We discuss the following problem: given n points in the plane (the "sites"), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem, and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the tgpology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror-image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams. O. IntroductionOne of the ft,ndamental data structures of computational geometry is the Voronoi diagram. This diagram arises from consideration of the following natural problem. Let n points in the plane be given, called sites. We wish to preprocess them into a data structure, so that given a new query point q, we can efficiently locate the nearest neighbor of q among the sites. The n sites in fact partition the plane into a collection of n regions, each associated with one of the sites. If region P is associated with site p, then P is the locus of all points in the plane closer to p than to any of the other n --1 sites.This partition is known as the Voronoi diagram (or the Dirichlet, or Thiessen, tesselation) determined by the given sites.The closest site problem is can therefore be solved by constructing the Voronoi diagram, and then locating the query point in it. Using the currently best available algorithms, the Voronoi diagram of n points can be computed in O(nlgn) time and stored in O(n) space; these bounds have been shown to be optimal in the worst case [Sh]. Once we have the Voronoi diagram, we can construct in linear further time a structure with which we can do point location in a planar subdivision in O(lg n) time [Kil].The work of Jorge Stolfi. who is on leave from the University of Sit Paulo (Sit Paulo, Brazil) was partially s'upponed by a grant from Conselho Nacional de Desenvolvimento Cienlffioo e Tecnol6gico (CNPq).Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.© 1983 ACM 0-89791-099-0/83/004/0221 $00.75Shamos [Sh] first pointed out that ...

Point location, often known in graphics as "hit detection," is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594-606] for locating a point in 5e based on separating chains. The new data structure, called a layered dag, can be built in O(m) time, uses O(m) storage, and makes possible point location in O(log rn) time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.

No abstract

No abstract

Abstract.This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen among n points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocess n points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimal O(k+logn) time algorithm for answering such queries, where k is the number of points to be reported. The algorithm requires O(n) space and O(n log n) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

No abstract

Given n points in the plane, we present algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points. Our algorithms work in linear space and time O(knlgn-I-nlg 2 n).For the special case k : 3 we give O(nlgn) algorithms for these problems. Several related issues are discussed.

No abstract