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97
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 65 (2 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
A survey of the theory of hypercube graphs
 Computers and Mathematics with Applications 15
, 1988
"... AlmtractWe present acomprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the ncube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced ..."
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Cited by 64 (1 self)
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AlmtractWe present acomprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the ncube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the determination of the cubical dimension of a given cubical graph. I.
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 44 (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.
Wiener number of vertexweighted graphs and a chemical application
 Discrete Appl. Math
, 1997
"... application ..."
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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 24 (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.
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 23 (8 self)
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A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic
A simple O(mn) algorithm for recognizing Hamming graphs
 Bull. Inst. Comb. Appl
, 1993
"... We show that any isometric irredundant embedding of a graph into a product of complete graphs is the canonical isometric embedding. This result is used to design a simple O(mn) algorithm for recognizing Hamming graphs. 1 ..."
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Cited by 20 (11 self)
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We show that any isometric irredundant embedding of a graph into a product of complete graphs is the canonical isometric embedding. This result is used to design a simple O(mn) algorithm for recognizing Hamming graphs. 1
Structure of Fibonacci cubes: a survey
, 2011
"... The Fibonacci cube Γn is the subgraph of the ncube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper, a survey on Fibonacci cubes is gi ..."
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Cited by 18 (2 self)
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The Fibonacci cube Γn is the subgraph of the ncube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper, a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.
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 conv ..."
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Cited by 18 (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
HAPPY ENDINGS FOR FLIP GRAPHS
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2010
"... We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets with no empty pentagon include intersections of lattices with convex sets, points on two lines, and seve ..."
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Cited by 17 (2 self)
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We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets with no empty pentagon include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.