3,144 research outputs found
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Minimum Convex Partitions and Maximum Empty Polytopes
Let be a set of points in . A Steiner convex partition
is a tiling of with empty convex bodies. For every integer ,
we show that admits a Steiner convex partition with at most tiles. This bound is the best possible for points in general
position in the plane, and it is best possible apart from constant factors in
every fixed dimension . We also give the first constant-factor
approximation algorithm for computing a minimum Steiner convex partition of a
planar point set in general position. Establishing a tight lower bound for the
maximum volume of a tile in a Steiner convex partition of any points in the
unit cube is equivalent to a famous problem of Danzer and Rogers. It is
conjectured that the volume of the largest tile is .
Here we give a -approximation algorithm for computing the
maximum volume of an empty convex body amidst given points in the
-dimensional unit box .Comment: 16 pages, 4 figures; revised write-up with some running times
improve
A Method of Rendering CSG-Type Solids Using a Hybrid of Conventional Rendering Methods and Ray Tracing Techniques
This thesis describes a fast, efficient and innovative algorithm for producing shaded, still images of complex objects, built using constructive solid geometry ( CSG ) techniques. The algorithm uses a hybrid of conventional rendering methods and ray tracing techniques. A description of existing modelling and rendering methods is given in chapters 1, 2 and 3, with emphasis on the data structures and rendering techniques selected for incorporation in the hybrid method. Chapter 4 gives a general description of the hybrid method. This method processes data in the screen coordinate system and generates images in scan-line order. Scan lines are divided into spans (or segments) using the bounding rectangles of primitives calculated in screen coordinates. Conventional rendering methods and ray tracing techniques are used interchangeably along each scan-line. The method used is detennined by the number of primitives associated with a particular span. Conventional rendering methods are used when only one primitive is associated with a span, ray tracing techniques are used for hidden surface removal when two or more primitives are involved. In the latter case each pixel in the span is evaluated by accessing the polygon that is visible within each primitive associated with the span. The depth values (i. e. z-coordinates derived from the 3-dimensional definition) of the polygons involved are deduced for the pixel's position using linear interpolation. These values are used to determine the visible polygon. The CSG tree is accessed from the bottom upwards via an ordered index that enables the 'visible' primitives on any particular scan-line to be efficiently located. Within each primitive an ordered path through the data structure provides the polygons potentially visible on a particular scan-line. Lists of the active primitives and paths to potentially visible polygons are maintained throughout the rendering step and enable span coherence and scan-line coherence to be fully utilised. The results of tests with a range of typical objects and scenes are provided in chapter 5. These results show that the hybrid algorithm is significantly faster than full ray tracing algorithms
JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere
An algorithm for the generation of non-uniform, locally-orthogonal staggered
unstructured spheroidal grids is described. This technique is designed to
generate very high-quality staggered Voronoi/Delaunay meshes appropriate for
general circulation modelling on the sphere, including applications to
atmospheric simulation, ocean-modelling and numerical weather prediction. Using
a recently developed Frontal-Delaunay refinement technique, a method for the
construction of high-quality unstructured spheroidal Delaunay triangulations is
introduced. A locally-orthogonal polygonal grid, derived from the associated
Voronoi diagram, is computed as the staggered dual. It is shown that use of the
Frontal-Delaunay refinement technique allows for the generation of very
high-quality unstructured triangulations, satisfying a-priori bounds on element
size and shape. Grid-quality is further improved through the application of
hill-climbing type optimisation techniques. Overall, the algorithm is shown to
produce grids with very high element quality and smooth grading
characteristics, while imposing relatively low computational expense. A
selection of uniform and non-uniform spheroidal grids appropriate for
high-resolution, multi-scale general circulation modelling are presented. These
grids are shown to satisfy the geometric constraints associated with
contemporary unstructured C-grid type finite-volume models, including the Model
for Prediction Across Scales (MPAS-O). The use of user-defined mesh-spacing
functions to generate smoothly graded, non-uniform grids for multi-resolution
type studies is discussed in detail.Comment: Final revisions, as per: Engwirda, D.: JIGSAW-GEO (1.0): locally
orthogonal staggered unstructured grid generation for general circulation
modelling on the sphere, Geosci. Model Dev., 10, 2117-2140,
https://doi.org/10.5194/gmd-10-2117-2017, 201
- …