The naive approach is obvious but they're not asking for that.
Start by clarifying the sparse vector representation and constraints, then propose a data structure that stores only non-zero elements (e.g., index-value pairs). Implement the dot product by iterating through the smaller non-zero list and looking up indices in the other, or by using a two-pointer merge if both are sorted by index. Finally, analyze time and space complexity in terms of the number of non-zero elements.
Pro tip: Mention that the optimal approach depends on whether the vectors are sorted by index; if not, you can sort them or use a hash map, but discuss the trade-offs. Also, proactively discuss edge cases like different lengths or all-zero vectors to show thoroughness.
Ask about the expected size of vectors, sparsity level, whether indices are sorted, and if the vectors are immutable. This guides the choice of data structure and algorithm.
Propose storing only non-zero elements as a list of (index, value) pairs or two parallel arrays. Mention that this reduces space from O(n) to O(k) where k is the number of non-zeros.
If indices are sorted, use a two-pointer approach to iterate through both lists in O(k1 + k2) time. If not, iterate through the smaller list and use a hash map for O(1) lookups, achieving O(min(k1, k2)) time on average.
Clearly state the time complexity based on the chosen approach and the space complexity of the data structure. Compare with the naive O(n) approach to highlight efficiency.
Mention scenarios where one approach is better (e.g., sorted vs unsorted, very sparse vs moderately sparse). Also cover edge cases like empty vectors, different lengths, and negative values.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.