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Removing Local Extrema from Imprecise Terrains
"... In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, finding an assignment of one height to each ve ..."
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In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, finding an assignment of one height to each vertex, within its error interval, so that the resulting terrain has minimum number of local extrema. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, the problem of finding a height assignment that minimizes the total number of local extrema (minima as well as maxima) is NPhard, and is even hard to approximate within a factor of O(log log n) unless P = NP. Moreover, we show that even a simplified version of the problem where we can have only three different types of intervals for the vertices is already NPhard, a result we obtain by proving hardness of a special case of 2Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graphalgorithms community. 1
Embedding Rivers in Triangulated Irregular Networks with Linear Programming
"... Data conflation is a major issue in GIS: different geospatial data sets covering overlapping regions, possibly obtained from different sources and using different acquisition techniques, need to be combined into one single consistent data set before the data can be analyzed. The most common occurren ..."
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Data conflation is a major issue in GIS: different geospatial data sets covering overlapping regions, possibly obtained from different sources and using different acquisition techniques, need to be combined into one single consistent data set before the data can be analyzed. The most common occurrence for hydrological applications is conflation of a digital elevation model and rivers. We assume that a triangulated irregular network (TIN) is given, and a subset of its edges are designated as river edges, each with a flow direction. The goal is to obtain a terrain where the rivers flow along valley edges, in the specified direction, while preserving the original terrain as much as possible. We study the problem of changing the elevations of the vertices to ensure that all the river edges become valley edges, while minimizing the total elevation change. We show that this problem can be solved using linear programming. However, several types of artifacts can occur in an optimal solution. We analyze which other criteria, relevant for hydrological applications, can be captured by linear constraints as well, in order to eliminate such artifacts. We implemented and tested the approach on real terrain and river data, and describe the results obtained with different variants of the algorithm. 1
Smoothing imprecise 1dimensional terrains ∗
"... An imprecise 1dimensional terrain is an xmonotone polyline where the ycoordinate of each vertex is not fixed but only constrained to a given interval. In this paper we study four different optimization measures for imprecise 1dimensional terrains, related to obtaining smooth terrains. In particu ..."
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An imprecise 1dimensional terrain is an xmonotone polyline where the ycoordinate of each vertex is not fixed but only constrained to a given interval. In this paper we study four different optimization measures for imprecise 1dimensional terrains, related to obtaining smooth terrains. In particular, we present algorithms to minimize the largest and total turning angle, and to maximize the smallest and total turning angle. 1