APPROXIMATE WEIGHTED FARTHEST NEIGHBORS AND MINIMUM DILATION STARS

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find approximate farthest neighbors that are farther than a factor (1 - ∊) of optimal in time O( log n) per query in D-dimensional Euclidean space for any constants D and ∊. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

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A Helly-type theorem on a sphere

Journal of the Australian Mathematical Society ◽

10.1017/s144678870000416x ◽

1967 ◽

Vol 7 (3) ◽

pp. 323-326 ◽

Cited By ~ 1

Author(s):

M. J. C. Baker

Keyword(s):

Euclidean Space ◽

Convex Sets ◽

Type Theorem ◽

Dimensional Euclidean Space ◽

Dimensional Sphere ◽

Strongly Convex ◽

Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

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ON THE STEINER RATIO IN $\mathcal{R}_{n}$

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001358 ◽

2011 ◽

Vol 03 (04) ◽

pp. 473-489

Author(s):

HAI DU ◽

WEILI WU ◽

ZAIXIN LU ◽

YINFENG XU

Keyword(s):

Euclidean Space ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Dimensional Euclidean Space ◽

Minimum Length ◽

Steiner Ratio ◽

Line Segments ◽

Finite Set ◽

Steiner Minimum Tree ◽

Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

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Minkowski spacetime

10.1093/oso/9780198786399.003.0019 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Euclidean Space ◽

Special Relativity ◽

Three Dimensional ◽

Minkowski Spacetime ◽

Dimensional Euclidean Space ◽

Fourth Dimension ◽

Relativistic Dynamics ◽

Newtonian Theory ◽

Space And Time ◽

Set Of Points

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical extension of three-dimensional Euclidean space. In special relativity, ‘relative, apparent, and common’ (in the words of Newton) space and time are represented by a mathematical set of points called events, which constitute the Minkowski spacetime. This chapter also stresses the interpretation of the fourth dimension, which in special relativity is time. Here, time now loses the ‘universal’ and ‘absolute’ nature that it had in the Newtonian theory.

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Bounding the Size of an Almost-Equidistant Set in Euclidean Space

Combinatorics Probability Computing ◽

10.1017/s0963548318000287 ◽

2018 ◽

Vol 28 (2) ◽

pp. 280-286 ◽

Cited By ~ 3

Author(s):

ANDREY KUPAVSKII ◽

NABIL H. MUSTAFA ◽

KONRAD J. SWANEPOEL

Keyword(s):

Euclidean Space ◽

Dimensional Euclidean Space ◽

Set Of Points

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

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ALGORITHMS FOR POINT SET MATCHING WITH k-DIFFERENCES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054106004170 ◽

2006 ◽

Vol 17 (04) ◽

pp. 903-917

Author(s):

TATSUYA AKUTSU

Keyword(s):

Euclidean Space ◽

Cell Complexes, Valuations, and the Euler Relation

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-030-9 ◽

1970 ◽

Vol 22 (2) ◽

pp. 235-241 ◽

Cited By ~ 8

Author(s):

M. A. Perles ◽

G. T. Sallee

Keyword(s):

Euclidean Space ◽

Hausdorff Metric ◽

Dimensional Euclidean Space ◽

Cell Complexes ◽

Finite Dimensional ◽

Finite Set ◽

Euler Relation ◽

Set Of Points ◽

The Relationship ◽

Dimensional Face

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

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On the Irreducibility of Convex Bodies

Canadian Journal of Mathematics ◽

10.4153/cjm-1959-027-3 ◽

1959 ◽

Vol 11 ◽

pp. 256-261 ◽

Cited By ~ 3

Author(s):

A. C. Woods

Keyword(s):

Euclidean Space ◽

Convex Bodies ◽

Dimensional Euclidean Space ◽

Star Body ◽

Closed Set ◽

Absolute Value ◽

Linearly Independent ◽

The Absolute ◽

Set Of Points ◽

Rational Integral

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

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The Relaxation Method for Linear Inequalities

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-038-x ◽

1954 ◽

Vol 6 ◽

pp. 393-404 ◽

Cited By ~ 289

Author(s):

T. S. Motzkin ◽

I. J. Schoenberg

Keyword(s):

Euclidean Space ◽

Relaxation Method ◽

Linear Inequalities ◽

Dimensional Euclidean Space ◽

Closed Set ◽

Image Position ◽

Set Of Points

Let A be a closed set of points in the n-dimensional euclidean space En. If p and p1 are points of En such that1.1then p1 is said to be point-wise closer than p to the set A. If p is such that there is no point p1 which is point-wise closer than p to A, then p is called a closest point to the set A.

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Factoring Euclidean isometries

International Journal of Algebra and Computation ◽

10.1142/s0218196715400135 ◽

2015 ◽

Vol 25 (01n02) ◽

pp. 325-347 ◽

Cited By ~ 5

Author(s):

Noel Brady ◽

Jon McCammond

Keyword(s):

Euclidean Space ◽

Metric Space ◽

Isometry Group ◽

Dimensional Euclidean Space ◽

Minimum Length ◽

Finite Dimensional ◽

Combinatorial Model ◽

Full Isometry Group

Every isometry of a finite-dimensional Euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.

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Immersions of Metric Spaces into Euclidean Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-095-6 ◽

1965 ◽

Vol 17 ◽

pp. 1015-1018

Author(s):

Takeo Akasaki

Keyword(s):

Euclidean Space ◽

Topological Space ◽

Metric Space ◽

Metric Spaces ◽

Necessary Condition ◽

Dimensional Euclidean Space ◽

Compact Metric Space ◽

Singular Cohomology ◽

Smith Theory ◽

Combinatorial Methods

In a recent paper on isotopy invariants (1), S. T. Hu denned the enveloping space Em(X) of any given topological space X for each integer m > 1. By an application of the Smith theory to the singular cohomology of the enveloping space Em(X), he obtained his immersion classes for every n = 1, 2, 3, . . . and proved (3) the main theorem that a necessary condition for a compact metric space X to be immersible into the ^-dimensional Euclidean space Rn is . This theorem was proved earlier by W. T. Wu (4) for finitely triangulable spaces X using purely combinatorial methods.

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