My first instinct was a standard union-find but the product IDs are strings not integers so I had to map them to indices first.
Model the problem as a graph where product IDs are nodes and each pair is an undirected edge. Use Union-Find (Disjoint Set Union) to efficiently merge connected components, then count the size of each component and sort the sizes in increasing order.
Pro tip: Mention that Union-Find with path compression and union by rank achieves near O(1) per operation, making it ideal for large datasets. Also, clarify that the output should be the sorted sizes, not the category IDs, to avoid misinterpretation.
Confirm that transitivity means if A~B and B~C, then A~C, so all connected IDs form one category. Ask if product IDs are integers or strings, and if the output should be sizes sorted ascending.
Select Union-Find (Disjoint Set Union) for efficient merging and finding connected components. Alternatively, mention DFS/BFS on a graph if the number of nodes is small, but highlight Union-Find's efficiency for large m.
Initialize each unique product ID as its own parent. For each pair, union the two sets using path compression and union by rank/size to keep trees shallow.
After processing all pairs, traverse all unique IDs, find their root, and tally the size of each root's component. Collect these sizes and sort them in increasing order.
State time complexity: O(m α(n) + k log k) where α is inverse Ackermann, k is number of categories. Handle edge cases: no pairs (each ID its own category), duplicate pairs, and IDs that appear only once.
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