## I/O-Efficient Computation of Water Flow Across a Terrain

### BibTeX

@MISC{Arge_i/o-efficientcomputation,

author = {Lars Arge and Morten Revsbæk and Norbert Zeh},

title = {I/O-Efficient Computation of Water Flow Across a Terrain},

year = {}

}

### OpenURL

### Abstract

Consider rain falling at a uniform rate onto a terrain T represented as a triangular irregular network. Over time, water collects in the basins of T, forming lakes that spill into adjacent basins. Our goal is to compute, for each terrain vertex, the time this vertex is flooded (covered by water). We present an I/O-efficient algorithm that solves this problem using O(sort(X) log(X/M) + sort(N)) I/Os, where N is the number of terrain vertices, X is the number of pits of the terrain, sort(N) is the cost of sorting N data items, and M is the size of the computer’s main memory. Our algorithm assumes that the volumes and watersheds of the basins of T have been precomputed using existing methods.

### Citations

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- 1988
(Show Context)
Citation Context ...flooding times of all terrain vertices. This algorithm makes use of an I/O-efficient meldable priority queue, which we discuss in Section 4. 1.1 I/O Model We use the I/O model with one (logical) disk =-=[2]-=- to design and analyze our algorithm. In this model, the computer is equipped with a two-level memory hierarchy consisting of an internal memory and a (disk-based) external memory. The internal memory... |

235 | Topological persistence and simplification
- Edelsbrunner, Letscher, et al.
(Show Context)
Citation Context ... height, volume or area. Agarwal, Arge and Yi [1] described an O(sort(N)) I/O partial flooding algorithm that removes all basins of small height. This is done by computing the topological persistence =-=[10,11]-=- of each basin and removing the ones with persistence below a small threshold. The key to obtaining an O(sort(N)) I/O solution to this problem is an algorithm that can process a sequence of N Union an... |

172 | External-memory graph algorithms
- Chiang, Goodrich, et al.
- 1995
(Show Context)
Citation Context ...ries of the basins β1, β2, . . . , βk takes O(sort(k + |Rβ|)) = O(sort(X)) I/Os, where X := |Mβ|+|Rβ|. The path P can be computed using O(sort(X)) I/Os using the Euler tour technique and list ranking =-=[6]-=-. This gives the following result. Theorem 1. The spill times of all basins of a terrain T can be computed using O(sort(X) log(X/M)) I/Os, where X is the number of elementary basins of T . Proof. We p... |

104 | Extracting Topographic Structure from Digital Elevation Data for Geographic Information System Analysis”, Photogrammetric Engineering and Remote
- Jenson, Domingue
- 1988
(Show Context)
Citation Context ... getting caught in small local basins, most flow modelling approaches first remove all basins by flooding the terrain, that is, conceptually pouring water onto the terrain until all basins are filled =-=[4,13,15]-=-. However, this often leads to unrealistic flow patterns, since many important geographical features are removed. Recent papers [1,5] suggested partial flooding algorithms that flood only“small”basins... |

93 |
The extraction of drainage networks from digital elevation data, Computer vision, graphics and image processing
- O’Callaghan, Mark
- 1984
(Show Context)
Citation Context ... getting caught in small local basins, most flow modelling approaches first remove all basins by flooding the terrain, that is, conceptually pouring water onto the terrain until all basins are filled =-=[4,13,15]-=-. However, this often leads to unrealistic flow patterns, since many important geographical features are removed. Recent papers [1,5] suggested partial flooding algorithms that flood only“small”basins... |

55 |
The buffer tree: A technique for designing batched external data structures
- Arge
(Show Context)
Citation Context ... spill events, the set of Union operations is known, but their order depends on the spill times of the basins. Using an internal-memory union-find structure and an I/O-efficient priority queue (e.g., =-=[3,12]-=-), a cost of O(Xα(X) + sort(N)) disk accesses can be achieved, where α(·) is the inverse of Ackermann’s function, sort(N) ≪ N is the I/O complexity of sorting N data items (see next section), and X is... |

47 | Hierarchical Morse complexes for piecewise linear 2-manifolds
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(Show Context)
Citation Context ... height, volume or area. Agarwal, Arge and Yi [1] described an O(sort(N)) I/O partial flooding algorithm that removes all basins of small height. This is done by computing the topological persistence =-=[10,11]-=- of each basin and removing the ones with persistence below a small threshold. The key to obtaining an O(sort(N)) I/O solution to this problem is an algorithm that can process a sequence of N Union an... |

20 |
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- Fadel, Jakobsen, et al.
- 1999
(Show Context)
Citation Context ... spill events, the set of Union operations is known, but their order depends on the spill times of the basins. Using an internal-memory union-find structure and an I/O-efficient priority queue (e.g., =-=[3,12]-=-), a cost of O(Xα(X) + sort(N)) disk accesses can be achieved, where α(·) is the inverse of Ackermann’s function, sort(N) ≪ N is the I/O complexity of sorting N data items (see next section), and X is... |

18 | I/O-efficient batched union-find and its applications to terrain analysis
- Agarwal, Arge, et al.
- 2006
(Show Context)
Citation Context ...ion of this approach using a priority queue and a union-find structure. This simulation approach does not translate into an I/Oefficient algorithm, as the only known I/O-efficient unionfind structure =-=[1]-=- requires the sequence of Union operations to be known in advance. In simulating the sequence of spill events, the set of Union operations is known, but their order depends on the spill times of the b... |

18 | Artifacts of TINBased Surface Flow Modeling - Theobald, Goodchild - 1990 |

14 | Flow computation on massive grid terrains
- Arge, Chase, et al.
(Show Context)
Citation Context ... getting caught in small local basins, most flow modelling approaches first remove all basins by flooding the terrain, that is, conceptually pouring water onto the terrain until all basins are filled =-=[4,13,15]-=-. However, this often leads to unrealistic flow patterns, since many important geographical features are removed. Recent papers [1,5] suggested partial flooding algorithms that flood only“small”basins... |

11 | TerraStream: from elevation data to watershed hierarchies
- Danner, Mølhave, et al.
- 2007
(Show Context)
Citation Context ...his implementation is needed as part of our I/O-efficient algorithm. We also require a tree structure that represents the nesting of the basins of T . This tree has been termed the merge tree of T in =-=[7,8]-=- (also see Section 2) and can be computed using O(sort(N)) I/Os using the topological persistence algorithm of [1]. This algorithm is easily augmented to compute the lowest saddle on the boundary of e... |

8 |
Flooding triangulated terrain
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- 2004
(Show Context)
Citation Context ...ersheds of all basins that yet have to spill, as well as predicted spill times for these basins based on their current watersheds. An internal-memory solutionbased on this idea has been presented in =-=[14]-=-. In Section 3.2, we discuss an O(N log N) time implementation of this approach using a priority queue and a union-find structure. This simulation approach does not translate into an I/Oefficient algo... |

8 | An object-oriented framework for hydrologic and geomorphic modeling using triangulated irregular networks - TUCKER, LANCASTER, et al. |

7 | Geometric analysis of surface runoff and the computation order of unit elements in distributed hydrological models - Palacios-Vélez, Gandoy-Bernasconi, et al. - 1998 |

4 |
I/O Efficient Algorithms and Applications in Geographic Information Systems
- Danner
- 2006
(Show Context)
Citation Context ...his implementation is needed as part of our I/O-efficient algorithm. We also require a tree structure that represents the nesting of the basins of T . This tree has been termed the merge tree of T in =-=[7,8]-=- (also see Section 2) and can be computed using O(sort(N)) I/Os using the topological persistence algorithm of [1]. This algorithm is easily augmented to compute the lowest saddle on the boundary of e... |

3 |
I/O-efficient contour tree simplification
- Arge, Revsbaek
- 2009
(Show Context)
Citation Context ...lly pouring water onto the terrain until all basins are filled [4,13,15]. However, this often leads to unrealistic flow patterns, since many important geographical features are removed. Recent papers =-=[1,5]-=- suggested partial flooding algorithms that flood only“small”basins, where the size of a basin can be defined using different measures, such as height, volume or area. Agarwal, Arge and Yi [1] describ... |

2 | I/O-Efficient Flow Modeling on Fat Terrains
- Berg, Cheong, et al.
- 2007
(Show Context)
Citation Context ...atershed sizes of all pits by summing watershed sizes bottom up in the merge tree. However, the only I/Oefficient algorithm for computing the watersheds of all pits of a terrain exactly is the one of =-=[9]-=-. This algorithm performs O(sort(N + S)) I/Os for fat terrains, where S is the size of the terrain’s strip map. The size of the strip map of a fat terrain is Θ(N 2 ) in the worst case [9], in which ca... |