Points In Euclidean Space Research Articles

Distances in benzenoid systems: Further developments

In this note we present some new results on distances in benzenoids. An algorithm is presented which, for a given benzenoid system G bounded by a simple circuit Z with n vertices, computes the Wiener index of G in O( n) time. Also we show that benzenoid systems have a convenient dismantling scheme, which can be derived by applying breadth-first search to their dual graphs. Our last result deals with the clustering problem of sets of atoms of benzenoids systems. We show how the k-means clustering algorithm (for points in Euclidean space) can be efficiently implemented in the case of benzenoids.

Testing simple polygons

We consider the problem of verifying a simple polygon in the plane using “test points”. A test point is a geometric probe that takes as input a point in Euclidean space, and returns “+” if the point is inside the object being probed or “−” if it is outside. A verification procedure takes as input a description of a target object, including its location and orientation, and it produces a set of test points that are used to verify whether a test object matches the description. We give a procedure for verifying an n-sided, non-degenerate, simple target polygon using 5 n test points. This testing strategy works even if the test polygon has n + 1 vertices, and we show a lower bound of 3 n + 1 test points for this case. We also give algorithms using O( n) test points for simple polygons that may be degenerate and for test polygons that may have up to n + 2 vertices. All of these algorithms work for polygons with holes. We also discuss extensions of our results to higher dimensions.

On configuration spaces, their homology, and Lie algebras

Homological and geometric properties of configuration spaces of points in Euclidean space are described. Connections to non-abelian analogues of exterior algebras and mapping class groups are also given.

A Probabilistic Model for the Spatial Distribution of Party Support in Multiparty Electorates

Abstract Spatial models of electoral competition locate voters and parties at points in euclidean space—representing issue positions—and specify utility of voters for parties as functions of these positions. Utility functions may also have stochastic components unassociated with issues. In this article probabilistic models are compared in which the utility function incorporates distance between voter and party positions (proximity model) or a scalar product (directional model). Model specification is significant because of its relation to party strategy and the resulting spatial distribution of parties. Maximum likelihood is used to estimate parameters of a mixed directional and proximity model—with stochastic and strategic components—from data in Norwegian and Swedish election studies. Expected spatial distributions of voters by party support are determined for the multiparty electorates of Norway and Sweden. Unlike previous deterministic work, which strongly favors the directional model, the results obt...

Constructive Heuristics and Lower Bounds for Graph Partitioning Based on a Principal-Components Approximation

This paper addresses the problem of partitioning the vertex set of an edge-weighted undirected graph into two parts of specified sizes so as to minimize the sum of the weights on edges joining Vertices in different parts. This problem is NP-hard and has several important applications in which the graph size is typically large and the brute-force approach (of listing all feasible partitions and comparing costs) is computationally prohibitive. In this paper a new class of algorithms is developed on the basis of a transformation of the graph problem to a geometric problem of clustering a set of points in Euclidean space. Instead of searching through all feasible partitions that meet the size specifications, it is shown that the search can be confined to a set of $n^{p( p + 1 )/2} $ partitions, where n is the number of vertices in the graph and p is the rank of the $n \times n$ graph connection matrix. Procedures are developed for constructing all such partitions in $O( n^{p( p + 3 ) /2} )$ time. For matrices...

A divide-and-conquer algorithm for constructing relative neighborhood graph

AnO(n logn) divide-and-conquer algorithm for finding the relative neighborhood graph RNG(V) of a set V ofn points in Euclidean space is presented. If implemented in parallel, its time complexity isO(n) and it requiresO(logn) processors.

Eigenvectors of Distance-Regular Graphs

The objective of this work is to find properties of a distance-regular graph G that are expressed in the eigenvectors of its adjacency matrix. The approach is to consider the rows of a matrix of orthogonal eigencolumns as (coordinates of) points in Euclidean space, each one corresponding to a vertex of G. For the second eigenvalue, the symmetry group of the points is isomorphic to the automorphism group of G. Adjacency of vertices is related to linear dependence, linear independence, and proximity of points. Relative position of points is studied by way of the polytope that is their convex hull. Several families of examples are included.

The Longest Chain Among Random Points in Euclidean Space

The Longest Chain Among Random Points in Euclidean Space

The longest chain among random points in Euclidean space

Let n n random points be chosen independently from the uniform distribution on the unit k k -cube [ 0 , 1 ] k {[0,1]^k} . Order the points coordinate-wise and let H k ( n ) {{\mathbf {H}}_k}\left ( n \right ) be the cardinality of the largest chain in the resulting partially ordered set. We show that there are constants c 1 , c 2 , … {c_1},{c_2}, \ldots such that c k > e , lim k → ∞ c k = e {c_k} > e,\;{\lim _{k \to \infty }}{c_k} = e , and lim n → ∞ H k ( n ) / n 1 / k = c k {\lim _{n \to \infty }}{{\mathbf {H}}_k}\left ( n \right )/{n^{1/k}} = {c_k} in probability. This generalizes results of Hammersley, Kingman and others on Ulam’s ascending subsequence problem, and settles a conjecture of Steele.

On the Complexity of Nonconvex Covering

We study the problem of covering given points in Euclidean space with a minimum number of nonconvex objects of a given type. We concentrate on the one-dimensional case of this problem, whose computational complexity was previously unknown. We define a natural measure for the “degree of nonconvexity” of a nonconvex object. Our results show that for any fixed bound on the degree of nonconvexity of the covering objects the one-dimensional nonconvex covering problem can be solved in polynomial time. On the other hand without such bound on the degree of nonconvexity the one-dimensional nonconvex covering problem is NP-complete. We also consider the capacitated version of the nonconvex covering problem and we exhibit a useful property of minimum coverings by objects whose degree of nonconvexity is low.

Infinite coverings of cages

A family of infinite cubic graphs Y s , s = 2, 3, 4, 5, is constructed. The vertices of Y s are lattice points in Euclidean space of dimension 2 s −1 , the girth of Y s is 6s - 6, and Y s has a group of automorphisms which acts regularly on the s -arcs.

Finding the convex hull facet by facet

Finding the convex hull of a finite set of points in Euclidean space was one of the first problems explored in the field of computational geometry. In two dimensions a variety of algorithms have been developed and analyzed. For higher dimensions the problems are less well understood. Several “convex hull” problems are defined, solved, and analyzed here in terms of the size of the input and output. In all cases these solutions are the best of the known algorthms. The problems include enumerating the facets of the convex hull, computing the facial lattice of the convex hull and producing a new compact structure representing the combinatorial type of the convex hull. In addition, deciding the combinatorial equivalence of two polytopes is shown to be coNP-hard.

A Central Limit Theorem for $k$-Means Clustering

A set of $n$ points in Euclidean space is partitioned into the $k$ groups that minimize the within groups sum of squares. Under the assumption that the $n$ points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the $k$ groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.

Basic program for a significance test for two clusters in euclidean space as measured by their overlap

Abstract A BASIC program, OVCLUST, is given for a significance test for the distinctness of a pair of clusters of points in Euclidean space. The input data may be given in one of three forms: (1) the coordinates of the points in the full space; (2) the distances of the points to the cluster centroids together with the intercentroid distance; or (3) the intercentroid distance and the standard deviations of projections of points onto the intercentroid axis. The method is based on an index of disjunction W that corresponds to a degree of overlap V. The test consists of ascertaining whether the observed overlap is significantly less than an expected overlap, by determining whether a t statistic derived from the observed overlap exceeds a noncentral t value for the expected degree of overlap.

A note on Rabin's nearest-neighbor algorithm

Rabin has proposed a probabilistic algorithm for finding the closest pair of a set of points in Euclidean space. His algorithm is asymtomatically linear whereas the best of the known deterministic algorithms take order $n \log n$ time. We show that at least part of the speed up is due to the model rather than the probabilistic nature of the algorithm.

On the regular increase of the variance of the number of random points in euclidean space

On the regular increase of the variance of the number of random points in euclidean space

On the regular increase of the variance of the number of random points in euclidean space

On the regular increase of the variance of the number of random points in euclidean space

An Algorithm for Euclidean Sum of Squares Classification

A ni analysis of surface pollen samples to discover iJ'they fall naturally in1to distinct groups oJ' simiiilar samiples is an example of a classification problem. In Euclidean classification, a set of n objects can be represented as n points in Euclidean space of p dimensions. The sum oJ'squares criterion defines the optimal partition of the points in1to g disjoint groups to be the partition which mninimzizes the total within-group sumn of squared distances about the g centroids. It is not usually Jeasible to examiine all possible partitions oJ'the objects into g groups. A critical review is mnade of algorithmiis which have been proposedfor seeking optimal partitions. The problem is reformulated itn non-linear programming terms, and a new algorithm for seeking the minimum sumi1 oJ'squares is described. The performance of this algorithm in analyzing the pollen data is Jound to compare vell vith the perJormance of three oJ the existing algorithms. An efficient hybrid algorithmi is introduced.

Multivariate Clustering Procedures with Variable Metrics

In this work we discuss several methods used in the clustering of obj ects which can be represented as points in Euclidean space. Moreover, only those procedures are considered that lead to partition of a sample into a preestablished number of groups, although some ideas are also valid for hierarchical schemes (Lance and Williams [1966]). There are two major approaches to this problem. In the first one (which could be called metric) a measure of distance is defined between each point x of the sample and each subset G thereof (an (i, k)-measure in the terminology of Lance and Williams [1968]). An initial partition is generated (for instance at random, or based on some external considerations) and then the points are successively reassigned to the nearest group, until no more reallocations are possible (or desired). In the second approach, each partition is assigned a numerical value, usually measuring the reduction of uncertainty due to grouping, and a search is made for partitions that optimize this functional (Friedman and Rubin [1967], Rubin [1967]). The simplest and most commonly used metric is the Euclidean distance between x and the mean of G (Ball and Hall [1967], Forgy [1965], MacQueen [1967]). In spite of its practical advantages, it has the drawback that the results it produces are not invariant under nonsingular linear transformations of the data (e.g., changes in the measurement units). One could avoid this drawback if he knew the within groups covariance matrix W corresponding to the correct partition he was looking for. Then the most natural metric would be Mahalanobis distance based on W. A possible solution to this dilemma is at each step to use the Mahalanobis metric induced by the matrix W corresponding to the current partition. A second conceivable improvement is obtained when one tries to overcome the fact that in the two former cases the metric is the same for all groups. If he suspects that the true clusters may have very different covariance structures, it would be reasonable to let each cluster have its own metric, based on its current covariance matrix. This was advocated by Chernoff [1970] and Rohlf [1970].

Multivariate Procedures Invariant Under Linear Transformations

Many well-known procedures in multivariate data analysis are invariant under the group, $L(p)$, of translations and nonsingular linear transformations. New maximal $L(p)$ invariant statistics are derived and are shown to have the geometrical interpretation of a scatter of points in Euclidean space. The distribution of maximal $L(p)$ invariants for the case of a single multivariate normal population is shown to follow from a result of James (1954). Finally we consider tests of the null hypothesis that $k > 1$ populations are identical and show that optimal $L(p)$ invariant tests are similar tests of randomness.