945 research outputs found
Further Results on Arithmetic Filters for Geometric Predicates
An efficient technique to solve precision problems consists in using exact
computations. For geometric predicates, using systematically expensive exact
computations can be avoided by the use of filters. The predicate is first
evaluated using rounding computations, and an error estimation gives a
certificate of the validity of the result. In this note, we studies the
statistical efficiency of filters for cosphericity predicate with an assumption
of regular distribution of the points. We prove that the expected value of the
polynomial corresponding to the in sphere test is greater than epsilon with
probability O(epsilon log 1/epsilon) improving the results of a previous paper
by the same authors.Comment: 7 pages 2 figures presented at the 15th European Workshop Comput.
Geom., 113--116, 1999 improve previous results (in other paper
On Deletion in Delaunay Triangulation
This paper presents how the space of spheres and shelling may be used to
delete a point from a -dimensional triangulation efficiently. In dimension
two, if k is the degree of the deleted vertex, the complexity is O(k log k),
but we notice that this number only applies to low cost operations, while time
consuming computations are only done a linear number of times.
This algorithm may be viewed as a variation of Heller's algorithm, which is
popular in the geographic information system community. Unfortunately, Heller
algorithm is false, as explained in this paper.Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom.,
181--188, 199
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Recent progress in exact geometric computation
AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
TetGen, towards a quality tetrahedral mesh generator
TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation
Voronoi diagrams in the max-norm: algorithms, implementation, and applications
Voronoi diagrams and their numerous variants are well-established objects in computational geometry. They have proven to be extremely useful to tackle geometric problems in various domains such as VLSI CAD, Computer Graphics, Pattern Recognition, Information Retrieval, etc. In this dissertation, we study generalized Voronoi diagram of line segments as motivated by applications in VLSI Computer Aided Design. Our work has three directions: algorithms, implementation, and applications of the line-segment Voronoi diagrams. Our results are as follows: (1) Algorithms for the farthest Voronoi diagram of line segments in the Lp metric, 1 ≤ p ≤ ∞. Our main interest is the L2 (Euclidean) and the L∞ metric. We first introduce the farthest line-segment hull and its Gaussian map to characterize the regions of the farthest line-segment Voronoi diagram at infinity. We then adapt well-known techniques for the construction of a convex hull to compute the farthest line-segment hull, and therefore, the farthest segment Voronoi diagram. Our approach unifies techniques to compute farthest Voronoi diagrams for points and line segments. (2) The implementation of the L∞ Voronoi diagram of line segments in the Computational Geometry Algorithms Library (CGAL). Our software (approximately 17K lines of C++ code) is built on top of the existing CGAL package on the L2 (Euclidean) Voronoi diagram of line segments. It is accepted and integrated in the upcoming version of the library CGAL-4.7 and will be released in september 2015. We performed the implementation in the L∞ metric because we target applications in VLSI design, where shapes are predominantly rectilinear, and the L∞ segment Voronoi diagram is computationally simpler. (3) The application of our Voronoi software to tackle proximity-related problems in VLSI pattern analysis. In particular, we use the Voronoi diagram to identify critical locations in patterns of VLSI layout, which can be faulty during the printing process of a VLSI chip. We present experiments involving layout pieces that were provided by IBM Research, Zurich. Our Voronoi-based method was able to find all problematic locations in the provided layout pieces, very fast, and without any manual intervention
TPC-H Analyzed: Hidden Messages and Lessons Learned from an Influential Benchmark
The TPC-D benchmark was developed almost 20 years ago, and even though its current existence as TPC H could be considered superseded by TPC-DS, one can still learn from it. We focus on the technical level, summarizing the challenges posed by the TPC-H workload as we now understand them, which w
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