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LowDistortion Embeddings of Finite Metric Spaces
 in Handbook of Discrete and Computational Geometry
, 2004
"... INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their diss ..."
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Cited by 51 (0 self)
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INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is dicult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way. For example, it would be very nice if we could assign to each x 2 X a point f(x) in the plane in such a way that D(x; y) equals the Euclidean distance of f(x) and f(y). Such a representation would allow us to see the structure of the metric space: tight clusters, isolated points, and so on. Another advantage would be that the metric would now be represented by only 2n real numbers, the coordinates of the n points in the plane, instead of numbers as before. Moreover, many quantities concern
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 42 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
Hypermetric spaces and the Hamming cone
 Canad. J. Math
, 1981
"... 1. Definitions and preliminary results. We denote by d = (d12} •. • , din} d2z,..., dni>n) a vector of I 9 I distances between n points. Such a vector d is called a metric if it satisfies the triangle inequalities (1) d{j + djk ^ dik IS i,j, k S n. The set of all metrics on n points forms a convex ..."
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Cited by 16 (0 self)
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1. Definitions and preliminary results. We denote by d = (d12} •. • , din} d2z,..., dni>n) a vector of I 9 I distances between n points. Such a vector d is called a metric if it satisfies the triangle inequalities (1) d{j + djk ^ dik IS i,j, k S n. The set of all metrics on n points forms a convex polyhedral cone, the extremal properties of which are discussed in [4]. We will be concerned with a subcone that is spanned by metrics of the form (2) dtJ(t) = { * \t JeV
Hypermetrics in Geometry of Numbers
, 1993
"... . A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connection ..."
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Cited by 11 (3 self)
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. A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connections with lattices and applications. 2 M. Deza, V.P. Grishukhin and M. Laurent Contents 1 Introduction 2 Preliminaries 2.1 Distance spaces Metric notions Operations on distance spaces Preliminary results on distance spaces 2.2 Lattices and Lpolytopes Lattices Lpolytopes Lpolytopes and Voronoi polytopes Lattices and positive quadratic forms Lpolytopes and empty ellipsoids Basic facts on Lpolytopes Construction of Lpolytopes Lpolytopes in dimension k 4 2.3 Finiteness of the number of types of Lpolytopes in given dimension 3 Hypermetrics and Lpolytopes 3.1 The connection between hypermetrics and Lpolytopes 3.2 Polyhedrality of the hypermetric cone 3.3 Lpolytopes in root lattic...
A Convexity Lemma and Expansion Procedures for Bipartite Graphs
 EUROPEAN J. COMBIN
, 1998
"... A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic ..."
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Cited by 11 (4 self)
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A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic
A Multifacility Location Problem on Median Spaces
 Discrete Applied Mathematics
, 1996
"... This paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is pre ..."
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Cited by 7 (1 self)
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This paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimumcut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with jV j points is O(jV j 3 + jV j/(n)), where /(n) is the complexity of the applied maximumflow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N+k(logN+logk+/(n))) time and O(N+k/(n)) time in the case of the vertexrestricted multifacility location problem. 1.
Recognizing Partial Cubes in Quadratic Time
, 2007
"... We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distancepreserving embedding of the graph into a hypercube, in the nearoptimal time bound O(n²), improving previous O(nm)time solutions. ..."
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Cited by 7 (1 self)
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We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distancepreserving embedding of the graph into a hypercube, in the nearoptimal time bound O(n²), improving previous O(nm)time solutions.
On Median Graphs and Median Grid Graphs
 Discrete Math
, 1996
"... Let e be an edge of a median graph G which is contained in exactly one 4cycle. ..."
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Cited by 5 (5 self)
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Let e be an edge of a median graph G which is contained in exactly one 4cycle.
The lattice dimension of a tree
, 2004
"... The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. In this note we prove that the lattice dimension of a tree with n leaves is ⌈n/2⌉. In the paper, T is a tree with q edges and n leaves. By the Djoković theorem [3] (see also [1, 2 ..."
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Cited by 5 (0 self)
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The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. In this note we prove that the lattice dimension of a tree with n leaves is ⌈n/2⌉. In the paper, T is a tree with q edges and n leaves. By the Djoković theorem [3] (see also [1, 2, 5]), T can be isometrically embedded into the q–cube Qq and cannot be embedded into a cube of a smaller dimension. The number q is the isometric dimension dimI(T) of the tree T [2]. Thus the vertices of T can be labeled with 0/1 q–dimensional vectors in such a way that the distance between two edges is the Hamming distance between the corresponding vectors. This labeling is also known as an addressing scheme for T [1]. Let Z d be the edge skeleton of the standard cubic lattice of dimension d. Simple examples show that, generally speaking, T can be isometrically embedded into Z d of a smaller dimension than dimI(T). In a more general setting, one can consider the following problem: for a given graph G, find the minimum possible