Space-efficient planar convex hull algorithms

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. . .

Procedia Computer Science

Computing the convex hull of a set of points is a fundamental issue in many fields, including geometric computing, computer graphics, and computer vision. This problem is computationally challenging, especially when the number of points is past the millions. In this paper, we propose a fast filtering technique that reduces the computational cost for computing a convex hull for a large set of points. The proposed method preprocesses the input set and filters all points inside a four-vertex polygon. The experimental results showed the proposed filtering approach achieved a speedup of up to 77 and 12 times faster than the standard Graham scan and Jarvis march algorithms, respectively.

Computing in Science & Engineering, 2009

International Journal of Computational Geometry & Applications, 1996

We present a parallel algorithm for finding the convex hull of a sorted point set. The algorithm runs in O( log log n) (doubly logarithmic) time using n/ log log n processors on a Common CRCW PRAM. To break the Ω( log n/ log log n) time barrier required to output the convex hull in a contiguous array, we introduce a novel data structure for representing the convex hull. The algorithm is optimal in two respects: (1) the time-processor product of the algorithm, which is linear, cannot be improved, and (2) the running time, which is doubly logarithmic, cannot be improved even by using a linear number of processors. The algorithm demonstrates the power of the “the divide-and-conquer doubly logarithmic paradigm” by presenting a non-trivial extension to situations that previously were known to have only slower algorithms.

Proceedings of the 1993 ACM/SIGAPP symposium on Applied computing states of the art and practice - SAC '93, 1993

IEEE Transactions on Computers, 1988

A bstract-In this paper, we present parallel algorithms to identifv (i.e., detect and enumerate) the extreme points of the convex hull of a set of planar points using a hypercube, pyramid, tree, mesh-of-trees, mesh with reconfigurable bus, EREW PRAM, and a modified AKS network. It is known that the problem of identifying the convex hull for a set of planar points given arbitrarily cannot be solved faster than sorting. For the situation where the input set of n planar points is given ordered (by x-coordinate) one per processor on a machine with 8 (n) processors, we introduce a worst case hypercube algorithm that finishes in 80og n) time, a worst case algorithm for the pyramid, tree, and mesh-of-trees that finishes in 8(log3 n/(log log n)*) time, and a worst case algorithm for the mesh with a reconfigurable bus that finishes in 8(log2 n) time. Notice that for ordered data the sorting bound does not apply. We also show that our 80og n) time hypercube algorithm for ordered data extends to yield an optimal time and processor 80og n) worst case time EREW PRAM algorithm for the case where the set of planar points is distributed arbitrarily one point per processor. We also show that this algorithm can be extended to run in worst case 80og n) time on a modified AKS network, giving the first optimal 80og n) time algorithm for solving the convex hull problem for arbitrary planar input on a fixed degree network.

Bit Numerical Mathematics, 1990

We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn 1/3 divide-and-conquer and uses a simple pointer-based data structure.

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.

Journal of Algorithms, 1997

In this paper we present a truly practical and provably optimal O(n logh) time outputsensitive algorithm for the planar convex hull problem. The basic algorithm is similar to the algorithm presented in Chan, Snoeyink and Yap 2] where the median-nding step is replaced by an approximate median. We analyze two such schemes and show that for both methods, the algorithm runs in expected O(n log h) time. The expected number of comparisons can be made smaller than 5n logh for the upper-hull. We further show that the probability of deviation from expected running time approaches 0 rapidly with increasing values of n and h for any input. Our experiments suggest that this algorithm is a practical alternative to the worstcase O(n log n) algorithms like Graham's and especially faster for small output-sizes. Our approach bears some resemblance to a recent algorithm of Wenger 13] but our analysis is substantially di erent. The planar convex hull problem is perhaps the most studied problem in computational geometry and a large body of literature deals with computing convex hulls. Graham 5] was the rst to present an O(n log n) worst-case time algorithm. This algorithm is optimal as Yao 14] showed that (n log n) is the lower bound of the convex hull problem for the worst-case input. Some simple algorithms have O(n) expected time for known distributions of points such as uniform in a box, normal, etc. The rst output-sensitive algorithm was proposed by Chand and Kapur 3]. The two-dimensional version of their algorithm is known as the rope fence method and was independently reported by Jarvis 6]. The rope fence method takes O(nh) time to compute h extreme edges of the convex hull. Kirkpatrick and Seidel 8] proved an (n logh) lower bound when both input and output sizes are considered, so Yao's lower-bound is a special case when log h 2 (log n). They also proposed an O(n log h) optimal algorithm based on the prune-and-search technique developed by Dyer 4] and Megiddo 9]. However, it has high constants and is considered prohibitively complicated for implementation. Very recently, in 1], two O(n log h) algorithms have been proposed. One uses the linear-time median nding algorithm and the other uses a clever grouping technique. Although the latter algorithm does not have any expensive median-nding step it relies on a sophisticated logarithmic time tangent-nding routine.

Annales Mathematicae et Informaticae, 2007

In this paper, in our modification of Graham scan for determining the convex hull of a finite planar set, we show a restricted area of the examination of points and its advantage. The actual run times of our scan and Graham scan on the set of random points shows that ...