4 research outputs found
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
distinct radii.
For the upper bound, we reduce the sphere convex hull problem to the problem
of computing the worst-case combinatorial complexity of the convex hull of a
set of -dimensional convex polytopes lying on parallel hyperplanes
in , where odd, a problem which is of independent
interest. More precisely, we show that the worst-case combinatorial complexity
of the convex hull of a set
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
We end with algorithmic considerations, and we show how our tight bounds for
the parallel polytope convex hull problem, yield tight bounds on the
combinatorial complexity of the Minkowski sum of two convex polytopes in
.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of
the convex hull of parallel polytopes (the new proof gives upper bounds for
all face numbers of the convex hull of the parallel polytopes
The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs
The Voronoi diagram is a certain geometric data structure which has numerous
applications in various scientific and technological fields. The theory of
algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich
and useful, with several different and important algorithms. However, this
theory has been quite steady during the last few decades in the sense that no
essentially new algorithms have entered the game. In addition, most of the
known algorithms are serial in nature and hence cast inherent difficulties on
the possibility to compute the diagram in parallel. In this paper we present
the projector algorithm: a new and simple algorithm which enables the
(combinatorial) computation of 2D Voronoi diagrams. The algorithm is
significantly different from previous ones and some of the involved concepts in
it are in the spirit of linear programming and optics. Parallel implementation
is naturally supported since each Voronoi cell can be computed independently of
the other cells. A new combinatorial structure for representing the cells (and
any convex polytope) is described along the way and the computation of the
induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of
some parts; correction of several inaccuracies; improvement of some proofs
and figures; added references; modification of the title; the paper is long
but more than half of it is composed of proofs and references: it is
sufficient to look at pages 5, 7--11 in order to understand the algorith
Computing Volumes and Convex Hulls: Variations and Extensions
Geometric techniques are frequently utilized to analyze and reason about multi-dimensional data. When confronted with large quantities of such data, simplifying geometric statistics or summaries are often a necessary first step. In this thesis, we make contributions to two such fundamental concepts of computational geometry: Klee's Measure and Convex Hulls. The former is concerned with computing the total volume occupied by a set of overlapping rectangular boxes in d-dimensional space, while the latter is concerned with identifying extreme vertices in a multi-dimensional set of points. Both problems are frequently used to analyze optimal solutions to multi-objective optimization problems: a variant of Klee's problem called the Hypervolume Indicator gives a quantitative measure for the quality of a discrete Pareto Optimal set, while the Convex Hull represents the subset of solutions that are optimal with respect to at least one linear optimization function.In the first part of the thesis, we investigate several practical and natural variations of Klee's Measure Problem. We develop a specialized algorithm for a specific case of Klee's problem called the βgroundedβ case, which also solves the Hypervolume Indicator problem faster than any earlier solution for certain dimensions. Next, we extend Klee's problem to an uncertainty setting where the existence of the input boxes are defined probabilistically, and study computing the expectation of the volume. Additionally, we develop efficient algorithms for a discrete version of the problem, where the volume of a box is redefined to be the cardinality of its overlap with a given point set.The second part of the thesis investigates the convex hull problem on uncertain input. To this extent, we examine two probabilistic uncertainty models for point sets. The first model incorporates uncertainty in the existence of the input points. The second model extends the first one by incorporating locational uncertainty. For both models, we study the problem of computing the probability that a given point is contained in the convex hull of the uncertain points. We also consider the problem of finding the most likely convex hull, i.e., the mode of the convex hull random variable