst case, or O(flops) where flops is the number of scalar multiplications. For transpose: O(nnz).
Mention that CSR is efficient for row-wise access and SpMV, while CSC is better for column-wise operations; COO is simple but less efficient for arithmetic. Conversion between formats costs O(nnz).
Relate the complexity to real-world ML workloads (e.g., sparse embeddings, graph neural networks) and note that memory bandwidth and irregular access patterns often make sparse ops slower than dense equivalents despite lower asymptotic complexity.
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