Given a set of n points in d-dimensional Euclidean space, S ae E d , and a query point q 2 E d , we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as efficiently as possible. We assume that the dimension d is a constant independent of n. Although reasonably good solutions to this problem exist when d is small, as d increases the performance of these algorithms degrades rapidly. We present a randomized algorithm for approximate nearest neighbor searching. Given any set of n points S ae E d , and a constant ffl ? 0, we produce a data structure, such that given any query point, a point of S will be reported whose distance from the query point is at most a factor of (1 + ffl) from that of the true nearest neighbor. Our algorithm runs in O(log 3 n) expected time and requires O(n log n) space. The data structure can be built in O(n 2 ) expe...

This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem --- the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ff-hulls. Thus our work is pot...

Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximum-degree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum vertex degree in a minimum-degree L p MST; towards this end, we define the MST number, which is closely related to the Hadwiger number. We bound Hadwiger and MST numbers for arbitrary L p metrics, and focus on the L 1 metric, where little was known. We show that the MST number of a diamond is 4, and that for the octahedron the Hadwiger number is 18 and the MST number is either 13 or 14. We also give an exponential lower bound on the MST number for an L p unit ball. Implications to L p minimum spanning trees and related problems are explored.

The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th problem. An example of a

We present a new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem. This problem concerns dividing a given polygon P into n polygonal pieces, each of a specified area and each containing a certain point (site) on its boundary. We first present the algorithm for the case when P is convex and contains no holes. Then the generalized version that handles nonconvex and nonsimply connected polygons is presented. The algorithm uses sweep-line and divide-andconquer techniques to construct the polygon partition. The algorithm assumes the input polygon P has been divided into a set of p convex pieces (p = 1 when P is convex) and runs in time O(pn 2 + vn), where v is the sum of the number of vertices of the convex pieces. Keywords -- polygon decomposition, area partition, divide and conquer, sweep line, robot workspace partition, robotics, terrain covering 1 Introduction The polygon decomposition problem...

In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius ε. In particular, we tighten the bounds currently known for the ε-entropy of Hamming spheres of an arbitrary radius r. New bounds for the ε-entropy of Hamming balls are also derived. If both parameters ε and r are linear in dimension n, then the upper bounds exceed the lower ones by an additive term of order log n. We also present the uniform bounds valid for all values of ε and r. In the second part of the paper, new sufficient conditions are obtained, which allow one to verify the validity of the asymptotic formula for the size of an ellipsoid in a Hamming space. Finally, we survey recent results concerning coverings of ellipsoids in Hamming and Euclidean spaces.

This chapter introduces the three basic types of grids used to represent a digital pattern: triangular, hexagonal and square. The notions of 4-connectedness and 8-connectedness are defined and their impact on tracing the boundary of a digital pattern is analyzed. 1. Tessellations A tessellation of the plane is essentially a partitioning of the plane into regions. Typical examples of bounded regions that form tessellations are a quilted bedspread and a tiled bathroom floor. There exists considerable variation in the terminology of tessellations. In the mathematical literature the words tiling and mosaic are often used and one also encounters the words paving and parqueting [GS87]. More formally we say that a planar tessellation T is a countable family of closed sets T = {T 1 , T 2 ,...} which cover the plane without gaps or overlaps. As in a well constructed bathroom floor we would not want some tiles to be missing (gaps) or to be lying on top of others (overlaps). More precise...

Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximum-degree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum vertex degree in a minimum-degree L p MST; towards this end, we define the MST number, which is closely related to the Hadwiger number. We bound Hadwiger and MST numbers for arbitrary L p metrics, and focus on the L 1 metric, where little was known. We show that the MST number of a diamond is 4, and that for the octahedron the Hadwiger number is 18 and the MST number is either 13 or 14. We also give an exponential lower bound on the MST number for an L p unit ball. Implications to L p minimum spanning trees and related problems are explored.

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