This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

We consider the problem of bounding the complexity of the k-th level in an arrangement of n curves or surfaces, a problem dual to, and extending, the well-known k-set problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity of the k-th level in an arrangement of n lines. (b) We derive an improved version of Lov'asz Lemma in any dimension, and use it to prove a new bound, O(n 2

For any n, k, n 2k > 0, we construct a set of n points in the plane with ne p log k k-sets. This improves the bounds of Erd}os, Lovasz, et al. As a consequence, we also improve the lower bound of Edelsbrunner for the number of halving hyperplanes in higher dimensions. 1 Introduction For a set P of n points in the d-dimensional space, a k-set is subset P 0 P such that P 0 = P \H for some half-space H, and jP 0 j = k. The problem is to determine the maximum number of k-sets of an n-point set in the d-dimensional space. Even in the most studied two dimensional case, we are very far from the solution, and in higher dimensions even much less is known. The rst results in the two dimensional case are due to Erd}os, Lovasz, Simmons and Straus [L71], [ELSS73]. They established an upper bound O(n p k), and a lower bound (n log k). Despite great interest in this problem [W86], [E87], [S91], [EVW97], [AACS98], partly due to its importance in the analysis of geometric alg...

Let be a set of points in in general position, i.e., no points on a common-flat,. A-set of is a set of points in that can be separated from by a hyperplane. A-facet of is an oriented-simplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar point set and is even, a halving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number! "$ # of-sets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number of-sets we show that

Leonard J. Schulman College of Computing Georgia Institute of Technology Atlanta GA 30332-0280 ABSTRACT We address the problem of partitioning a set of n points into clusters, so as to minimize the sum, over all intracluster pairs of points, of the cost associated with each pair. We obtain a randomized approximation algorithm for this problem, for the cost functions ` 2 2 ; `1 and `2 , as well as any cost function isometrically embeddable in ` 2 2 .

We prove that the maximum number of k-sets in a set S of n points in IR 3 is O(nk 3=2 ). This improves substantially the previous best known upper bound of O(nk 5=3 ) (see [7] and [1]).

A geometric hypergraph H is a collection of i-dimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)-tuples of a vertex set V in general position in d-space. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the k-set problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (i-simplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges...

Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

We prove an O(nk 1=3 ) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general. 1 Introduction The problem of determining the optimum asymptotic bound on the number of k-sets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer k n, a k-set is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains elusive even in ! ...