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46
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 ..."
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Cited by 323 (28 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 260 (13 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.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 143 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Distance Approximating Trees for Chordal and Dually Chordal Graphs
, 1999
"... In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a ..."
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Cited by 32 (17 self)
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In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree T of G that is an additive 3spanner as well as a multiplicative 4spanner of G. In all cases the tree T can be computed in linear time
NPCompleteness Results for Minimum Planar Spanners
"... For any fixed parameter t _> 1, a tspanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum tspanner is a tspanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this ..."
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Cited by 25 (0 self)
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For any fixed parameter t _> 1, a tspanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum tspanner is a tspanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the AlPhardness of finding minimum tspanners for planar weighted graphs and digraphs if t _> 3, and for planar unweighted graphs and digraphs if t _> 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum planar tspanners and establish its Alphardness for similar fixed values of t.
Approximating minimum maxstretch spanning trees on unweighted graphs
 In Proc. ACMSIAM Symposium on Discrete Algorithms
, 2004
"... Given a graph G and a spanning tree T of G, we say that T is a tree tspanner of G if the distance between every pair of vertices in T is at most t times their distance in G. The problem of finding a tree tspanner minimizing t is referred to as the Minimum MaxStretch spanning Tree (MMST) problem. ..."
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Cited by 23 (0 self)
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Given a graph G and a spanning tree T of G, we say that T is a tree tspanner of G if the distance between every pair of vertices in T is at most t times their distance in G. The problem of finding a tree tspanner minimizing t is referred to as the Minimum MaxStretch spanning Tree (MMST) problem. This paper concerns the MMST problem on unweighted graphs. The problem is known to be NPhard, and the paper presents an O(log n)approximation algorithm for it. Furthermore, it is established that unless P = NP, the problem cannot be approximated additively by any o(n) factor.
Additive Tree Spanners
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1998
"... A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some consta ..."
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Cited by 14 (0 self)
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A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some constant k such that every member of the class has some kadditive tree spanner. On the other hand, there are chordal graphs without kadditive tree spanner for arbitrary large k.
Treedecompositions with bags of small diameter
, 2007
"... This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right ..."
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Cited by 14 (1 self)
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This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded treelength graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, ATfree graphs, etc.). For instance, we show that the treelength of any outerplanar graph is ⌈k/3⌉, where k is the chordality of the graph, and we compute the treelength of meshes. More fundamentally we show that any algorithm computing a treedecomposition approximating the treewidth (or the treelength) of an nvertex graph by a factor α or less does not give an αapproximation of the treelength (resp. the treewidth) unless if α = Ω(n 1/5). We complete these results presenting several polynomial time constant approximate algorithms for the treelength. The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with nearoptimal route length, and by the construction of sparse additive spanners.
Approximation algorithms for embedding general metrics into trees
 In 18th Symposium on Discrete Algorithms
, 2007
"... ..."
I.: Collective tree spanners of graphs
 SIAM J. Discrete Math
, 2006
"... Abstract. In this paper we introduce a new notion of collective tree spanners. We say that a graph G =(V,E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈T(G) exists suc ..."
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Cited by 13 (11 self)
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Abstract. In this paper we introduce a new notion of collective tree spanners. We say that a graph G =(V,E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈T(G) exists such that dT (x, y) ≤ dG(x, y) +r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2–spanners and any cchordal graph admits a system of at most log 2 n collective additive tree (2⌊c/2⌋)–spanners. Towards establishing these results, we present a general property for graphs, called (α, r)– decomposition, and show that any (α, r)–decomposable graph G with n vertices admits a system of at most log 1/α n collective additive tree 2r– spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs. 1