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45
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 266 (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.
Sublinear Time Algorithms for Metric Space Problems
"... In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k- median, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms i ..."
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Cited by 90 (2 self)
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In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k- median, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms is that their running time is linear in the number of metric space points. As the full specification o`f an n-point metric space is of size \Theta(n 2 ), the complexity of our algorithms is sublinear with respect to the input size. All previous algorithms (exact or approximate) for the problems we consider have running time\Omega\Gamma n 2 ). We believe that our techniques can be applied to get similar bounds for other problems. 1 Introduction In recent years there has been a dramatic growth of interest in algorithms operating on massive data sets. This poses new challenges for algorithm design, as algorithms quite efficient on small inputs (for example, having quadratic running time) ...
Approximating a Finite Metric by a Small Number of Tree Metrics
- In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science
, 1998
"... Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms f ..."
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Cited by 88 (10 self)
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Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his
Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and k-median
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Advances in metric embedding theory
- IN STOC ’06: PROCEEDINGS OF THE THIRTY-EIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2006
"... Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians ..."
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Cited by 36 (13 self)
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Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓq-distortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im-
Cuts, trees and l1-embeddings of graphs
, 2002
"... Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ` 1 space. The main results are: 1. Explicit constant-distortion embeddings of all s ..."
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Cited by 30 (3 self)
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Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ` 1 space. The main results are: 1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are the rst natural families known to have constant dis-tortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor. 2. A constant-distortion embedding of outerplanar graphs into the restricted class of ` 1 metrics known as \dominating tree metrics". A lower bound of (logn) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics
Cuts, Trees and ℓ1-Embeddings of Graphs
, 2002
"... Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ℓ1 space. The main results are: 1. Explicit constant-distortion embeddings of all ..."
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Cited by 29 (8 self)
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Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ℓ1 space. The main results are: 1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor. 2. A constant-distortion embedding of outerplanar graphs into the restricted class of ℓ1-
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion, 2006. Arxiv
"... This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling dist ..."
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Cited by 18 (8 self)
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This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of ǫ, with the guarantee that for each ǫ the distortion of a fraction 1−ǫ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓq-distortions are small. Specifically, our embeddings have constant average distortion and O ( √ log n) ℓ2-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O ( √ 1/ǫ). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O ( √ 1/ǫ). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of Õ(log 2 (1/ǫ)), which implies constant ℓq-distortion for every fixed q < ∞. 1
Efficient distributed approximation algorithms via probabilistic tree embeddings
- IN: PROC. OF THE 27TH SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING
, 2008
"... We present a uniform approach to design efficient distributed approximation algorithms for various network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol, Rao, and Talwar [10] (FRT embedding). We show how to efficiently compute an ..."
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Cited by 18 (3 self)
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We present a uniform approach to design efficient distributed approximation algorithms for various network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol, Rao, and Talwar [10] (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain expected O(log n)-approximate distributed algorithms for the generalized Steiner forest problem, the minimum routing cost spanning tree problem, and the k-source shortest paths problem in arbitrary networks. The time complexities of our algorithms are within a polylogarithmic factor of the optimum. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that might be of independent interest. Assuming a global order on the nodes of a network, the LE list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen [3], Cohen and Kaplan [4]). Assuming a random order on the nodes, we give an almost-optimal distributed algorithm for computing LE lists on weighted graphs. For unweighted graphs, our LE lists computation has asymptotically optimal time complexity O(D), where D is the diameter of the network. As a byproduct, we get an improved synchronous leader election algorithm for general networks.
