Planar Voronoi Diagrams Research Articles

Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include L_p-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses O(n log ^3 n) storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required O(n^{varepsilon }) time for an update and O(log n) time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an O(n^{varepsilon }) factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in {{mathbb {R}}}^3. Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

Automated Generalization of Facility Points-of-Interest With Service Area Delimitation

Point cluster generalization is a major concern in cartography. This problem is complicated by how to preserve the distribution patterns (e.g., density) of points during the process of deriving small-scale maps from a large-scale map. To address this problem, existing methods adopt geometric models such as planar Voronoi diagram, which considers the geographic space as an ideal Euclidean one. The associated operations determine the relationships of points based on the Euclidean straight-line distance. However, many map features are in different geographic environments. For example, as one of the most significant features on a map, urban facilities rely on the transport function of street networks to compete for service areas, and thus their distributions, in reality, are constrained to the spatial layout of street networks. To preserve such characteristic after generalization, this paper establishes a network-constrained Voronoi model for delimitating service areas of facility points-of-interest (POIs). With the help of Voronoi model, the proposed method further treats the generalization process of facility POIs as a compete and merge iterative process of service areas: a facility that has a relatively small service area faces more intense competition from its neighbors and by deleting it from the resulting small-scale map in the first priority its service area will be partitioned and merged to the neighbors. The experimental results demonstrate that our method considers the service patterns of facility POIs is particularly useful in applications related to navigation.

On the Expected Complexity of Voronoi Diagrams on Terrains

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et al. [2008] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ( n + m √ n ) in the worst case, where n denotes the number of triangles that define the terrain and m denotes the number of Voronoi sites. We prove that, under a relaxed set of assumptions, the Voronoi diagram has expected complexity O ( n + m ), given that the sites are sampled uniformly at random from the domain of the terrain (or the surface of the terrain). Furthermore, we present a construction of a terrain that implies a lower bound of Ω( nm 2/3 ) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

Spectral Experiments+

ABSTRACTWe describe extensive computational experiments on spectral properties of random objects—random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere. We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.

Finding Optimal Allocation of Constrained Cloud Capacity Using Hyperbolic Voronoi Diagrams on the Sphere

We consider a network of computer data centers on the earth surface delivering computing as a service to a big number of users. The problem is to assign users to data centers to minimize the total communication distance between compu-ting resources and their users in the face of capacity constrained datacenters. In this paper, we extend the classical pla-nar Voronoi Diagram to a hyperbolic Voronoi Diagram on the sphere. We show that a solution to the distance minimi-zation problem under capacity constraints is given by a hyperbolic spherical Voronoi Diagram of data centers. We also present numerical algorithms, computer implementation and results of simulations illustrating our solution. We note applicability of our solution to other important assignment problems, including the assignment of population to regional trauma centers, location of airbases, the distribution of the telecommunication centers for mobile telephones in global telephone companies, and others.

Computing with liquid crystal fingers: Models of geometric and logical computation

When a voltage is applied across a thin layer of cholesteric liquid crystal, fingers of cholesteric alignment can form and propagate in the layer. In computer simulation, based on experimental laboratory results, we demonstrate that these cholesteric fingers can solve selected problems of computational geometry, logic, and arithmetics. We show that branching fingers approximate a planar Voronoi diagram, and nonbranching fingers produce a convex subdivision of concave polygons. We also provide a detailed blueprint and simulation of a one-bit half-adder functioning on the principles of collision-based computing, where the implementation is via collision of liquid crystal fingers with obstacles and other fingers.

Vesicle computers: Approximating a Voronoi diagram using Voronoi automata

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata – finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting, excited, refractory and precipitate. A resting cell excites if it has at least one neighbour in an excited state. The cell precipitates if the ratio of excited cells in its neighbourhood versus the number of neighbours exceeds a certain threshold. To approximate a Voronoi diagram on Voronoi automata we project a planar set onto the automaton lattice, thus cells corresponding to data-points are excited. Excitation waves propagate across the Voronoi automaton, interact with each other and form precipitate at the points of interaction. The configuration of the precipitate represents the edges of an approximated Voronoi diagram. We discover the relationship between the quality of the Voronoi diagram approximation and the precipitation threshold, and demonstrate the feasibility of our model in approximating Voronoi diagrams of arbitrary-shaped objects and in constructing a skeleton of a planar shape.

Aspects of the Facet‐Edge Data Structure for the Construction of Voronoi Cells

AbstractThe construction of Voronoi diagrams has two main aspects: The construction algorithm and the data structure. For the construction of planar Voronoi diagrams e.g. the well known plane sweep algorithm can be applied. Another efficient method is the incremental construction of the Delaunay triangulation by using the quad‐edge data structure. Within this data structure the Delaunay triangulation is treated as a planar graph. Incremental construction implies that manipulations of the triangulation are allowed. Its Voronoi diagram is obtained simply by accessing the triangulation's dual graph. We extend this method to tree dimensions, by implementing a 3D Delaunay triangulation on the facet‐edge data structure. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hot ice computer

We experimentally demonstrate that supersaturated solution of sodium acetate, commonly called ‘hot ice’, is a massively-parallel unconventional computer. In the hot ice computer data are represented by a spatial configuration of crystallization induction sites and physical obstacles immersed in the experimental container. Computation is implemented by propagation and interaction of growing crystals initiated at the data-sites. We discuss experimental prototypes of hot ice processors which compute planar Voronoi diagram, shortest collision-free paths and implement and and or logical gates.

Generalized network Voronoi diagrams: Concepts, computational methods, and applications

In the real world, there are many phenomena that occur on a network or alongside a network; for example, traffic accidents on highways and retail stores along streets in an urbanized area. In the literature, these phenomena are analysed under the assumption that distance is measured with Euclidean distance on a plane. This paper first examines this assumption and shows an empirical finding that Euclidean distance is significantly different from the shortest path distance in an urbanized area if the distance is less than 500 m. This implies that service areas in urbanized areas cannot be well represented by Voronoi diagrams defined on a plane with Euclidean distance, termed generalized planar Voronoi diagrams. To overcome this limitation, second, this paper formulates six types of Voronoi diagrams defined on a network, termed generalized network Voronoi diagrams, whose generators are given by points, sets of points, lines and polygons embedded in a network, and whose distances are given by inward/outward distances, and additively/multiplicatively weighted shortest path distances. Third, in comparison with the generalized planar Voronoi diagrams, the paper empirically shows that the generalized network Voronoi diagrams can more precisely represent the service areas in urbanized areas than the corresponding planar Voronoi diagrams. Fourth, because the computational methods for constructing the generalized planar Voronoi diagrams in the literature cannot be applied to constructing the generalized network Voronoi diagrams, the paper provides newly developed efficient algorithms using the ‘extended’ shortest path trees. Last, the paper develops user‐friendly tools (that are included in SANET, a toolbox for spatial analysis on a network) for executing these computational methods in a GIS environment.

Fast Planar-Oriented Ripple Search Algorithm for Hyperspace VQ Codebook

This paper presents a fast codebook search method for improving the quantization complexity of full-search vector quantization (VQ). The proposed method is built on the planar Voronoi diagram to label a ripple search domain. Then, the appropriate codeword can easily be found just by searching the local region instead of global exploration. In order to take a step further and obtain the close result full-search VQ would, we equip the proposed method with a duplication mechanism that helps to bring down the possible quantizing distortion to its lowest level. According to the experimental results, the proposed method is indeed capable of providing better outcome at a faster quantization speed than the existing partial-search methods. Moreover, the proposed method only requires a little extra storage for duplication.

Abstract Voronoi diagram in 3-space

We propose a class of abstract Voronoi diagrams in 3-space that generalizes the planar abstract Voronoi diagram of Klein. Our class of abstract Voronoi diagrams includes the Voronoi diagram of point sites in general position under any convex distance function. To characterize the abstract Voronoi diagram in 3-space, we introduce the notion of intersection characteristic. We determine the intersection characteristic for the simplex, the L∞, and the Lp distance function. We find that the intersection characteristic in case of the simplex distance function is similar to that of the usual Euclidean distance. This enables us to give a randomized incremental algorithm for computing the Voronoi diagram under the simplex distance function in quadratic expected time.

Voronoi diagrams on the sphere

Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O( nlog n) algorithm for the spherical Voronoi diagram of a set of circular arcs on the sphere, and an O( nlog n) algorithm for the furthest-site Voronoi diagram for a set of circular arcs in the plane.

A randomized parallel algorithm for Voronoi diagrams based on symmetric convex distance functions

This paper presents a randomized parallel algorithm for computing planar Voronoi diagrams of n point sites according to symmetric convex distance functions. This algorithm uses, with high probability, O(n log(n)/p) local computation time and O(n/p) space on the processors plus time for O(1) communication rounds on a coarse grained multicomputer with p processors.

Randomized incremental construction of Delaunay and Voronoi diagrams

In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

FORTRAN programs to construct the planar Voronoi diagram

Abstract The Voronoi diagram and its geometric dual, the Delaunay triangulation, both have been applied extensively in spatial modeling and analysis. This paper gives two FORTRAN programs which use a new algorithm that creates the Voronoi diagram and Delaunay triangulation rapidly and efficiently for large data sets ( N > 1000). The first program applies to the special situation of data points lying either in an open domain or in a simply connected closed domain bounded by rheir convex hull: under certain conditions the running time of this program is linear in the number of data points. The second program applies to the general situation of data points lying in a domain that may be open or closed, simply or multiply connected, and have convex or reentrant boundaries. A GKS-based plotting program to give graphic output also is provided.

A straightforward iterative algorithm for the planar Voronoi diagram

Several published algorithms construct the planar N-point Voronoi diagram by adding the points in turn to a diagram already constructed for a subset of those points. The algorithm described here works in this way, but stipulates that every added point be outside the convex hull of the already constructed partial diagram. This stipulation, which is implemented by presorting the points by increasing x-coordinate, ensures that the core of the algorithm runs in almost linear time. The algorithm is inherently iterative, not recursive, and provides a rapid, efficient way to construct the Voronoi diagram for large data sets.

On the two-dimensional davenport-schinzel problem

We analyse the combinatorial complexity κ( F ) of the minimum M( x,y ) of a collection F of n continuous bivariate functions f 1 ( x,y ), ... , f n ( x,y ), such that each triple of function graphs intersect in at most s points, and each pair of functions intersect in a curve having at most t singular points. The following is proved. (1) If the intersection curve of each pair of functions intersects each plane x = const in exactly one point and s = 1 (but not if s = 2) then κ( F ) is at most O ( n ), and can be calculated in time 0 ( n log n ) by a method extending Shamos' algorithm for the calculation of planar Voronoi diagrams, (2) If s = 2 and the intersection of each pair of functions is connected then κ( F )= 0 ( n 2 ). (3) If the intersection curve of each pair of functions intersects every plane x = const in at most two points, then κ( F ) is at most O (nλ s+2 ( n )), where the constant of proportionality depends on s and t , and where λ,( q ) is the (almost linear) maximum length of a ( q,r ) Davenport-Schinzel sequence. We also present an algorithm for calculating M in this case, running in time O (nλ s+2 ( n ) log n). (4) Finally, we present some geometric applications of these results.

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.

Voronoi diagrams from convex hulls

The problem of construction of planar Voronoi diagrams arises in many areas, one of the most important of which is in nearest neighbor problems. This includes clustering [ 141, contour maps [6] and (Euclidean) minimum spanning trees [23]. Shamos [22] gives several more applications. An JZ(N log N) time worst case lower bound can be shown for this problem by reducing it to sorting [2 11. The challenge is to construct an O(N log N) time algorithm. Shamos [213 and Shamos anti Hoey [23] describe an O(N log N) time divide-and-conquer algorithm for construction of the planar Euclidean Voronoi diagram. Lee and Wong [ 161 describe an O(N log N) time algorithm for the L1 and L, metrics in the plane, and Drysdale pnd Lee [8] present an O(N@g N)l/*) t’ rme algorithm for the Voronoi diagram of N line segments (which they have since improved to O(N(log N)*) time). Shamos [2 11, Lee and Preparata [ 151, and Lipton and Tarjan [ 171 have produced fast algorithms for searching a Voronoi diagram (or any other straight-line planar graph). In this paper we describe an O(N log N) time algorithm for constructing a planar Euclidean Voronoi diagram which extends straightforwardly to higher dimensions. The fundamental result is that a K-dimensional Euclidean Voronoi diagram of N points can be constructed by transforming the points to K + I-space,