My first instinct was pure BFS and I wasted a few minutes going down that road before realizing I needed to think about this differently.
Recognize this as a maximin path problem and solve it using binary search on the answer combined with BFS/DFS reachability. For a candidate distance D, treat cells with distance to nearest thief < D as blocked and check if a path exists from start to end. Precompute distances to nearest thief using multi-source BFS.
Pro tip: Mention that you would first clarify edge cases (e.g., start or end is a thief, no thieves, multiple thieves) and discuss trade-offs between binary search + BFS and a modified Dijkstra approach. This shows thoroughness and practical engineering judgment.
Confirm the problem constraints: grid size, thief positions, movement allowed (4-directional?), and what 'distance' means (Manhattan). Ask about edge cases like start/end being thieves or no thieves present.
Use multi-source BFS from all thief cells to compute the Manhattan distance from every cell to the nearest thief. This gives a distance map in O(n^2) time.
Binary search the maximum possible minimum distance D. For each D, check if there is a path from start to end using only cells with distance >= D.
For a given D, perform BFS/DFS from start to end, only moving through cells with distance >= D. If reachable, D is feasible; otherwise, not.
Discuss time complexity: O(n^2 log n) for binary search over distances (max distance O(n)) with O(n^2) BFS each. Mention possible optimizations like using union-find or sorting cells by distance.
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