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 $Sigma$, 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 $Sigma$ 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 $Sigma$ in $O(rho k+|phi| log n)$ time, where $rho$ is the number of sinks in $Sigma$, $k$ is the number of times the rain distribution changes, and $|phi|$ 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 $Sigma$ into a linear-size data structure so that for a rain distribution $R$, it can compute the flow-rate function of all edges using $O(text{Sort}(|phi|))$ I/Os and $O(|phi| log |phi|)$ internal computation time. (iii) $Sigma$ can be preprocessed into a linear-size data structure so that for a given rain distribution $R$, the flow-rate function of an edge $(q,r) in Sigma$ 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 Mannings 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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