No abstract

A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that, if the Delaunay triangulation has the ratio property introduced in Miller et al. [1995], then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement algorithm using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. Guaranteed bounds on the aspect ratios, normal variation and dihedral angles are also provided. To this end, we present new concepts and results including a new definition of local feature size and a proof for a generalized topological ball property.

Querying uncertain data has emerged as an important problem in data management due to the imprecise nature of many measurement data. In this paper we study answering range queries over uncertain data. Specifically, we are given a collection P of n points in R, each represented by its one-dimensional probability density function (pdf). The goal is to build an index on P such that given a query interval I and a probability threshold τ , we can quickly report all points of P that lie in I with probability at least τ . We present various indexing schemes with linear or near-linear space and logarithmic query time. Our schemes support pdf's that are either histograms or more complex ones such as Gaussian or piecewise algebraic. They also extend to the external memory model in which the goal is to minimize the number of disk accesses when querying the index.