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 \emph{flow-query} related problems: Given a terrain $Σ$, represented as a triangulated $xy$-monotone surface with $n$ vertices, 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) We present an internal-memory algorithm that preprocesses $Σ$ into a linear-size data structure that for a (possibly time varying) rain distribution $R$ can return the flow-rate functions of all edges of $Σ$ in $O(ρk+|ϕ| \log n)$ time, where $ρ$ is the number of sinks in $Σ$, $k$ is the number of times the rain distribution changes, and $|ϕ|$ is the total complexity of the flow-rate functions that have non-zero values; (ii) We also present an I/O-efficient algorithm for preprocessing $Σ$ into a linear-size data structure so that for a rain distribution $R$, it can compute the flow-rate function of all edges using $O(\text{Sort}(|ϕ|))$ I/Os and $O(|ϕ| \log |ϕ|)$ internal computation time. (iii) $Σ$ can be preprocessed into a linear-size data structure so that for a given rain distribution $R$, the flow-rate function of an edge $(q,r) \in Σ$ under the single-flow direction (SFD) model can be computed more efficiently. (iv) We present an algorithm for computing the two-dimensional channel along which water flows using Manning's equation; a widely used empirical equation that relates the flow-rate of water in an open channel to the geometry of the channel along with the height of water in the channel.

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