This is the widest path problem. The reference "On the bottleneck shortest path problem" by Kaibel and Peinhardt in the bottom of the page seems nice.
Given a topographical map that lists the altitude along the direct road between any two neighboring cities, and two cities a and b, how would you develop a linear time algorithm that finds a route from a to b that minimizes the maximum altitude? Roads can be traveled in both directions.
I have no idea how to begin this problem, can anyone help?