Space subdivision to speed-up convex hull construction in E3

Abstract

a b s t r a c tConvex hulls are fundamental geometric tools used in a number of algorithms. This paper presents a fast, simple to implement and robust Smart Convex Hull (S-CH) algorithm for computing the convex hull of a set of points in E 3 . This algorithm is based on "spherical" space subdivision. The main idea of the S-CH algorithm is to eliminate as many input points as possible before the convex hull construction. The experimental results show that only a very small number of points are used for the final convex hull calculation. Experiments made also proved that the proposed S-CH algorithm achieves a better time complexity in comparison with other algorithms in E 3 .

LPCN: Least polar-angle connected node algorithm to find a polygon hull in a connected euclidean graph

“…The time complexity of this algorithm is not given, but it depends on the number of samples and features, the population size (input parameter k), the number of iterations, the number of vertices of the convex hull, and on the distribution of the samples in the dataset. 9) S-CH algorithm [38]: The Smart Convex Hull (S-CH) algorithm starts by applying a space subdivision method in order to eliminate a maximum of the initial points. Then, it determines the convex hull over the remaining points by applying, for instance, any standard convex hull algorithm.…”

“…Figure 2(c)). In the same way, we define the least polar angle point with respect to E and C which is point H. The last point defined [17] O(n log n) -2D Jarvis [18] O(nh) -2D Quick-hull [6] O(n 2 ) -2D Incremental [20] O(n (d+1)/2 ) -Multidimensional TORCH [16] O(n log n) -2D NICP [40] O(n + n log n) -2D Requires the Quick-hull algorithm Mei [27] O(n log n) -2D GPU-accelerated convex hull algorithm Ruano et al [35] [not given] -Multidimensional Complexity depends on many parameters S-CH [38] O(n log n) -3D Requires a standard convex hull algorithm Split and Merge [14] O(nh) Not Optimal 2D Starts from the convex hull of points Alpha-Shape [10] O(n log n) Not Optimal 2D Depends on a parameter α PBE [8] O(n) Not Optimal 2D KNN [30] O(nh 2 ) Not Optimal 2D CM [32] O(n log n + rn) Not Optimal Multidimensional Braune et al [7] O(n log h) Not Optimal 2D Starts from the convex hull of points Gheibi et al [15] O(n log n) Not Optimal 2D Starts from the convex hull of points EC-Shape [29] O(n log n) Not Optimal 2D Requires Delaunay Triangulation (DT) Gift Opening [34] O(n) Not Optimal 2D Starts from the convex hull of points RGH [21] O after H is A which was the starting point for the first iteration. Hence, the convex hull is found (cf.…”

CudaCHPre2D: A straightforward preprocessing approach for accelerating 2D convex hull computations on the GPU

“…The simplest case in two dimensions is to form a convex quadrilateral using four extreme points with min or max x or y coordinates and then check each point to determine whether it locates in the quadrilateral . Recently, several other strategies are also introduced to efficiently discard interior points …”

“…Compared to the related preprocessing algorithms, the most important feature of the proposed preprocessing algorithm CudaCHPre2D is that it is quite straightforward and easy to implement in practice. Note that a short conference paper describing the proposed preprocessing algorithm, CudaCHPre2D, has been published in the 26th Euromicro International Conference on Parallel, Distributed and Network‐based Processing (PDP).…”