Minimize maximum height along a grid path

You are given an `m x n` grid `heights` of **distinct** integers representing terrain heights. You start at the top-left cell `(0,0)` and want to reach the bottom-right cell `(m-1,n-1)` by moving **4-directionally** (up, down, left, right) within the grid. Define the **cost** of a path as the **maximum height value** among all cells on that path. ### Task Return the **minimum possible path cost**, i.e., among all valid paths from `(0,0)` to `(m-1,n-1)`, minimize the maximum height encountered. ### Input - `heights`: `m x n` integer grid ### Output - An integer: the minimum achievable value of `max(height on path)` ### Notes / Follow-ups discussed - A typical approach is to model this as a shortest-path variant (e.g., Dijkstra where path “distance” is `minimize max-so-far`). - Follow-up: Can this be implemented with DFS? If yes, what approach would you use (plain DFS vs DFS + binary search vs memoization), and what is the time complexity?