Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

Three techniques in computational geometry are explored: Scaling solves a problem by viewing it at increasing levels of numerical precision; activation is a restricted type of update operation, useful in sweep algorithms; the Cartesian tree is a data structure for problems involving maximums and minimums. These techniques solve the minimum spanning tree problem in Rk1 and Rk

Scaling and related techniques for geometry problems

We give a survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations. We also provide a rather comprehensive and up-to-date bibliography which covers more than three hundred items.

Steiner tree problems

We propose a uniform and general framework for defining and dealing with Voronoi diagrams. In this framework a Voronoi diagram is a partition of a domainD induced by a finite number of real valued functions onD. Valuable insight can be gained when one considers how these real valued functions partitionD ×R. With this view it turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.

https://link.springer.com/content/pdf/10.1007/BF02187681.pdf

Voronoi diagrams and arrangements

LetX be a given set ofn circular and straight line segments in the plane where two segments may interest only at their endpoints. We introduce a new technique that computes the Voronoi diagram ofX inO(n logn) time. This result improves on several previous algorithms for special cases of the problem. The new algorithm is relatively simple, an important factor for the numerous practical applications of the Voronoi diagram.

/pdf/ano-n-logn-algorithm-for-the-voronoi-diagram-of-a-set-of-58qm4qnqgr.pdf

AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments

In this paper we study the problem of scheduling the read/write head movement to handle a batch of $nI/O$ requests in a 2-dimensional secondary storage device in minimum time. Two models of storage systems are assumed in which the access time of a record (being proportional to the “distance” between the position of the record and that of the read/write head) is measured in terms of $L_1 $ and $L_\infty $ metrics, respectively. The scheduling problem, referred to as the Open Path Problem (OPP), is equivalent to finding a shortest Hamiltonian path with a specified end point in a complete graph with n vertices. We first show in this paper that there exists a natural isometry between the $L_1 $ and $L_\infty $ metrics. Consequently, the existence of a polynomial time algorithm for the OPP in one metric implies the existence of a polynomial time algorithm for the same problem in the other metric. Based on a result by Garey, Graham and Johnson, it is easy to show that the OPP in $L_1 $ (hence in $L_\infty $) me...

C. K. Wong

Papers

Voronoi Diagrams in L1 (L∞) Metrics with 2-Dimensional Storage Applications