Results 11  20
of
20
Algorithms for Manifolds and Simplicial Complexes in Euclidean 3Space
, 1996
"... A new approach to analyze simplicial complexes in Euclidean 3space R 3 is described. First, methods from topology are used to analyze triangulated 3manifolds in R 3 . Then it is shown that these methods can in fact be applied to arbitrary simplicial complexes in R 3 after (simulating) the pro ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
A new approach to analyze simplicial complexes in Euclidean 3space R 3 is described. First, methods from topology are used to analyze triangulated 3manifolds in R 3 . Then it is shown that these methods can in fact be applied to arbitrary simplicial complexes in R 3 after (simulating) the process of thickening a complex to a 3manifold homotopic to it. As a consequence considerable structural information about the complex can be determined and certain discrete problems solved as well. For example, it is shown how to determine the homology groups, as well as concrete representations of their generators, for a given complex K. Further, given a 1cycle or 2cycle in K it is shown how to express this cycle in terms of the generators of a homology group, which solves the problem of classifying cycles up to their homology class. An application is to the classification of simplicial maps up to their actions on homology groups. Recent developments in analyzing molecular structures thr...
SPHERE RECOGNITION LIES IN NP
"... Abstract. We prove that the threesphere recognition problem lies in the complexity class NP. Our work relies on Thompson’s original proof that the problem is decidable [Math. Res. Let., 1994], Casson’s version of her algorithm, and recent results of Agol, Hass, and Thurston [STOC, 2002]. 1. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that the threesphere recognition problem lies in the complexity class NP. Our work relies on Thompson’s original proof that the problem is decidable [Math. Res. Let., 1994], Casson’s version of her algorithm, and recent results of Agol, Hass, and Thurston [STOC, 2002]. 1.
Curvature Aware Fundamental Cycles
"... We present a graph algorithm to find fundamental cycles aligned with the principal curvature directions of a surface. Specifically, we use the treecotree decomposition of graphs embedded in manifolds, guided with edge weights, in order to produce these cycles. Our algorithm is very quick compared t ..."
Abstract
 Add to MetaCart
(Show Context)
We present a graph algorithm to find fundamental cycles aligned with the principal curvature directions of a surface. Specifically, we use the treecotree decomposition of graphs embedded in manifolds, guided with edge weights, in order to produce these cycles. Our algorithm is very quick compared to existing methods, with a worst case running time of O(nlogn + gn) where n is the number of faces and g is the surface genus. Further, its flexibility to accommodate different weighting functions and to handle boundaries may be used to produce cycles suitable for a variety of applications and models. Categories and Subject Descriptors (according to ACM CCS): Generation—Line and curve generation
UNO is an Equal Opportunity / Affirmative Action Institution. Interacting With Surfaces In Four Dimensions Using Computer Graphics
, 1993
"... Highspeed, highquality computer graphics enables a user to interactively manipulate surfaces in four dimensions and see them on a computer screen. Surfaces in 4space exhibit properties that are prohibited in 3space. For example, nonorientable surfaces may be free of selfintersections in 4spac ..."
Abstract
 Add to MetaCart
Highspeed, highquality computer graphics enables a user to interactively manipulate surfaces in four dimensions and see them on a computer screen. Surfaces in 4space exhibit properties that are prohibited in 3space. For example, nonorientable surfaces may be free of selfintersections in 4space. Can a user actually make sense of the shapes of surfaces in a largerdimensional space than the familiar 3D world? Experiment shows he can. A prototype system called Fourphront, running on the graphics engine PixelPlanes 5 (developed at UNCChapel Hill) allows the user to perform interactive algorithms in order to determine some of the properties of a surface in 4space. This dissertation describes solutions to several problems associated with manipulating surfaces in 4space. It shows how the user in 3space can control a surface in 4space in an intuitive way. It describes how to extend the common illumination models to large numbers of dimensions. And it presents visualization techniques for conveying 4D depth information, for calculating intersections, and for calculating silhouettes. iii Acknowledgements Many minds, many hands, and many pockets contributed to this research. My sincerest thanks go to Brice Tebbs and Greg Turk for teaching me about computer graphics and showing me how to program PixelPlanes; Howard Good (callback functions, onesided polygon picking), Marc Olano (conditional executes,
Efficient Homology Computations on Multicore and Manycore Systems
"... Abstract—Homology computations form an important step in topological data analysis that helps to identify connected components, holes, and voids in multidimensional data. Our work focuses on algorithms for homology computations of large simplicial complexes on multicore machines and on GPUs. This p ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract—Homology computations form an important step in topological data analysis that helps to identify connected components, holes, and voids in multidimensional data. Our work focuses on algorithms for homology computations of large simplicial complexes on multicore machines and on GPUs. This paper presents two parallel algorithms to compute homology. A core component of both algorithms is the algebraic reduction of a cell with respect to one of its faces while preserving the homology of the original simplicial complex. The first algorithm is a parallel version of an existing sequential implementation using OpenMP. The algorithm processes and reduces cells within each partition of the complex in parallel while minimizing sequential reductions on the partition boundaries. Cache misses are reduced by ensuring data locality for data in the same partition. We observe a linear speedup on algebraic reductions and an overall speedup of up to 4.9 × with 16 cores over sequential reductions. The second algorithm is based on a novel approach for homology computations on manycore/GPU architectures. This GPU algorithm is memory efficient and capable of extremely fast computation of homology for simplicial complexes with millions of simplices. We observe up to 40 × speedup in runtime over sequential reductions and up to 4.5 × speedup over REDHOM library, which includes the sequential algebraic reductions together with other advanced homology engines supported in the software. I.
Algorithms for Modeling and Measuring Proteins
, 1995
"... In this paper we investigate efficient algorithms for computing the volume and surface area of protein molecules and for graphically displaying the molecules in real time. Protein molecules are modeled by sets of overlapping spheres in R³. We summarize and critique three papers in the field ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper we investigate efficient algorithms for computing the volume and surface area of protein molecules and for graphically displaying the molecules in real time. Protein molecules are modeled by sets of overlapping spheres in R&sup3;. We summarize and critique three papers in the field, and we add several new contributions of our own. First, we present and discuss a new randomized algorithm for computing volumes of proteins. This algorithm is faster than any previouslyknown algorithm. We also suggest several extensions to this research, including ideas for detecting errors in the xray crystallography data used as input. Finally, we propose applying a recent machinelearning result [BCGS95] to determine the tolerance of errors in the data.
Z2HOMOLOGY OF WEAK (n − 2)FACELESS nPSEUDOMANIFOLDS MAY BE COMPUTED IN O(n) TIME
"... Abstract. We consider the class of weak (n−2)faceless npseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with Z2 coefficients may be computed in time O(n) and the Z2 homology generators in time O(nm) where n denotes the cardinality of the npseu ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We consider the class of weak (n−2)faceless npseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with Z2 coefficients may be computed in time O(n) and the Z2 homology generators in time O(nm) where n denotes the cardinality of the npseudomanifold on input and m is the number of homology generators. 1.
A NEW APPROACH TO CRUSHING 3MANIFOLD TRIANGULATIONS
"... Abstract. The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3manifold topology: it enabled the first practical implementations of 3sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in stateoftheart algorithms ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3manifold topology: it enabled the first practical implementations of 3sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in stateoftheart algorithms for unknot recognition and testing for essential surfaces. Although the crushing operation will always reduce the size of a triangulation, it might alter its topology, and so it requires a careful theoretical analysis for the settings in which it is used. The aim of this short paper is to make the crushing operation more accessible to practitioners, and easier to generalise to new settings. When the crushing operation was first introduced, the analysis was powerful but extremely complex. Here we give a new treatment that reduces the crushing process to a sequential combination of three “atomic ” operations on a cell decomposition, all of which are simple to analyse. As an application, we generalise the crushing operation to the setting of nonorientable 3manifolds, where we obtain a new practical and robust algorithm for nonorientable prime decomposition. We also apply our crushing techniques to the study of nonorientable minimal triangulations. 1.