1D and 2D Flow Routing on a Terrain

An important problem in terrain analysis is modeling how water flows across a terrain creating floods by forming channels and filling depressions. In this paper we study a number of flow-query related problems: given a terrain Σ represented as a triangulated xy-monotone surface with n vertices, and a rain distribution R which may vary over time, determine how much water is flowing over a given edge as a function of time. We develop internal-memory as well as I/O-efficient algorithms for flow queries. This paper contains four main results: (i) An internal-memory algorithm for answering terrain-flow queries: preprocess Σ into a linear-size data structure so that given a rain distribution R the flow-rate functions of all edges of Σ can be reported quickly. (ii) I/O-efficient algorithms for answering terrain-flow queries. (iii) An internal memory algorithm for answering edge-flow queries: preprocess Σ into a linear-size data structure so that given a rain distribution R, the flow-rate function of an edge under the single-flow direction (SFD) model can be computed quickly. (iv) We present an efficient algorithm that given a path in Σ computes the two-dimensional channel along which water flows.