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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 49 (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
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.
Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 17 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 17 (4 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
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...
Decomposition and l_1Embedding of Weakly Median Graphs
, 1998
"... . Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasimedian graphs as well as pseudomedian graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5wheel ..."
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Cited by 9 (6 self)
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. Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasimedian graphs as well as pseudomedian graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2connected bridged graphs not containing K4 or K1;1;3 as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l 1 embeddable, that is, it embeds as a metric subspace into some R n equipped with the 1norm. In this paper we continue to elaborate on a structure theory of graphs based on two fundamental operations, viz., Cartesian multiplication and gated amalgamation. While Cartesian multiplication is a standard operation, gated amalgamation seems to appear only in the context of median graphs and their generalizations; cf. [4, 6, 8, 23, 27]. An induced subgraph H of a grap...
Cellular Bipartite Graphs
, 1996
"... this paper we investigate the graphs that are obtained from single edges and even cycles by successive gated amalgamations. These "cellular" graphs are characterized among bipartite graphs by having a totally decomposable shortestpath metric, and can be recognized by a quadratic time algorithm ..."
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Cited by 8 (6 self)
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this paper we investigate the graphs that are obtained from single edges and even cycles by successive gated amalgamations. These "cellular" graphs are characterized among bipartite graphs by having a totally decomposable shortestpath metric, and can be recognized by a quadratic time algorithm
On Equicut Graphs
 QUARTERLY JOURNAL OF MATHEMATICS OXFORD
, 2000
"... The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equic ..."
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Cited by 7 (3 self)
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The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equicut graph, that is, admits an `1 embedding f that for any 1 i n f satis es x2V f(x) i 2 fdv=2e; bv=2cg. Basic properties of equicut graphs are investigated. A construction of equicut graphs from `1graphs via a natural doubling construction is given. It generalizes several wellknown constructions of polytopes and distanceregular graphs. Finally, large families of examples, mostly related to polytopes and distanceregular graphs, are presented.
Uniform Partitions of 3space, their Relatives and Embedding
 European J. of Combinatorics
, 2000
"... We review 28 uniform partitions of 3space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4polytopes of ConwayGuy, noncompact uniform par ..."
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Cited by 7 (2 self)
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We review 28 uniform partitions of 3space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4polytopes of ConwayGuy, noncompact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last ones we indicate two continuums of aperiodic tilings by semiregular 3prisms with cubes or with regular tetrahedra and regular octahedra. On the way many new partitions are added to incomplete cases considered here. 1
Recognition of the l 1 graphs with Complexity O(nm), or Football in a Hypercube
, 1995
"... We fill in the details of the algorithm sketched in [Sh] and determine its complexity. As a part of this main algorithm, we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes. We discuss possible further applications of the same ideas and give a nice exa ..."
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Cited by 5 (5 self)
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We fill in the details of the algorithm sketched in [Sh] and determine its complexity. As a part of this main algorithm, we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes. We discuss possible further applications of the same ideas and give a nice example of a non ` 1 graph allowing a highly isometric embedding into a halved cube. 1 Introduction For a set \Omega\Gamma let 2\Omega denote the set of all the subsets of \Omega\Gamma We turn 2\Omega into an ndimensional cube graph Q k , where k = j\Omega j, by making two subsets A and B adjacent whenever the symmetric difference A4B has size 1. The graph Q k is bipartite, the bipartite half of Q k being known as the halved cube graph (we denote it HQ k ). Therefore, HQ k can be defined as the graph on the even size subsets in 2\Omega , in which two such subsets A and B are adjacent whenever jA4Bj = 2. We usually identify a graph \Gamma with its set of vertices, and we use the same notati...