## The Complexity of Flow on Fat Terrains and its I/O-Efficient Computation

Citations: | 2 - 1 self |

### BibTeX

@MISC{Berg_thecomplexity,

author = {Mark Berg and Otfried Cheong and Herman Haverkort and Jung-gun Lim and Laura Toma},

title = {The Complexity of Flow on Fat Terrains and its I/O-Efficient Computation},

year = {}

}

### OpenURL

### Abstract

We study the complexity and the I/O-efficient computation of flow on triangulated terrains. 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 in O(Sort(d + r)) i/o’s, where r is the complexity of the river network and d of the descent graph. Furthermore we describe a data structure based on the subdivision of the terrain induced by the edges of the triangulation and paths of steepest ascent and descent from its vertices. This data structure can be used to report the boundary of the watershed of a query point q or the flow path from q in O(l(s) + Scan(k)) i/o’s, where s is the complexity of the subdivision underlying the data structure, l(s) is the number of i/o’s used for planar point location in this subdivision, and k is the size of the reported output. On α-fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by α, we show that the worst-case complexity of the descent graph and of any path of steepest descent is O(n/α 2), where n is the number of triangles in the terrain. The worst-case complexity of the river network and the above-mentioned data structure on such terrains is O(n 2 /α 2). When α is a positive constant this improves the corresponding bounds for arbitrary terrains by a linear factor. We prove that similar bounds cannot be proven for Delaunay triangulations: these can have river networks of complexity Θ(n 3). 1

### Citations

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Citation Context ...e study of river networks and watersheds on tins, and the design of i/o-efficient algorithms for computing these structures. We analyze our algorithms with the model introduced by Aggarwal and Vitter =-=[3]-=-, which has become the standard model for i/o-efficient algorithms. In this model, a computer has an internal memory of size M and an arbitrarily large external memory (disk) where data is stored in b... |

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29 | I/O-efficient dynamic planar point location
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19 |
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18 |
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16 |
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Citation Context .... This approach is appealing because of its simplicity; it is problematic, however, because it discretizes flow and tends to lead to inconsistencies when the triangles in the tin differ a lot in size =-=[23, 25]-=-. The approach taken in the computational-geometry literature considers the tin as a continuous surface on which water always flows in the direction of steepest descent. De Berg et al. [11], McAlliste... |

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Citation Context ...ional-geometry literature considers the tin as a continuous surface on which water always flows in the direction of steepest descent. De Berg et al. [11], McAllister [16, 17], McAllister and Snoeyink =-=[18]-=- and Yu and Snoeyink [25] study the structure and the complexity of the river network and other structures on tins under this model. In particular, de Berg et al. [11] prove that the complexity of the... |

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Citation Context ...sible to report watershed boundaries without going up and down dead ends (as explained below); • the river network, preprocessed for fast downstream traversals (using the results of Hutchinson et al. =-=[13]-=-); 4 McAllister [16] uses a graph with fewer arcs: his graph models all ridges and the up-paths from a certain subset of the vertices. However, defining a subset of vertices with which the approach wi... |

8 | A Watershed Algorithm for Triangulated Terrains - McAllister - 1999 |

8 |
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Citation Context ...isk) takes Scan(n) = Θ(n/B) i/o’s, and sorting Sort(n) = Θ((n/B) logM/B(n/B)) i/o’s in the worst case. Related work. The previous work on modeling flow on tins falls into two classes. Most gis papers =-=[14, 19, 20, 21, 22]-=- adopt a discrete approach and route flow from a triangle to one of its three neighbor triangles using the direction of steepest descent, for example from the centre of the triangle. This approach is ... |

8 |
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Citation Context ...e unlikely to occur in real life. In computational geometry such discrepancies between worst case and practice have led to the study of input models that resemble realistic inputs better. Moet et al. =-=[24]-=- studied visibility and distance problems on realistic terrains. In this paper we consider flow modeling on fat terrains, that is, terrains where the minimum angle of any triangle is bounded from belo... |

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Citation Context |

5 | The Computational Geometry of Hydrology Data in Geographic Information Systems
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Citation Context ...The approach taken in the computational-geometry literature considers the tin as a continuous surface on which water always flows in the direction of steepest descent. De Berg et al. [11], McAllister =-=[16, 17]-=-, McAllister and Snoeyink [18] and Yu and Snoeyink [25] study the structure and the complexity of the river network and other structures on tins under this model. In particular, de Berg et al. [11] pr... |

2 | Sha-Mayn Teh. I/Oefficient point location using persistent B-trees - Arge, Danner |