I recognized this as a min-bottleneck path problem pretty quickly, which helped.
This is a minimax path problem that can be solved by binary searching the answer and checking connectivity, or by using a priority queue to always expand the cell with the smallest maximum height so far. Start by clarifying the problem and edge cases, then present the binary search + BFS/DFS approach as it's intuitive and efficient. Alternatively, mention Dijkstra-like approach as a more direct solution.
Pro tip: Discuss the trade-offs between binary search + BFS (O(mn log(maxHeight))) and Dijkstra-like (O(mn log(mn))) approaches, and mention that the latter can be more efficient if heights are large. Also, note that the problem is equivalent to finding the minimum bottleneck path.
Confirm that the grid has unique heights, moves are 4-directional, and we need to minimize the maximum height along the path. Ask about grid size limits and height range to choose the optimal algorithm.
Binary search over the possible maximum height values (from min to max in grid). For each candidate, check if there's a path from start to end where all cells have height <= candidate using BFS/DFS.
Alternatively, use a min-heap to always expand the cell with the smallest maximum height so far (Dijkstra-like). This directly finds the minimized maximum without binary search.
Binary search + BFS: O(mn log(maxHeight)) time, O(mn) space. Dijkstra-like: O(mn log(mn)) time, O(mn) space. Discuss which is better based on constraints.
Consider 1x1 grid, start or end being the maximum, and unreachable cases (though grid is fully connected). Summarize the chosen approach and its correctness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the problem: we need to find the minimum height threshold such that a path exists from start to end where all cells have height ≤ threshold. Then, describe the binary search over the sorted unique heights, and for each threshold, perform a DFS from the start to check if the end is reachable. Finally, analyze the time complexity as O(MN log(MN)) and compare with other approaches like BFS or union-find.
Pro tip: Mention that DFS may cause stack overflow on large grids, so an iterative DFS or BFS is often preferred in practice. Also, note that binary search on the answer is a common pattern for optimization problems with monotonic properties.
Restate the problem: given a grid of heights, find the minimum height threshold such that there is a path from top-left to bottom-right moving up/down/left/right, where all cells on the path have height ≤ threshold. Confirm grid dimensions and that heights are integers.
Sort the unique heights and binary search over the possible thresholds. For each mid threshold, check if a path exists using DFS. The monotonic property: if a path exists at threshold T, it also exists for any T' > T.
Implement DFS from the start cell, only visiting cells with height ≤ threshold and not yet visited. Use a visited matrix or set to avoid cycles. If the end cell is reached, return true; otherwise false.
Each DFS takes O(MN) time. Binary search over up to MN unique heights takes O(log(MN)) iterations. Total time: O(MN log(MN)). Space: O(MN) for visited and recursion stack.
Mention that BFS is similar but uses a queue; union-find can solve in O(MN log(MN)) or O(MN α(MN)) with sorting. DFS may have recursion depth issues; iterative DFS or BFS is safer. Binary search adds a log factor but is simple.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.