On the performance of peeling algorithms

The peeling of a d-dimensional set of points is usually performed with successive calls to a convex hull algorithm; the optimal worst-case convex hull algorithm, known to have an O(n^˙ Log (n)) execution time, may give an O(n^˙n^˙ Log (n)) to peel all the set; an O(n^˙n) convex hull algorithm, m being the number of extremal points, is shown to peel every set with an O(n-n) time, and proved to be optimal; an implementation of this algorithm is given for planar sets and spatial sets, but the latter give only an approximate O(n^˙n) performance.     

Optimal convex hull algorithms on enhanced meshes

In this paper we propose time-optimal convex hull algorithms for two classes of enhanced meshes. Our first algorithm computes the convex hull of an arbitrary set ofn points in the plane inO (logn) time on a mesh with multiple broadcasting of sizen×n. The second algorithm shows that the same problem can be solved inO (1) time on a reconfigurable mesh of sizen×n. Both algorithms achieve time lower bounds for their respective model of computation.This work was supported by NASA under grant NCCI-99.Additional support by the National Science Foundation under grant CCR-8909996 is gratefully acknowledged.     

An optimal convex hull algorithm in any fixed dimension

We present a deterministic algorithm for computing the convex hull ofn points inE ^d in optimalO(n logn+n ^⌞d/2⌟ ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n ^⌜d/2⌝ ) time. This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.     

A geometric inequality and the complexity of computing volume

The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2 ⁿ . This implies that no polynomial time algorithm can compute the volume of a convex set (given by an oracle) with less than exponential relative error. A lower bound on the complexity of computing width can also be deduced.Dedicated to my teacher Kõváry Károly     

Optimal output-sensitive convex hull algorithms in two and three dimensions

We present simple output-sensitive algorithms that construct the convex hull of a set ofn points in two or three dimensions in worst-case optimalO (n logh) time andO (n) space, whereh denotes the number of vertices of the convex hull. This research was supported by a Killam Predoctoral Fellowship and an NSERC Postgraduate Scholarship.     

Delaunay triangulation and the convex hull ofn points in expected linear time

An algorithm is presented which produces a Delaunay triangulation ofn points in the Euclidean plane in expected linear time. The expected execution time is achieved when the data are (not too far from) uniformly distributed. A modification of the algorithm discussed in the appendix treats most of the non-uniform distributions. The basis of this algorithm is a geographical partitioning of the plane into boxes by the well-known Radix-sort algorithm. This partitioning is also used as a basis for a linear time algorithm for finding the convex hull ofn points in the Euclidean plane.     

Primal Dividing and Dual Pruning: Output-Sensitive Construction of Four-Dimensional Polytopes and Three-Dimensional Voronoi Diagrams

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E ⁴ . Our algorithm runs in time and uses O(n+f) space. This is the first algorithm within a polylogarithmic factor of optimal time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E ³ and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the ``ultimate convex hull algorithm' of Kirkpatrick and Seidel in E ² and also leads to improved output-sensitive results on constructing convex hulls in E ^d for any even constant d > 4. Received August 3, 1995, and in revised form September 19, 1996.     

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^ε) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and ε > 0 is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^ε).     

Applications of random sampling in computational geometry,II

We use random sampling for several new geometric algorithms. The algorithms are Las Vegas, and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.     

Generalizations of Heilbronn’s triangle problem

For given integers d,j≥2 and any positive integers n, distributions of n points in the d-dimensional unit cube [0,1]^d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d,k≥2 the existence of a configuration of n points in [0,1]^d is shown, such that, simultaneously for j=2,…,k, the volume of the convex hull of any j points among these n points is Ω(1/n^{(j−1)/(1+|d−j+1|)}). Moreover, a deterministic algorithm is given achieving this lower bound, provided that d+1≤j≤k.     

A sublogarithmic convex hull algorithm

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.Part of this work was done when the last three authors were at the Department of Computer and Information Science, Linköping University. The research of the second author was supported by the Academy of Finland.     

Minimization of convex functions on the convex hull of a point set

A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R ⁿ is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in R ^m , which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution.     

Applications of a semi-dynamic convex hull algorithm

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.     

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body. (Received 25 January 2000)     

Extreme points for convex mappings ofB n

We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ ⁿ forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials of degree 2 in ℂ ⁿ⁻¹, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the closed convex hull of the set of extreme points here exhibited.     

Counting the onion

Iteratively computing and discarding a set of convex hulls creates a structure known as an “onion.” In this paper, we show that the expected number of layers of a convex hull onion for n uniformly and independently distributed points in a disk is Θ(n^2/3). Additionally, we show that in general the bound is Θ(n^2/(d+1)) for points distributed in a d‐dimensional ball. Further, we show that this bound holds more generally for any fixed, bounded, full‐dimensional shape with a nonempty interior. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body.     

An efficient algorithm for the three-dimensional diameter problem

We explore a new approach for computing the diameter of n points in \Bbb R ³ that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi diagram has linear complexity. We present a deterministic algorithm with O(nlog ² n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the chromatic diameter and all-furthest neighbors in \Bbb R ³ can be found in the same running time. Received February 18, 2000, and in revised form June 27, 2000. Online publication January 17, 2001.     

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.