This paper presents a high-performance method for the k-nearest neighbourhood search. Starting from a point cloud, first the method carries out the space division by the typical cubic grid partition of the bounding box; then a new data structure is constructed. Based on these two previous steps, an efficient implementation of the k-nearest neighbourhood is proposed. The performance of the metho...

A class of constrained LQ optimal control problems can be abstractly formulated as seeking a nearest point in a `slice' S \ C where S is an aane subspace of a Hilbert space H and C is the (possibly unbounded) closed convex set corresponding to imposition of the constraint. Subject to a slackness condition, it is shown that the optimal control must then be a `nearest point in C' for some control...

Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo& point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-media problems relative to both the...

This paper explores a nearest neighbour pattern recognition method for time-series forecasting. A nearest neighbour method (FNNM) based on fuzzy membership values is developed. The main aim of the forecasting algorithm is to make single point forecasts into the future on the basis of past nearest neighbours. The nearest neighbours are selected using a membership threshold value. The results inc...

This paper deals with the kth nearest rectilinear distance of two regular point patterns: square and diamond lattices. The probability density functions of the kth nearest rectilinear distance are theoretically derived for k = 1, 2, . . . , 8. Upper and lower bounds of the kth nearest distance are also derived. As an application of the kth nearest distance, we consider a facility location probl...

In this paper, we present an evaluation of nearest neighbour searching using the Area Code tree. The Area Code tree is a trie-type structure that organizes area code representations of each point of interest (POI) in a data set. This data structure provides a fast method for locating an actual or approximate nearest neighbour POI for a query point. We first summarize the area code generation, i...

The on-line nearest-neighbour graph on a sequence of n uniform random points in (0, 1)d (d ∈ N) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α ∈ (0, d/2], we prove O(max{n1−(2α/d), log n}) upper bounds on the variance. On the other hand, we give an n → ∞ large-sample convergence r...

Distances of several nearest neighbors of a given point in a multidimensional space play an important role in some tasks of data mining. Here we analyze these distances as random variables defined to be functions of a given point and its k-th nearest neighbor. We prove that if there is a constant q such that the mean k-th neighbor distance to this constant power is proportional to the near neig...

We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest and next-nearest neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins S1 > S2, using 1/S spin-wave expansions, density-matrix renormalization group, and exact-diagonalization techniques. It is argued that the zero-point spin fluctuations completely des...

Definition 1.1. Nearest Neighbor Search: Given a set of points {x1, . . . , xn} ∈ R preprocess them into a data structure X of size poly(n, d) in time poly(n, d) such that nearest neighbor queries can be performed in logarithmic time. In other words, given a search point q a radius r and X one can return an xi such the ||q − xi|| ≤ r or nothing if no such point exists. The search for xi should ...