The BFS part I got through fine, pretty standard.
Start by explaining the baseline BFS/DFS solution for static reachability, then analyze the dynamic update scenario. Compare recomputing from scratch (O(V+E) per update) against incremental approaches like maintaining a reachability tree or using union-find with rollback, discussing trade-offs in time, space, and implementation complexity.
Pro tip: Emphasize that the choice depends on the frequency of updates and queries; for a single update, recomputation is often simpler and fast enough, but for many updates, incremental structures pay off. Mention that in practice, you'd also consider the grid size and whether blockers are permanent.
Ask about grid size, number of updates, whether blockers are permanent, and if queries are interleaved with updates. This determines the appropriate algorithm.
Explain that for a static grid, BFS or DFS from start to end works in O(V+E) time. For a single update, you can simply re-run BFS/DFS.
Discuss that adding a blocker can only disconnect the graph if the blocker lies on all paths. Recomputing from scratch is O(V+E) per update, which may be acceptable for few updates but inefficient for many.
Describe maintaining a reachability tree (e.g., BFS tree) and checking if the new blocker is a bridge; if so, recompute only affected parts. Alternatively, use union-find with rollback or dynamic connectivity data structures, noting their complexity.
Weigh recomputation (simple, O(V+E) per update) against incremental (complex, potentially faster updates). Recommend based on expected update frequency and grid size.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.