This is the classic tree diameter problem.
Clarify that the problem is about finding the diameter of a tree. Present the two-pass DFS/BFS algorithm: pick any node, find the farthest node A, then from A find the farthest node B; the distance between A and B is the diameter. Alternatively, describe a single DFS that computes the diameter by tracking the two longest downward paths at each node.
Pro tip: Mention that the two-pass BFS/DFS works because trees are acyclic and connected, and that the single DFS approach is more efficient for large trees as it avoids two traversals. Also, discuss edge cases like a single node (diameter 0) and skewed trees.
Confirm that the graph is a tree (connected, acyclic) and that the longest path is the tree's diameter. Define the problem clearly and ask about constraints (e.g., n up to 10^5).
Decide between two-pass BFS/DFS or single DFS. Explain the two-pass method: from any node, find farthest node A; from A, find farthest node B; distance A-B is diameter. Or explain single DFS: for each node, compute the two longest downward paths and update the diameter.
Walk through the chosen algorithm step-by-step. For two-pass: perform BFS/DFS to find farthest node, then repeat. For single DFS: recursively compute heights and track max sum of two longest child paths.
State time complexity: O(n) for both approaches (each node visited once or twice). Space complexity: O(n) for recursion stack or queue, plus adjacency list storage.
Mention edge cases: n=1 (diameter 0), skewed tree (diameter n-1). Compare approaches: two-pass is simpler to implement, single DFS is more efficient for very large trees but requires careful recursion.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Part A was straightforward BFS, no surprises.
Start by clearly explaining the BFS approach for the shortest path, including how to handle obstacles and track visited cells. Then, for the follow-up, describe a backtracking DFS to enumerate all simple paths, and analyze the exponential time complexity and practical limits such as grid size and path explosion.
Pro tip: Mention that BFS is optimal for unweighted grids, but for all paths, pruning techniques like early termination or memoization can help in some cases, though the problem remains exponential. Also, note that in practice, enumerating all paths is infeasible for large grids, so you'd use it only for small inputs or with additional constraints.
Confirm the grid size, movement directions (usually 4-directional), and that obstacles are impassable. Ask if diagonal moves are allowed or if there are any other constraints.
Describe using a queue to explore level by level, marking visited cells to avoid cycles, and tracking the distance or path. Mention that BFS guarantees the shortest path in an unweighted grid.
Explain using DFS with backtracking to explore all possible paths without revisiting cells. Emphasize that this is a brute-force approach and will be exponential in the worst case.
For BFS, time and space are O(n^2). For all paths, time is exponential (e.g., O(4^(n^2)) in the worst case) and space is O(n^2) for recursion stack. Discuss that it's only feasible for very small grids (e.g., n <= 5 or 6).
For all paths, suggest pruning (e.g., if only need count, use DP; if need paths, use backtracking with early exit). For shortest path, mention that BFS is optimal, but A* could be used if heuristic available.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.