Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a “black box”. The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in low-dimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a “kd-tree ” approach in the metric space setting, using Voronoi regions of a subset in place of axis-aligned boxes. 1

In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macro-molecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity to develop geometric structures associated with necklaces that are useful in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres built on a necklace. Such a hierarchy is easy to compute and is suitable for maintenance when the necklace deforms, as our theoretical and experimental results show. This hierarchy can be used for collision and self-collision detection. In particular, we achieve an upper bound of O(nlog n) in two dimensions and O(n 2−2/d) in d-dimensions, d ≥ 3, for collision checking. To our knowledge, this is the first sub-quadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be also used to detect self-collisions of a necklace in certain ways complementary to the sphere hierarchy.

For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers. 1

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in may be quadratic in the worst-case, we show in this paper that it is only linear when the points are distributed on a fixed number of well-sampled facets of (e.g. the facets of a polyhedron). Our bound is deterministic and the constants are explicitly given. Categories and Subject Descriptors I.3.5 [Computing Methodologies]: Computational Geometry and

Abstract. We extend the classic notion of well-separated pair decomposition [10] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c ≥ 1, there exists a c-wellseparated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k ≥ 3, there exists a c-wellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric. Keywords Well separated pair decomposition, Unit-disk graph, Approximation algorithm

Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.

We define and study a geometric graph over points in the plane that captures the local behavior of Delaunay triangulations of points on smooth surfaces in IR 3. Two points in a planar point set P are neighbors in the empty-ellipse graph if they lie on an axis-aligned ellipse with no point of P in its interior. The empty-ellipse graph can be a clique in the worst case, but it is usually much less dense. Specifically, the emptyellipse graph of n points has complexity Θ(∆n) in the worst case, where ∆ is the ratio between the largest and smallest pairwise distances. For points generated uniformly at random in a rectangle, the empty-ellipse graph has expected complexity Θ(n log n). As an application of our proof techniques, we show that the Delaunay triangulation of n random points on a circular cylinder has expected complexity Θ(n log n).

In this paper, we give the definition for the voronoi diagram and its dual graph -- Delaunay triangulation for 3D cuboids. We prove properties of the 3D Delaunay triangulation, and provide algorithms to construct and update the Delaunay triangulation. The Delaunay triangulation data structure is used to perform proximity searches for both static and kinetic cases. We describe experimental results that show how the Delaunay triangulation is used on a mobile robot to model, understand and reason about the spatial information of the environment.

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Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.