Went with BFS and it passed, but I knew halfway through they were probably expecting union-find.
Model the problem as a graph where products are nodes and pairs are edges, then find connected components using Union-Find (Disjoint Set Union) for near-linear time. After processing all pairs, count the distinct roots and the size of each component to get the number of categories and product counts.
Pro tip: Mention that Union-Find with path compression and union by rank achieves almost O(1) per operation, making it ideal for large datasets. Also, clarify edge cases like duplicate pairs or self-loops, and discuss how to handle products that appear only in pairs versus those that might be listed separately.
Confirm that categories are transitive, meaning if A-B and B-C are pairs, then A, B, and C are in the same category. Ask if the input list can contain duplicate pairs or if products are only given via pairs.
Select Union-Find (Disjoint Set Union) to efficiently group products into categories. Explain that it supports union and find operations with near-constant time complexity when optimized.
Iterate through each pair, perform union on the two products, and ensure each product is added as a separate set initially. Use a hash map to map product IDs to parent pointers if products are not integers.
After processing, traverse all products, find their root, and aggregate counts per root. The number of distinct roots is the number of categories, and the count per root is the number of products in that category.
State that time complexity is O(N α(N)) where N is the number of products (or pairs), and space is O(N). Discuss handling of duplicate pairs, self-loops, and products that appear only once (if applicable).
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