An improved algorithm for approximating the radii of point sets

Computational Geometry: Theory and Applications, 2006

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in P , and the smallest such set S that contains P . More precisely, for any ε > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 − ε)|S| and we find an axially symmetric convex polygon Q containing C with area |Q | < (1 + ε)|S |. We assume that C is given in a data structure that allows to answer the following two types of query in time TC : given a direction u, find an extreme point of C in direction u, and given a line , find C ∩ . For instance, if C is a convex n-gon and its vertices are given in a sorted array, then TC = O(log n). Then we can find Q in time O(TCε −1/2 + ε −3/2 ) and we can find Q in time O(TC ε −1/2 + ε −3/2 log(ε −1 )). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing restangle and smallest enclosing circle of C in time O(TC ε −1/2 ).

Journal of Algorithms, 1991

Let S be a set consisting of n points in the plane. We consider the problem of finding k points of S that form a "small" set under some given measure, and present efficient algorithms for several natural measures including the diameter and the variance. ej 1991 Academic Press, 1nc.

Discrete &amp; Computational Geometry, 2001

We describe a deterministic algorithm for computing the diameter of a finite set of points in R 3 , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(n log n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their groundbreaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(n log 2 n). The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time.

We present an optimization algorithm to determine a partition of the convex hull of a nite set of ponts in the plane. The partition uses the points as corners of convex polygonal cells, each cell having at most K sides. We minimize the total number of cells that are obtained. The algorithm runs in polynomial time when the points lie on a xed number of (almost) parallel lines.

Fundamentals of Computation Theory, 2019

Let P be a set of n points in the plane. We consider a variation of the classical Erdős-Szekeres problem, presenting efficient algorithms with O(n 3) running time and O(n 2) space complexity that compute: (1) A subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P , (2) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P and its interior contains no element of P , (3) a subset S of P such that the rectilinear convex hull of S has maximum area and its interior contains no element of P , and (4) when each point of P is assigned a weight, positive or negative, a subset S of P that maximizes the total weight of the points in the rectilinear convex hull of S.

Discrete Applied Mathematics, 2001

A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is a convex partition of S such that the number of convex polygons is minimised. In this paper, we will present a polynomial time algorithm to nd a minimum convex partition with respect to a point set S where S is constrained to lie on the boundaries of a xed number of nested convex hulls.

Discrete Applied Mathematics, 1995

We present a random polynomial time algorithm for well-rounding convex bodies K in the following sense: Given K G R" and E > 0, the algorithm, with probability at least 1 -E, computes two simplices A* and A**, where A** is the blow up of A* from its center by a factor of n + 3, such that

2014

The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n ≥ 1, every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n ≤ 3. In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.

Discrete &amp; Computational Geometry, 1993

For all n > d there exist n points in the Euclidean space E d such that not all points are in a hyperplane and all mutual distances are integral. It is proved that the minimum diameter of such integral point sets has an upper bound of 2 cl°gnl°gl°gn.

We consider approximation of diameter of a set S of n points in dimension m. Egecioglu and Kalantari [6] have shown that given any p ∈ S, by computing its farthest in S, say q, and in turn the farthest point of q, say q ′ , we have diam(S) ≤ √ 3 d(q, q ′). Furthermore, iteratively replacing p with an appropriately selected point on the line segment pq, in at most t ≤ n additional iterations, the constant bound factor is improved to c * = 5 − 2 √ 3 ≈ 1.24. Here we prove when m = 2, t = 1. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary m. The algorithms outperform many existing algorithms. On sets of data as large as 1, 000, 000 points, the proposed algorithms compute solutions to within an absolute error of 10 −4 .

Lecture Notes in Computer Science, 2013

Consider a set of d-dimensional points where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position si but only exists with some probability πi, for 0 < πi ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n 3) time for n points in d = 2 dimensions, but it is NP-hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP-hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull.

American Journal of Computational Mathematics, 2013

The algorithms of convex hull have been extensively studied in literature, principally because of their wide range of applications in different areas. This article presents an efficient algorithm to construct approximate convex hull from a set of n points in the plane in time, where k is the approximation error control parameter. The proposed algorithm is suitable for applications preferred to reduce the computation time in exchange of accuracy level such as animation and interaction in computer graphics where rapid and real-time graphics rendering is indispensable.

Numerical Algorithms, 2020

In this paper, we present an efficient improvement of gift wrapping algorithm for determining the convex hull of a finite set of points in R n space, applying the best restricted area technique inspired from the Method of Orienting Curves (this method was used successfully in computational geometry by An and Trang in Numerical Algorithms 59, 347-357 (2012), Optimization 62, 975-988 (2013)). The numerical experiments on the sets of random points in two-and three-dimensional space show that the running time of our algorithm is faster than the gift wrapping algorithm and the newest modified one. Keywords Convex hull • Convex polytope • Gift wrapping algorithm • Orienting curves Mathematics Subject Classification (2010) 51-52 Phan Thanh An and Nam Dũng Hoang equally contributed to this work.

2013

The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best approximation, and it also seems to have potential value in the study of robotics. 2010 Mathematics Subject Classification: 41A65, 52A27. Keywords and phrases: best approximation from convex sets, visible points in convex sets.

2005

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. We use new ideas, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10 −11 )π/2.