AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatio-temporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, which means the locality between objects in the multidimensional space being preserved in the linear space. It is widely believed that the Hilbert space-filling curve achieves the best clustering [1], [14]. In this paper, we analyze the clustering property of the Hilbert space-filling curve by deriving closed-form formulas for the number of clusters in a given query region of an arbitrary shape (e.g., polygons and polyhedra). Both the asymptotic solution for the general case and the exact solution for a special case generalize previous work [14]. They agree with the empirical results that the number of clusters depends on the hypersurface area of the query region and not on its hypervolume. We also show that the Hilbert curve achieves better clustering than the z curve. From a practical point of view, the formulas given in this paper provide a simple measure that can be used to predict the required disk access behaviors and, hence, the total access time.

The subject of this paper is a special class of configurations, or patterns, of intersecting circles in constant curvature surfaces. The combinatorial aspect of such a pattern is described by a cellular decomposition of the surface. The faces of the cellular decomposition correspond to circles and the vertices correspond to points where circles intersect. (See figures 1 and 2.) In the most general case that we consider, the surface may have cone-like singularities in the centers of the circles and in the points of intersection. In oarticular, we treat...

This paper is concerned not with space and spatial relations as objective entities of the world, but rather with human experience and perception of phenomena and relations in space. The goal arising from this concern is to identify models of space that can be used both in cognitive science and in the design and implementation of geographic information systems (GISs). Experiential models of the world are based on sensorimotor and visual experiences with environments, and form in individual minds as the associated bodies and senses experience their worlds. Formal models consist of axioms expressed in a formal language, together with mathematical rules to infer conclusions from them. The paper reviews both kinds of models, viewing them each as abstractions of the same 'real world.' The review of experiential models is grounded in recent developments in cognitive science, expounded by Rosch, Johnson, Talmy, and especially Lakoff. Among other things, these models suggest that perception and...

We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a self-adjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a max-degree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount. 1

We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spring-embedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.

Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it di#cult to support persistence, including "multiple futures", the ability to use a data structure in several unrelated ways in a given computation; "time travel", the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously a#ect end-to-end running time. Analytically, we present a new randomized algorithm for 3-dimensional Convex Hull based on our representations for which the running time matches the #(n lg n) lower-bound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3-dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persis...

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In this paper, we investigate topological watersheds.

We present a study of fifteen parallel thinning algorithms, based on the framework of critical kernels. We prove that ten among these fifteen algorithms indeed guarantee topology preservation, and give counter-examples for the five other ones. We also investigate, for some of these algorithms, the relation between the medial axis and the obtained homotopic skeleton.

Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3 Graphs 15 3.1 Paired directed multigraphs : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 3.2 Constructions on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Proof nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 4 Homology 27 4.1 General theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.2 Application on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31