TY - JOUR

T1 - The complexity of flow on fat terrains and its I/O-efficient computation

AU - Berg, de, M.

AU - Cheong, O.

AU - Haverkort, H.J.

AU - Lim, J.G.

AU - Toma, L.

PY - 2010

Y1 - 2010

N2 - 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 a-fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by a, we show that the worst-case complexity of the descent graph and of any path of steepest descent is O(n/a2), 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(n2/a2). When a 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 T(n3).

AB - 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 a-fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by a, we show that the worst-case complexity of the descent graph and of any path of steepest descent is O(n/a2), 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(n2/a2). When a 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 T(n3).

U2 - 10.1016/j.comgeo.2008.12.008

DO - 10.1016/j.comgeo.2008.12.008

M3 - Article

VL - 43

SP - 331

EP - 356

JO - Computational Geometry

JF - Computational Geometry

SN - 0925-7721

IS - 4

ER -