I went with a sorted list of (index, value) pairs and a two-pointer merge to compute the dot product.
Represent each sparse vector as a list of (index, value) pairs sorted by index, ignoring zeros. For the dot product, use a two-pointer technique to iterate through both lists, multiplying values only when indices match. This yields O(nnz1 + nnz2) time and O(nnz1 + nnz2) space, which is optimal for sparse data.
Pro tip: After presenting the two-pointer solution, proactively mention that if one vector is much sparser than the other, you could binary search each element of the sparser list in the denser list to reduce comparisons, showing you consider trade-offs. Also, clarify assumptions about input format (e.g., whether indices are sorted) and handle edge cases like empty vectors or no overlapping indices.
Ask about the expected size of vectors, sparsity level, whether indices are sorted, and if the dot product will be called multiple times. This informs the choice of representation and algorithm.
Propose storing non-zero elements as a list of (index, value) pairs, sorted by index. Mention alternatives like hash maps and justify why sorted lists are efficient for dot product.
Iterate through both lists simultaneously, advancing the pointer with the smaller index. When indices match, multiply values and add to the result. This avoids unnecessary multiplications by zero.
State that the time complexity is O(nnz1 + nnz2) and space is O(nnz1 + nnz2) for storage. Compare with dense array approach (O(n) time and space) to highlight efficiency for sparse data.
Anticipate extensions like handling unsorted indices, supporting updates, or optimizing for skewed sparsity. Discuss how to adapt the solution, e.g., using binary search or hash maps, and the trade-offs involved.
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