A Preprocessing Technique for Fast Convex Hull Computation

Journal of Computational and Applied Mathematics, 2019

This work presents an optimization technique that reduces the computational cost for building the Convex Hull from a set of points. The proposed method pre-processes the input set, filtering all points inside an eight-vertex polygon in O(n) time and returns a reduced set of candidate points, ordered and distributed across four priority queues. Experimental results show that for a normal distribution of points in two-dimensional space, the filtering approach in conjunction with the Graham scan is up to 10× faster than the qhull library, and between 1.7× to 10× faster than the Convex Hull methods available in the CGAL library. Results on the worst case scenario (when all points lie in the circumference) show that a slight random radial displacement of the points make this method the fastest one. Moreover, when increasing the magnitude of this displacement, the performance of the proposed method scales at a faster rate than the other methods. In terms of memory efficiency, the proposed implementation manages to use from 3× to 6× less memory than the other methods. The reason behind this memory improvement is because the proposed method stores indices of the input arrays, avoiding duplicates of the original floating points. Furthermore, the approach extends the problem size up to n ≤ 2 40 by employing 5-byte indices (instead of 8bytes) when n ≥ 2 32. The optimization technique presented in this work has shown to be significantly useful in accelerating the computation of the Convex Hull, and it is not limited just to the combination with the Graham scan, but it can also be used in conjunction with other Convex Hull algorithms.

Theoretical Computer Science, 2004

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

Cornell University - arXiv, 2022

Resumen-The convex hull is a fundamental geometrical structure for many applications where groups of points must be enclosed or represented by a convex polygon. Although efficient sequential convex hull algorithms exist, and are constantly being used in applications, their computation time is often considered an issue for time-sensitive tasks such as real-time collision detection, clustering or image processing for virtual reality, among others, where fast response times are required. In this work we propose a parallel GPU-based adaptation of heaphull, which is a state of the art CPU algorithm that computes the convex hull by first doing a efficient filtering stage followed by the actual convex hull computation. More specifically, this work parallelizes the filtering stage, adapting it to the GPU programming model as a series of parallel reductions. Experimental evaluation shows that the proposed implementation significantly improves the performance of the convex hull computation, reaching up to 4× of speedup over the sequential CPU-based heaphull and between 3× ∼ 4× over existing GPU based approaches.

An in-place algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others. . .

Pattern Recognition, 1983

In this paper, a linear time algorithm is described for finding the convex hull of a simple (i.e. nonself intersecting) polygon. Convex hull algorithm Computational complexity Simple polygon Convex polygon Ordered crossing polygon

Innovative Systems Design and Engineering, 2011

Virtual reality techniques have proved their importance in almost every field of knowledge, particularly in medical and architecture. Convex hull is an application of virtual reality which is used to draw the boundary of some object inside an image. In this paper a hybrid method is proposed to compute convex hull. The method is based on two already existing convex hull algorithms i.e. quick hull and grahams Scan algorithm. The proposed technique is an attempt to remove the deficiencies in the two above mentioned techniques of the convex hull.

Proceedings of the …, 2011

General purpose programming on the graphics processing units(GPGPU) has received a lot of attention in the parallel computing community as it promises to offer a large computational power at a very low price. GPGPU is best suited for regular data parallel algorithms. They are not directly amenable for algorithms which have irregular data access patterns such as convex hull, list ranking etc. In this paper, we present a GPU-optimized implementation for finding the convex hull of a two dimensional point set. Our implementation tries to minimize the impact of irregular data access patterns. Our implementation can find the convex hull of 10 million random points in less than 0.2 seconds and achieves a speedup of up to 14 over the standard sequential CPU implementation. We also discuss some of the practical issues relating to the implementation of convex hull algorithms on massively multithreaded architectures like that of the GPU.

2019

We study the following range searching problem: Preprocess a set P of n points in the plane with respect to a set O of k orientations for a constant, in the plane so that given an O-oriented convex polygon Q, the convex hull of P∩ Q can be computed efficiently, where an O-oriented polygon is a polygon whose edges have orientations in O. We present a data structure with O(nk^3log^2n) space and O(nk^3log^2n) construction time, and an O(h+slog^2 n)-time query algorithm for any query O-oriented convex s-gon Q, where h is the complexity of the convex hull. Also, we can compute the perimeter or area of the convex hull of P∩ Q in O(slog^2n) time using the data structure.

Computing in Science & Engineering, 2009

Highlights • A novel incremental convex hull algorithm is proposed with O (h 2) time complexity. • The convex hull is maintained as a set of line segments. • The proposed algorithm is faster than the existing algorithm when h < SQRT (n).