I/O-Efficient Flow Modeling on Fat Terrains

We study the flow of water on fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by a positive constant. We show that the worstcase complexity of any path of steepest descent on a fat terrain of n triangles is Θ(n), and that the worst-case complexity of the river network on such terrains is Θ(n). This improves the corresponding bounds for arbitrary terrains by a linear factor. We prove that in general similar bounds cannot be proven for Delaunay triangulations: these can have river networks of complexity Θ(n). Moreover, we present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/o-efficiently. We use the descent graph to obtain i/o-efficient algorithms for computing river networks and watershed-area maps of fat terrains in O(Sort(r+n)) i/o’s, where r is the complexity of the river network. We also describe a data structure for reporting the boundary of the watershed of a query point q (or the flow path from q) in O(l+Scan(k)) i/o’s, where l is the number of i/o’s used for planar point location and k is the size of the reported output.