I knew binary search had to be involved but my first instinct was just to test all pairs, which is O(N^2) and the interviewer's face said it all.
Model the problem as finding all edges in a graph where vertices are items and edges are bad pairs, using subset queries that test if a subset is an independent set. Use a divide-and-conquer strategy to identify edges efficiently, leveraging binary search to isolate bad pairs. Aim for O(N log N) queries by recursively splitting the set and testing subsets.
Pro tip: Start by clarifying the query model and constraints (e.g., can you query any subset? Is the function monotonic?); then propose a solution with a clear complexity analysis, and discuss trade-offs between query count and computational overhead.
Confirm that the black-box function returns true iff the subset contains no bad pairs, and that we need to find all bad pairs with minimal calls. Ask about N's size and any assumptions (e.g., graph is sparse).
Represent items as vertices and bad pairs as edges. The function tests if a subset is an independent set. The goal is to find all edges using independent set queries.
Recursively split the vertex set into two halves. For each half, test if it contains any bad pairs; if not, skip. To find edges across halves, use a binary search-like approach: for each vertex in one half, test subsets of the other half to identify its neighbors.
Show that the algorithm uses O(N log N) queries in the worst case. Discuss potential improvements for sparse graphs, such as using group testing or adaptive splitting to reduce calls.
Compare with naive O(N^2) approach. Mention handling of isolated vertices, complete graphs, and the impact of query cost. Consider if the function can be queried with large subsets efficiently.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.