Farthest-polygon Voronoi diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(nlog³n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k−1 connected components, but if one component is bounded, then it is equal to the entire region.     

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements.Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site.The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane.The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.     

Randomized incremental construction of simple abstract Voronoi diagrams in 3-space

We introduce the simple abstract Voronoi diagram in 3-space as an abstraction of the usual Voronoi diagram. We show that the 3-dimensional simple abstract Voronoi diagram of n sites can be computed in O(n²) expected time using O(n²) expected space by a randomized algorithm. The algorithm is based on the randomized incremental construction technique of Clarkson and Shor (1989). We apply the algorithm to some concrete types of such diagrams: power diagrams, diagrams under ellipsoid convex distance functions, and diagrams under the Hausdorff distance for sites that are parallel segments all having the same length.     

A “retraction” method for planning the motion of a disc

A new approach to certain motion-planning problems in robotics is introduced. This approach is based on the use of a generalized Voronoi diagram, and reduces the search for a collision-free continuous motion to a search for a connected path along the edges of such a diagram. This approach yields an O(n log n) algorithm for planning an obstacle-avoiding motion of a single circular disc amid polygonal obstacles. Later papers will show that extensions of the approach can solve other motion-planning problems, including those of moving a straight line segment or several coordinated discs in the plane amid polygonal obstacles.     

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.     

Facility Location on a Polyhedral Surface

Given an orientable genus-0 polyhedral surface defined by n triangles, and a set of m point sites} on it, we would like to identify its 1-center, i.e., the location on the surface that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the surface. To compute the 1-center, we compute the furthest-site Voronoi diagram of the sites on the polyhedral surface. We show that the diagram has maximum combinatorial complexity (m n ²), and present an algorithm that computes the diagram in O(m n ²log mlog n) expected time. The 1-center can then be identified in time linear in the size of the diagram.     

A generalization of Berwick's unit algorithm

In 1913 W. E. H. Berwick published an algorithm for finding the fundamental unit of a cubic field with negative discriminant. His method relied heavily on the geometry of such fields and was less efficient than the well-known algorithm of Voronoi. In the present paper we show that the use of cubic geometry is not necessary and also that Berwick's method can be generalized. We present a periodic algorithm for finding a maximal set of independent units in an arbitrary algebraic number field.     

Bregman Voronoi Diagrams

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.     

A linear-time algorithm for computing the voronoi diagram of a convex polygon

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in (n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.This research began while the first and fourth authors were visiting the Mathematical Sciences Research Institute in Berkeley, California. Work by the fourth author was supported in part by NSF Grant No. 8120790.     

A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams

We describe an algorithm for the construction of discretized Voronoi diagrams on a CPU within the context of a large scale numerical fluid simulation. The Discrete Voronoi Chain (DVC) algorithm is fast, flexible and robust. The algorithm stores the Voronoi diagram on a grid or lattice that may be structured or unstructured. The Voronoi diagram is computed by alternatively updating two lists of grid cells per particle to propagate a growth boundary of cells from the particle locations. Distance tests only occur when growth boundaries from different particles collide with each other, hence the number of distance tests is effectively minimized. We give some empirical results for two and three dimensions. The method generalizes to any dimension in a straight forward manner. The distance tests can be based any metric.     

Uniformly Distributed Circular Porous Pattern Generation on Surface for 3D Printing

We present an algorithm for uniformly distributed circular porous pattern generation on surface for three-dimensional (3D) printing using a phase-field model. The algorithm is based on the narrow band domain method for the nonlocal Cahn–Hilliard (CH) equation on surfaces. Surfaces are embedded in 3D grid and the narrow band domain is defined as the neighborhood of surface. It allows one can perform numerical computation using the standard discrete Laplacian in 3D instead of the discrete surface Laplacian. For complex surfaces, we reconstruct them from point cloud data and represent them as the zero-level set of their discrete signed distance functions. Using the proposed algorithm, we can generate uniformly distributed circular porous patterns on surfaces in 3D and print the resulting 3D models. Furthermore, we provide the test of accuracy and energy stability of the proposed method.     

Finding a Guard that Sees Most and a Shop that Sells Most

We present a near-quadratic time algorithm that computes a point inside a simple polygon P in the plane having approximately the largest visibility polygon inside P, and a near-linear time algorithm for finding the point that will have approximately the largest Voronoi region when added to an n-point set in the plane. We apply the same technique to find the translation that approximately maximizes the area of intersection of two polygonal regions in near-quadratic time, and the rigid motion doing so in near-cubic time.     

The Voronoi Diagram of Curved Objects

Voronoi diagrams of curved objects can show certain phenomena that are often considered artifacts: The Voronoi diagram is not connected; there are pairs of objects whose bisector is a closed curve or even a two-dimensional object; there are Voronoi edges between different parts of the same site (so-called self-Voronoi-edges); these self-Voronoi-edges may end at seemingly arbitrary points not on a site, and, in the case of a circular site, even degenerate to a single isolated point. We give a systematic study of these phenomena, characterizing their differential-geometric and topological properties. We show how a given set of curves can be refined such that the resulting curves define a “well-behaved” Voronoi diagram. We also give a randomized incremental algorithm to compute this diagram. The expected running time of this algorithm is O(n log n).     

On the construction of abstract voronoi diagrams

We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7]. This work was supported by the DFG, Me 620/6, and ESPRIT P3075 ALCOM. A preliminary version of this paper has been presented at STACS '90, Rouen, France.     

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matou?ek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et?al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.     

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations

In this first installment of a two-part paper, the underlying theory for an algorithm that computes the Voronoi diagram and medial axis of a planar domain bounded by free-form (polynomial or rational) curve segments is presented. An incremental approach to computing the Voronoi diagram is used, wherein a single boundary segment is added to an existing boundary-segment set at each step. The introduction of each new segment entails modifying the Voronoi regions of the existing boundary segments, and constructing the Voronoi region of the new segment. We accomplish this by (i) computing the bisector of the new segment with each of the current boundary segments; (ii) updating the Voronoi regions of the current boundary segments by partitioning them with these bisectors; and (iii) constructing the Voronoi region of the new segment as a union of regions obtained from the partitioning in (ii). When all boundary segments are included, and their Voronoi regions have been constructed, the Voronoi diagram of the boundary is obtained as the union of the Voronoi polygons for each boundary segment. To construct the medial axis of a planar domain, we first compute the Voronoi diagram of its boundary. The medial axis is then obtained from the Voronoi diagram by (i) removing certain edges of the Voronoi diagram that do not belong to the medial axis, and (ii) adding certain edges that do belong to the medial axis but are absent from the Voronoi diagram; unambiguous characterizations for edges in both these categories are given. Details of algorithms based on this theory are deferred to the second installment of this two-part paper.     

The Voronoi diagram for graphs and its application in the Sickle Cell Disease research

We propose using a graph-theoretic data structure, the Voronoi diagram for graphs, for analyzing the structure of biological networks. The Voronoi diagram for graphs (VDG) offers an efficient way to cluster the members of a network based on their distance to a subset of input nodes. We study the distance-based decomposition of the human erythrocyte interactome provided by VDG, seeking to elucidate the influence of sickle cell anemia on the function of the erythrocyte proteins. We also provide an efficient open-source Java package that computes VDG.     

Quickest Paths, Straight Skeletons, and the City Voronoi Diagram

The city Voronoi diagram is induced by quickest paths in the L ₁plane, made faster by an isothetic transportation network. We investigate the rich geometric and algorithmic properties of city Voronoi diagrams, and report on their use in processing quickest-path queries. In doing so, we revisit the fact that not every Voronoi-type diagram has interpretations in both the distance model and the wavefront model. Especially, straight skeletons are a relevant example where an interpretation in the former model is lacking. We clarify the relationship between these models, and further draw a connection to the bisector-defined abstract Voronoi diagram model, with the particular goal of computing the city Voronoi diagram efficiently.     

Locational optimization problems solved through Voronoi diagrams

This paper reviews a class of continuous locational optimization problems (where an optimal location or an optimal configuration of facilities is found in a continuum on a plane or a network) that can be solved through the Voronoi diagram. Eight types of continuous locational optimization problems are formulated, and these problems are solved through the ordinary Voronoi diagram, the farthest-point Voronoi diagram, the weighted Voronoi diagram, the network Voronoi diagram, the Voronoi diagram with a convex distance function, the line Voronoi diagram, and the area Voronoi diagram.