Results 1  10
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39
Probabilistic Approximation of Metric Spaces and its Algorithmic Applications
 In 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized ..."
Abstract

Cited by 316 (29 self)
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The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilisticallyapproximates another metric space. We prove that any metric space can be probabilisticallyapproximated by hierarchically wellseparated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divideandconquer algorithmic approach. The technique presented is of particular interest in the context of online computation. A large number of online al...
On Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hi ..."
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Cited by 254 (14 self)
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This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.
Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs
 Discrete & Computational Geometry
, 1996
"... The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any npoint metric space can be represented by a (weighted) graph with n vertices and n ..."
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Cited by 41 (4 self)
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The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any npoint metric space can be represented by a (weighted) graph with n vertices and n 1+O(1=r) edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g=4 \Gamma 3=2. In the special case when jV (G)j = jV (H)j and G has strictly less edges than H , an improved bound of g=3 \Gamma 1 is shown. In addition, we discuss the case (G) ! (H) \Gamma 1, as well as some interesting higher dimensional analogues. The proofs employ basic ...
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.
The Wiener index and the Szeged index of benzenoid systems in linear time
 J. Chem. Inf. Comput. Sci
, 1997
"... A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach y ..."
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Cited by 24 (5 self)
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A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach yields also a linear algorithm for computing the Szeged index of benzenoid systems. 1.
On Approximating Planar Metrics by Tree Metrics
 Information Processing Letters
, 2001
"... We connect the results of Bartal [4] on probabilistic approximation of metric spaces by tree metrics, and Klein, Plotkin and Rao [11] on decompositions of graphs without small Ks,s minors (such as planar graphs) to show that metrics induced by such graphs can be probabilistically approximated by tre ..."
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Cited by 21 (3 self)
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We connect the results of Bartal [4] on probabilistic approximation of metric spaces by tree metrics, and Klein, Plotkin and Rao [11] on decompositions of graphs without small Ks,s minors (such as planar graphs) to show that metrics induced by such graphs can be probabilistically approximated by tree metrics with an O(log d) distortion, where d is the diameter of the given graph. This improves upon Bartal's result that holds for general nnode metrics with a distortion of O(lognloglogn). The main ingredient of our proof is that weak probabilistic partitions suffice for the construction of tree metrics with low distortion, in contrast to strong partitions used by Bartal. We also show that our result is the best possible by providing a lower bound of (log d) for the distortion of any probabilistic approximation of the square grid by tree metrics.
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
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
On the conflict matrix of clausesets
, 2003
"... We study the asymmetric respectively symmetric conflict matrix of a multiclauseset F, where the entry at position (i, j) is the number of literals, which appear positively in clause Ci of F and negatively in clause Cj (at the same time), respectively the number of clashes (at all) between Ci and C ..."
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Cited by 7 (5 self)
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We study the asymmetric respectively symmetric conflict matrix of a multiclauseset F, where the entry at position (i, j) is the number of literals, which appear positively in clause Ci of F and negatively in clause Cj (at the same time), respectively the number of clashes (at all) between Ci and Cj. A central problem is the determination of the symmetric/asymmetric conflict number of a (symmetric) conflict matrix A, which is the minimal number of variables in a multiclauseset F with symmetric/asymmetric conflict matrix A. The problem of determining the symmetric conflict number of a (symmetric) conflict matrix has been introduced as the addressing problem by Graham and Pollak, which, given an undirected graph G, asks for a labelling of the nodes with code words over {0, 1, ∗} such that the distance of two nodes in G equals the distance of their code words (the Hamming distance after deletion of all positions with a ∗), and where the goal is to