Start with a clear baseline scalar implementation, then systematically optimize for parallelism by addressing SIMD, multithreading, memory alignment, and cache behavior. Finally, contrast with sparse vectors by discussing appropriate data structures, complexity changes, and parallelization challenges.
Pro tip: Emphasize that performance optimization is iterative and measurement-driven; mention that you'd profile to identify bottlenecks before applying optimizations, and that sparse operations often shift the bottleneck from compute to memory access.
Write a simple scalar dot product loop, noting its O(n) time complexity and sequential nature. This establishes a correctness reference and performance baseline.
Discuss SIMD vectorization (e.g., AVX, NEON) to process multiple elements per instruction, multithreading (e.g., OpenMP, TBB) to utilize multiple cores, memory alignment to enable efficient SIMD loads, and cache-friendly access patterns (e.g., blocking, prefetching) to reduce memory stalls.
Explain common sparse formats like CSR, CSC, or coordinate lists, and how they store only non-zero elements and indices. Discuss the impact on complexity: dot product becomes O(nnz) where nnz is the number of non-zeros, but with overhead for index lookups.
Address challenges such as load imbalance due to varying nnz per row/partition, irregular memory access, and the need for efficient merging or intersection of indices. Suggest strategies like partitioning by non-zero count, using hash-based or merge-based intersection, and considering hybrid approaches.
Summarize trade-offs: dense is simple and fast for high density, sparse saves memory and compute for low density but adds complexity. Mention that the choice depends on sparsity level, hardware, and whether the vectors are static or dynamic.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.