River networks and watershed maps of triangulated terrains revisited ∗

Triangulated surfaces are often used to represent terrains in geographic information systems. We investigate the complexity of river networks and watershed maps on such terrains under the assumption that water always follows the path of steepest descent. We show that the worst-case complexity is only Θ(n) if all triangles are non-obtuse or if all triangles are fat, that is, their minimum angles are bounded from below by a positive constant. Furthermore, we can compute the river networks and watershed maps by tracing paths in a directed acyclic graph representation of the triangulation—a property that can be exploited to do computations I/O-efficiently.