Showing papers on "Voronoi diagram published in 1993"

An optimal convex hull algorithm in any fixed dimension

Abstract: We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n?d/2?) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n?d/2?) time.

Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving Maps

Abstract: The problem of forming perfectly topology preserving maps of feature manifolds is studied. First, through introducing “masked Voronoi polyhedra” as a geometrical construct for determining neighborhood on manifolds, a rigorous definition of the term “topology preserving feature map” is given. Starting from this definition, it is shown that a network G of neural units i, i = 1, …, N has to have a lateral connectivity structure A, Aij ∈ {0, 1}, i, j = 1,…, N which corresponds to the “induced Delaunay triangulation” of the synaptic weight vectors wi ∈ ℜDin order to form a perfectly topology preserving map of a given manifold M ⊆ ℜD of features v ∈ M. The lateral connections determine the neighborhood relations between the units in the network, which have to match the neighborhood relations of the features on the manifold. If all the weight vectors wi are distributed over the given feature manifold M, and if this distribution resolves the shape of M, it can be shown that Hebbian learning with competition leads to lateral connections i —j (Aij = 1) that correspond to the edges of the “induced Delaunay triangulation” and, hence, leads to a network structure that forms a perfectly topology preserving map of M, independent of M’s topology. This yields a means for constructing perfectly topology preserving maps of arbitrarily structured feature manifolds.

Four results on randomized incremental constructions

Abstract: We prove four results on randomized incremental constructions (RICs): an analysis of the expected behavior under insertion and deletions, a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, a tail estimate for the space complexity of RICs, a lower bound on the complexity of a game related to RICs.

The upper envelope of voronoi surfaces and its applications

Abstract: Given a setS ofsources (points or segments) in ?211C;d, we consider the surface in ?211C;d+1 that is the graph of the functiond(x)=minp?S?(x, p) for some metric?. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metric?. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.

Backwards Analysis of Randomized Geometric Algorithms

Abstract: The theme of this chapter is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as “analyze a randomized algorithm as if it were running backwards in time, from output to input.” We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others.

Randomized incremental construction of abstract Voronoi diagrams

Abstract: Voronoi diagrams were introduced by R. Klein [14, 11, 12] as an axiomatic basis of Voronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(n log n) by a randomized algorithm; the algorithm is based on Clarkson and Shor's randomized incremental construction technique [6]. The new algorithm has the following advantages over previous algorithms: It can handle a much wider class of abstract Voronoi diagrams than the algorithms presented in [14, 17]. It can be adapted to a concrete kind of Voronoi diagram by providing a single basic operation, namely the construction of a Voronoi diagram of five sites. Moreover, all geometric decisions are confined to the basic operation, and using this operation, abstract Voronoi diagrams can be constructed in a purely combinatorial manner.

Detecting structure in two dimensions combining Voronoi tessellation and percolation

Abstract: Conventional source-detection algorithms in high-energy astrophysics and other fields mostly use spherical or quadratic sliding windows of varying size on two-dimensionally binned representations of spatial event distributions in order to detect statistically significant event enhancements (sources) within a given field. While this is a reasonably reliable technique for nearly pointlike sources with good statistics, poor and extended sources are likely to be incorrectly assessed or even missed at all, as the calculations are governed by nonphysical parameters like the bin size and the window geometry rather than by the actual data. The approach presented here does not introduce any artificial bias but makes full use of the unbinned two-dimensional event distribution. A Voronoi tessellation on a finite plane surface yields individual densities, or fluxes, for every single event, the distribution of which allows the determination of the contribution from a random Poissonian background field (noise). The application of a nonparametric percolation to the tessellation cells exceeding this noise level leads directly to a source list which is free of any assumptions about the source geometry. High-density fluctuations from the random background field will still be included in this tentative source list but can be easily eliminated, in most cases, by setting a lower threshold to the required number of events per source. Since no finite-size detection windows or the like have been used, this analysis yields automatically straightforward fluxes for every source finally accepted. The main disadvantage of this approach is the considerable CPU time required for the construction of the Voronoi tessellation---it is thus applicable only to either small fields or low-event density regions.

Iterated nearest neighbors and finding minimal polytypes

Abstract: We introduce a new method for finding several types of optimalk-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of theO(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, which were based on high-order Voronoi diagrams. Our technique allows us for the first time to maintain minimal sets efficiently as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convexk-vertex polygons and polytopes, and to improve many previous results. We achieve many of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in timeO(n logn+kn). We also demonstrate a related technique for finding minimum areak-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

The Voronoi polyhedra as tools for structure determination in simple disordered systems

Abstract: The structure of some simple fluids is analyzed through the Voronoi tessellation of the configurations generated by computer simulation. Special emphasis is put on the investigation of disordering, i.e., the deviation from solid-state properties. The models studied include a simple Lennard-Jones (LJ) liquid, two quenched states obtained from the liquid, and a system made of noninteracting particles (the ideal gas), while a state point corresponding to the LJ solid phase is used as reference. To perform the Voronoi construction, an algorithm specially suited for disordered systems is presented. The arrangement of particles has been studied through the geometrical features of their Voronoi polyhedra (VP)

Neural network design using Voronoi diagrams

Abstract: A novel approach is proposed which determines the number of layers, the number of neurons in each layer, and their connection weights for a particular implementation of a neural network, with the multilayer feedforward topology, designed to classify patterns in the multidimensional feature space. The approach is based on construction of a Voronoi diagram over the set of points representing patterns in feature space and this finds added usefulness in deriving alternate neural network structures for realizing the desired pattern classification. >

Arrangement of cells in Voronoi tesselations of monosize packing of discs

Abstract: We describe a two-dimensional mosaic obtained by the Voronoi tesselation of a monosize assembly of discs at different packing fractions The experimental device (hard discs moving on an air table) produces, for every concentration of the discs, a succession of mosaics in statistical equilibrium, which constitutes a statistical ensemble This ensemble is large enough for fluctuations from the most probable distributions to be negligible Both topological and metric properties show deviations from those of a totally random mosaic These deviations can be ascribed to steric exclusions In particular, distributions of the numbers of sides, of the perimeters and of the areas of the polygons differ from those observed in biological celi assemblies The Aboav law holds, but with a slope which can be as Iow as 4–66 The Lewis law is obeyed only for smali packing fractions The variance of the distribution of the number n of polygon sides is a universal function of p 6, the probability of finding a six-si

Correlations in topological models of 2D random cellular structures

Abstract: General relations and constraints which must be satisfied by the topological correlations in 2D space-filling random cellular structures are discussed and a topological short-range order coefficient is defined. Topological models of 2D structures are associated with planar tessellations with topologically unstable sites which belong to z)3 polygons. The stable configurations, called states, are obtained by replacing every vertex by z-3 added sides. The topological properties of the latter models are calculated exactly for a distribution of independent and equiprobable states on the various sites and for any value of z. The case of the structures associated with tilings by triangles is thoroughly considered. The calculated correlations are compared with the correlations in alumina cuts and in random Voronoi froths. The variability of the topological properties of 2D random cellular structures is discussed.

Robust Construction of the Voronoi Diagram of a Polyhedron.

Abstract: This paper describes a practical algorithm for the construction of the Voronoi diagram of a three dimensional polyhedron using approximate arithmetic. This algorithm is intended to be implemented in oating point arithmetic. The full two-dimensional version and sig-niicant portions of the three-dimensional version have been implemented and tested. The running time 1 of this algorithm is O(npnv log 2 b), where np is the size of the input polyhedron, nv is the size of the output Voronoi diagram, and b is the number of desired bits of precision. This algorithm can be made more practical with the use of a binary partition of space. In the worst case, binary partition does not improve the running time, but it should reduce the running time to O(nvb) on well-behaved inputs. Since b is constant, this eliminates a factor of np. The algorithm can be generalized to higher dimensions and the order k Voronoi diagram.

Properties of local coordinates based on Dirichlet tessellations

Abstract: Local coordinates based on the Dirichlet tessellation (Voronoi diagram) provide a means to express a point as a linear combination of certain fixed points by using ratios of areas (or volumes) of certain regions. We add insight into the structure of the local coordinates by proving some of their basic properties by deriving formulas for their gradients from some simple geometry and proving the properties of smoothness and linear precision.

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Abstract: Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k 3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ⌈(d+1)/2⌉+1 n ⌊(d+1)/2⌋) and space complexityO(k ⌈(d+1)/2⌉ n ⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

The furthest-site geodesic voronoi diagram

Abstract: We present anO((n+k) log(n+k))-time,O(n+k)-space algorithm for computing the furthest-site Voronoi diagram ofk point sites with respect to the geodesic metric within a simplen-sided polygon.

Randomized incremental construction of abstract Voronoi diagrams

Abstract: Voronoi diagrams were introduced by R.~Klein as an axiomatic basis of Voronoi diagrams. We show how to construct abstract Voronoi diagrams in time $O(n\log n)$ by a randomized algorithm, which is based on Clarkson and Shor's randomized incremental construction technique. The new algorithm has the following advantages over previous algorithms: \begin{itemize} \item It can handle a much wider class of abstract Voronoi diagrams than the algorithms presented in [Kle89b, MMO91]. \item It can be adapted to a concrete kind of Voronoi diagram by providing a single basic operation, namely the construction of a Voronoi diagram of five sites. Moreover, all geometric decisions are confined to the basic operation, and using this operation, abstract Voronoi diagrams can be constructed in a purely combinatorial manner.

Convex distance functions in 3-space are different

Abstract: We investigate the bisector systems of convex distance functions in 3-space and show that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. Namely, more than one sphere can pass through four points in general position. We show that in the L4-metric there exist quadrupels of points that lie on the surface of three L4-spheres, and that this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. Moreover, for each n ≥ 2 we construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly n d-spheres. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

The voronoi model and cultural space: applications to the social sciences and humanities

Abstract: The Voronoi model of space is becoming more and more important as a tool in the mathematical modelling of space for many application domains. In the Voronoi model, space is neither an empty void within which can be found occasional objects (the vector model), nor a lattice of arbitrary cells (the raster model). Rather, space is a continuous medium filled with proximity fields generated by objects. This representation of space has important implications for domains where geographic space is endowed with cultural characteristics or values. The Voronoi model of space concords fairly closely with the perceptual and linguistic spaces of humans and hence Voronoi zones around objects are meaningful. The Voronoi model of space is also a closer fit to qualitative data representation and analysis than other models. Finally, the Voronoi model permits nested hierarchical relations between entities which increases the richness of the querying capabilities. It is shown that the Voronoi model favors a rich qualitative database of the kind which will be found in culturally intensive application domains.

A new bound on the local density of sphere packings

Abstract: It is shown that a packing of unit spheres in three-dimensional Euclidean space can have density at most 0.773055..., and that a Voronoi polyhedron defined by such a packing must have volume at least 5.41848... These bounds are superior to the best bounds previously published [5] (0.77836 and 5.382, respectively), but are inferior to the tight bounds of 0.7404... and 5.550... claimed by Hsiang [2]. Our bounds are proved by cutting a Voronoi polyhedron into cones, one for each of its faces. A lower bound is established on the volume of each cone as a function of its solid angle. Convexity arguments then show that the sum of all the cone volume bounds is minimized when there are 13 faces each of solid angle 4?/13.

The Expected Number of k-Faces of a Voronoi Diagram

Abstract: The asymptotic behavior of the expected number of faces of all dimensions of the Voronoi diagram of n independent, uniformly distributed sites in R d is investigated. It is shown that, when k and d are fixed and n grows without bound, the expected number of k -dimensional faces grows as C k,d n . Further, for fixed k as d grows without bound, there exist constants α k such that C k,d = O (α d k ). Numerical estimates of C k,d are given for small values of k and d .

Constructing the Voronoi diagram of a set of line segments in parallel

Abstract: In this paper we give a parallel algorithm for constructing the Voronoi diagram of a polygonal scene, i.e., a set of line segments in the plane such that no two segments intersect except possibly at their endpoints. Our algorithm runs inO(log2n) time usingO(n) processors in the CREW PRAM model.

Periodic and quasi-periodic laguerre tilings

Abstract: The Laguerre construction extends and unifies the notions of Voronoi (or Dirichlet) and Delaunay complex. It is shown how the standard projection formalism for the generation of quasi-periodic tilings from higher-dimensional periodic “oblique” (or “klotz”) tilings associated to periodic Voronoi and Delaunay complexes also applies more generally to Laguerre complexes. It turns out that all quasi-periodic tilings obtained this way are Laguerre tilings themselves.