Flow Computations on Imprecise Terrains

We study the computation of the flow of water on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time

Optimistic and pessimistic shortest paths on uncertain terrains

In the Uncertain Terrain Shortest Path problem we consider a triangulated terrain with vertices having uncertain Z-coordinates: each vertex is denned as a (x,y,z―,z+) tuple, where the z coordinate of a vertex is uncertain and can be anywhere in the range from z― to z+. We are looking for a path (defined by its projection to the XY- plane) such that, over all possible terrains, the path is as short as possible. We look at both pessimistic (terrain arranges itself to maximize the length of the path that we choose) and optimistic (terrain takes the state that minimizes the length of our path) scenarios. We restrict ourselves to walk only along the edges of the terrain. The unrestricted problem (when we are allowed to walk on the faces of the terrain) has been proven to be NP-hard in both pessimistic and optimistic scenarios. We prove that the edge-restricted pessimistic problem is NP-hard by providing a reduction from the SUBSET-SUM problem and give a polynomial time algorithm for the edge-restricted optimistic problem.Science, Faculty ofComputer Science, Department ofGraduat

Flow computations on imprecise terrains

Abstract. We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x, y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time. Rose knew almost everything that water can do, there are an awful lot when you think what. Gertrude Stein, The World is Round

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop