Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4k?2)(n?k) vertices, (6k?3)(n?k) edges, and (2k?1)(n?itk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k ² n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.     

Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Ω(n²) lower bound and the O(n3+?) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered “cross-sections” of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+?) with lower bound Ω(n²).In this paper, we determine the combinatorial complexities of these four problems for some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n²) or O(n2+?) and lower bounds of Ω(n²).     

Some results on the computation of Voronoi diagrams on a mesh with multiple broadcasting

Voronoi diagram is one of the most fundamental and important geometric data structures. Voronoi diagram was historically defined for a set of points on the plane. The diagram partitions the plane into regions, one per site. The region of a site s consists of all points closer to s than to any other sites on the plane. Concepts of Voronoi diagram are often attributed to Voronoi (J. Reine Angew. Math. 133 (1907) 97) and Dirichlet (J. Reine Angew. Math. 40 (1850) 209). As a result of these early works, often the name Voronoi diagram and Dirichlet tessellation is used. Due to the importance of Voronoi diagrams, it is important that algorithms are devised to compute these structures in an efficient manner. Of course, this will create new opportunities for the applicability of these data structures. Towards this end, this paper presents new results for the computation of Voronoi diagrams for a set of n points, or n disjoint circles on the plane, on a mesh with multiple broadcasting (MMB) of size n×n. The algorithm runs in O(log² n) time.     

The Hausdorff Voronoi Diagram of Point Clusters in the Plane

We study the Hausdorff Voronoi diagram of point clusters in the plane, a generalization ofVoronoi diagrams based on the Hausdorff distance function. We derive a tight combinatorial bound on the structural complexity of this diagram and present a plane sweep algorithm for its construction. In particular, we show that the size of the Hausdorff Voronoi diagram is (n + m), where n is the number of points on the convex hulls of the given clusters, and m is the number of crucial supporting segments between pairs of crossing clusters. The plane sweep algorithm generalizes the standard plane sweep paradigm for the construction of Voronoi diagrams with the ability to handle disconnected Hausdorff Voronoi regions. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI layout, a measure reflecting the sensitivity of the VLSI design to spot defects during manufacturing.     

eulerForce: Force-directed layout for Euler diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time.     

Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4k–2)(n–k) vertices, (6k–3)(n–k) edges, and (2k–1)(n–itk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k ² n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.Work on this paper has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862.     

Ready, Set, Go! The Voronoi diagram of moving points that start from a line

It is an outstanding open problem of computational geometry to prove a near-quadratic upper bound on the number of combinatorial changes in the Voronoi diagram of points moving at a common constant speed along linear trajectories in the plane. In this note we observe that this quantity is Θ(n²) if the points start their movement from a common line.     

Venn and now

One of the most famous visual tools for logic is the Venn diagram, which most frequently shows the relationship between three object classes by drawing three overlapping circles. I discuss Venn diagrams, how they can be generalized to more than three sets of objects, and related tools for visual logic.     

Division-based analysis of symmetry and its application

A computational method, DAS, is proposed for symmetry analysis of a planar figure closed by a simply connected curve. DAS determines both the symmetric axis and the symmetric point pairs on the curve, consistently, based on the duality of two geometric plane divisions, the Delaunay triangulation and the Voronoi diagram     

Visualizing set concordance with permutation matrices and fan diagrams

Scientific problem solving often involves concordance (or discordance) analysis among the result sets from different approaches. For example, different scientific analysis methods with the same samples often lead to different or even conflicting conclusions. To reach a more judicious conclusion, it is crucial to consider different perspectives by checking concordance among those result sets by different methods. In this paper, we present an interactive visualization tool called ConSet, where users can effectively examine relationships among multiple sets at once. ConSet provides an overview using an improved permutation matrix to enable users to easily identify relationships among sets with a large number of elements. Not only do we use a standard Venn diagram, we also introduce a new diagram called Fan diagram that allows users to compare two or three sets without any inconsistencies that may exist in Venn diagrams. A qualitative user study was conducted to evaluate how our tool works in comparison with a traditional set visualization tool based on a Venn diagram. We observed that ConSet enabled users to complete more tasks with fewer errors than the traditional interface did and most users preferred ConSet.     

Sorting helps for voronoi diagrams

It is well known that, using standard models of computation, Ω(n logn) time is required to build a Voronoi diagram forn point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. However, if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard, types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built inO (n log logn) time after initially sorting then sites twice. Convex distance functions based on other polygons require more initial sorting.     

Minkowski-Type Theorems and Least-Squares Clustering

Dissecting Euclidean d -space with the power diagram of n weighted point sites partitions a given m -point set into clusters, one cluster for each region of the diagram. In this manner, an assignment of points to sites is induced. We show the equivalence of such assignments to constrained Euclidean least-squares assignments. As a corollary, there always exists a power diagram whose regions partition a given d -dimensional m -point set into clusters of prescribed sizes, no matter where the sites are placed. Another consequence is that constrained least-squares assignments can be computed by finding suitable weights for the sites. In the plane, this takes roughly O(n ² m) time and optimal space O(m) , which improves on previous methods. We further show that a constrained least-squares assignment can be computed by solving a specially structured linear program in n+1 dimensions. This leads to an algorithm for iteratively improving the weights, based on the gradient-descent method. Besides having the obvious optimization property, least-squares assignments are shown to be useful in solving a certain transportation problem, and in finding a least-squares fitting of two point sets where translation and scaling are allowed. Finally, we extend the concept of a constrained least-squares assignment to continuous distributions of points, thereby obtaining existence results for power diagrams with prescribed region volumes. These results are related to Minkowski's theorem for convex polytopes. The aforementioned iterative method for approximating the desired power diagram applies to continuous distributions as well. May 30, 1995; revised June 25, 1996.     

Exact and approximate area-proportional circular Venn and Euler diagrams

Scientists conducting microarray and other experiments use circular Venn and Euler diagrams to analyze and illustrate their results. As one solution to this problem, this paper introduces a statistical model for fitting area-proportional Venn and Euler diagrams to observed data. The statistical model outlined in this paper includes a statistical loss function and a minimization procedure that enables formal estimation of the Venn/Euler area-proportional model for the first time. A significance test of the null hypothesis is computed for the solution. Residuals from the model are available for inspection. As a result, this algorithm can be used for both exploration and inference on real data sets. A Java program implementing this algorithm is available under the Mozilla Public License. An R function venneuler() is available as a package in CRAN and a plugin is available in Cytoscape.     

The semantics of augmented constraint diagrams

Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They generalize Venn diagrams and Euler circles, and include syntax for quantification and navigation of relations. The notation was designed to complement the Unified Modelling Language in the development of software systems.Since symbols representing quantification in a diagrammatic language can be naturally ordered in multiple ways, some constraint diagrams have more than one intuitive meaning in first-order predicate logic. Any equally expressive notation which is based on Euler diagrams and conveys logical statements using explicit quantification will have to address this problem.We explicitly augment constraint diagrams with reading trees, which provides a partial ordering for the quantifiers (determining their scope as well as their relative ordering). Alternative approaches using spatial arrangements of components, or alphabetical ordering of symbols, for example, can be seen as implicit representations of a reading tree.Whether the reading tree accompanies the diagram explicitly (optimizing expressiveness) or implicitly (simplifying diagram syntax), we show how to construct unambiguous semantics for the augmented constraint diagram.     

“The big sweep”: On the power of the wavefront approach to Voronoi diagrams

We show that the wavefront approach to Voronoi diagrams (a deterministic line-sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worst-case optimal (O (n logn) time,O(n) space) algorithm that is valid for the full class of what has been callednice metrics in the plane. This also solves the previously open problem of providing anO (nlogn)-time plane-sweep algorithm for arbitraryL _k -metrics. Nice metrics include all convex distance functions but also distance measures like the Moscow metric, and composed metrics. The algorithm is conceptually simple, but it copes with all possible deformations of the diagram. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. Research partially supported by the Deutsche Forschungsgemeinschaft, Grant No. Kl 655/2-1.     

PaL diagrams: A linear diagram-based visual language

Linear diagrams have recently been shown to be more effective than Euler diagrams when used for set-based reasoning. However, unlike the growing corpus of knowledge about formal aspects of Euler and Venn diagrams, there has been no formalisation of linear diagrams. To fill this knowledge gap, we present and formalise Point and Line (PaL) diagrams, an extension of simple linear diagrams containing points, thus providing a formal foundation for an effective visual language. We prove that PaL diagrams are exactly as expressive as monadic first-order logic with equality, gaining, as a corollary, an equivalence with the Euler diagram extension called spider diagrams. The method of proof provides translations between PaL diagrams and sentences of monadic first-order logic.     

Set space diagrams

This paper introduces set space diagrams and defines their formal syntax and semantics. Conventional region based diagrams, like Euler circles and Venn diagrams, represent sets and their intersections by means of overlapping regions. By contrast, set space diagrams provide a certain layout that avoids overlapping geometrical entities. This enables the representation of a good deal of sets without getting diagrams which are cluttered due to overlapping regions. In particular, these diagrams can be employed for illustration purposes, e.g., for showing the laws of Boolean algebras. Additionally, cardinalities are represented and can be easily compared; inferences can be drawn to derive unknown cardinalities from a given knowledge base. The soundness of set space diagrams is shown with respect to their set-theoretic interpretation.     

Parametric graph drawing

A diagram is a drawing on the plane that represents a graph like structure, where nodes are represented by symbols and edges are represented by curves connecting pairs of symbols. An automatic layout facility is a tool that receives as input a graph like structure and is able to produce a diagram that nicely represents such a structure. Many systems use diagrams in the interaction with the users; thus, automatic layout facilities and algorithms for graphs layout have been extensively studied in the last years. We present a new approach in designing an automatic layout facility. Our approach is based on a modular management of a large collection of algorithms and on a strategy that, given the requirements of an application, selects a suitable algorithm for such requirements. The proposed approach has been used for designing the automatic layout facility of Diagram Server, a network server that offers to its clients several facilities for managing diagrams