gScan: Accelerating Graham Scan on the GPU

This paper presents a fast implementation of the Graham scan on the GPU. The proposed algorithm is composed of two stages: (1) two rounds of preprocessing performed on the GPU and (2) the finalization of finding the convex hull on the CPU. We first discard the interior points that locate inside a quadrilateral formed by four extreme points, sort the remaining points according to the angles, and then divide them into the left and the right regions. For each region, we perform a second round of filtering using the proposed preprocessing approach to discard the interior points in further. We finally obtain the expected convex hull by calculating the convex hull of the remaining points on the CPU. We directly employ the parallel sorting, reduction, and partitioning provided by the library Thrust for better efficiency and simplicity. Experimental results show that our implementation achieves 6x ~ 7x speedups over the Qhull implementation for 20M points.Comment: arXiv admin note: text overlap with arXiv:1508.0548

CudaChain: A Practical GPU-accelerated 2D Convex Hull Algorithm

This paper presents a practical GPU-accelerated convex hull algorithm and a novel Sorting-based Preprocessing Approach (SPA) for planar point sets. The proposed algorithm consists of two stages: (1) two rounds of preprocessing performed on the GPU and (2) the finalization of calculating the expected convex hull on the CPU. We first discard the interior points that locate inside a quadrilateral formed by four extreme points, and then distribute the remaining points into several (typically four) sub regions. For each subset of points, we first sort them in parallel, then perform the second round of discarding using SPA, and finally form a simple chain for the current remaining points. A simple polygon can be easily generated by directly connecting all the chains in sub regions. We at last obtain the expected convex hull of the input points by calculating the convex hull of the simple polygon. We use the library Thrust to realize the parallel sorting, reduction, and partitioning for better efficiency and simplicity. Experimental results show that our algorithm achieves 5x ~ 6x speedups over the Qhull implementation for 20M points. Thus, this algorithm is competitive in practical applications for its simplicity and satisfied efficiency

A Novel Implementation of QuickHull Algorithm on the GPU

We present a novel GPU-accelerated implementation of the QuickHull algorihtm for calculating convex hulls of planar point sets. We also describe a practical solution to demonstrate how to efficiently implement a typical Divide-and-Conquer algorithm on the GPU. We highly utilize the parallel primitives provided by the library Thrust such as the parallel segmented scan for better efficiency and simplicity. To evaluate the performance of our implementation, we carry out four groups of experimental tests using two groups of point sets in two modes on the GPU K20c. Experimental results indicate that: our implementation can achieve the speedups of up to 10.98x over the state-of-art CPU-based convex hull implementation Qhull [16]. In addition, our implementation can find the convex hull of 20M points in about 0.2 seconds.Comment: 10 pages, 5 figure

Plane-Sweep Incremental Algorithm: Computing Delaunay Tessellations of Large Datasets

We present the plane-sweep incremental algorithm, a hybrid approach for computing Delaunay tessellations of large point sets whose size exceeds the computer's main memory. This approach unites the simplicity of the incremental algorithms with the comparatively low memory requirements of plane-sweep approaches. The procedure is to first sort the point set along the first principal component and then to sequentially insert the points into the tessellation, essentially simulating a sweeping plane. The part of the tessellation that has been passed by the sweeping plane can be evicted from memory and written to disk, limiting the memory requirement of the program to the "thickness" of the data set along its first principal component. We implemented the algorithm and used it to compute the Delaunay tessellation and Voronoi partition of the Sloan Digital Sky Survey magnitude space consisting of 287 million points.Comment: Technical Report from 200

Beneath-and-Beyond revisited

It is shown how the Beneath-and-Beyond algorithm can be used to yield another proof of the equivalence of V- and H-representations of convex polytopes. In this sense this paper serves as the sketch of an introduction to polytope theory with a focus on algorithmic aspects. Moreover, computational results are presented to compare Beneath-and-Beyond to other convex hull implementations.Comment: 21 pages, 2 figures; v2: added the bibliography which was erroneously omitted in v

A Note on Experiments and Software For Multidimensional Order Statistics

In this note we describe experiments on an implementation of two methods proposed in the literature for computing regions that correspond to a notion of order statistics for multidimensional data. Our implementation, which works for any dimension greater than one, is the only that we know of to be publicly available. Experiments run using the software confirm that half-space peeling generally gives better results than directly peeling convex hulls, but at a computational cost

A geometric method of sector decomposition

We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method provides a deterministic algorithm and never falls into an infinite loop. The number of resulting sectors depends on the algorithm of triangulation. Our test implementation shows smaller number of sectors comparing with other existing methods with iterations.Comment: 17 pages, 2 eps figure

Optimal Compression of a Polyline with Segments and Arcs

This paper describes an efficient approach to constructing a resultant polyline with a minimum number of segments and arcs. While fitting an arc can be done with complexity O(1) (see [1] and [2]), the main complexity is in checking that the resultant arc is within the specified tolerance. There are additional tests to check for the ends and for changes in direction (see [3, section 3] and [4, sections II.C and II.D]). However, the most important part in reducing complexity is the ability to subdivide the polyline in order to limit the number of arc fittings [2]. The approach described in this paper finds a compressed polyline with a minimum number of segments and arcs.Comment: 40 pages, 34 figures, 3 table

Computing convex hulls and counting integer points with polymake

The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb".Comment: major revision: experiments repeated with new software versions; new experiments and additional software tested; new title; 38 pages including appendix, 10 figures, 9 table

Experience Report: Formal Methods in Material Science

Increased demands in the field of scientific computation require that algorithms be more efficiently implemented. Maintaining correctness in addition to efficiency is a challenge that software engineers in the field have to face. In this report we share our first impressions and experiences on the applicability of formal methods to such design challenges arising in the development of scientific computation software in the field of material science. We investigated two different algorithms, one for load distribution and one for the computation of convex hulls, and demonstrate how formal methods have been used to discover counterexamples to the correctness of the existing implementations as well as proving the correctness of a revised algorithm. The techniques employed for this include SMT solvers, and automatic and interactive verification tools.Comment: experience repor