On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of some parts; correction of several inaccuracies; improvement of some proofs and figures; added references; modification of the title; the paper is long but more than half of it is composed of proofs and references: it is sufficient to look at pages 5, 7--11 in order to understand the algorith

Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA09

On the Structure of Higher Order Voronoi Cells

The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given $k$-element subset of the set of sites consists of the $k$ closest sites. We study the structure of the $k$-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi cells for four points.Comment: Minor correction