Nearest neighbor searching is a fundamental computational problem. A set of n data points is given in real d-dimensional space, and the problem is to preprocess these points into a data structure, so that given a query point, the nearest data point to the query point can be reported efficiently. Because data sets can be quite large, we are primarily interested in data structures that use only O...

Geometric models impose an upper bound on the number of points that can share the same nearest neighbor. A much more restrictive bound is implied by the assumption that the data points represent a sample from some continuous distribution in a multidimensional Euclidean space. The analysis of 100 data sets shows that most perceptual data satisfy the geometric-statistical bound whereas many conce...

In this paper, we show that a pointwise symmetric pre-isotonic preclosure function is uniquely determined the pairs of sets it separates. We then show that when the preclosure function of the domain is pre-isotonic and the pre-closure function of the codomain is pre-isotonic and pointwise-pre-symmetric, functions which separate only those pairs of sets which are already separated are precontinu...

In this paper, we present our research on data analysis and nearest neighbor search problems. A nearest neighbor search problem is normally described as finding data point or data points from a data set that are closest to a given query point. It is used in many research and industrial fields. In our paper, we propose an approach that explores the meaning of K nearest neighbors from a new persp...

We present a new approach for approximate nearest neighbor queries for sets of high dimensional points under any L t-metric, t = 1; : : : ; 1. The proposed algorithm is eecient and simple to implement. The algorithm uses multiple shifted copies of the data points and stores them in up to (d + 1) B-trees where d is the dimensionality of the data, sorted according to their position along a space ...

In nearest neighbor searching we are given a set of n data points in real d-dimensional space, <d, and the problem is to preprocess these points into a data structure, so that given a query point, the nearest data point to the query point can be reported efficiently. Because data sets can be quite large, we are interested in data structures that use optimal O(dn) storage. In this paper we consi...

We introduce a new method for nding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(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 rst time to e ciently maintain minima...

A set T ⊂ R at which interpolation from Π≤n(R ) (polynomials of degree ≤ n) is uniquely possible is a GCn-set if the associated Lagrange fundamental polynomials have only linear factors. For such GCn-sets T in the plane, Gasca and Maeztu conjectured the existence of a line containing n + 1 points from T. It is shown here that, already in R, there exist GC2-sets T without any hyperplanes contain...