Results 11  20
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36
Volume distortion for subsets of Euclidean spaces
, 2006
"... In [Rao, SoCG 1999], it is shown that every npoint Euclidean metric with polynomial spread admits a Euclidean embedding with kdimensional distortion bounded by O ( √ log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the spread, and g ..."
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Cited by 6 (2 self)
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In [Rao, SoCG 1999], it is shown that every npoint Euclidean metric with polynomial spread admits a Euclidean embedding with kdimensional distortion bounded by O ( √ log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the spread, and give an improved bound of O ( √ log n(log k) 1/4). Our main result is an upper bound of O ( √ log n log log n) independent of the value of k, nearly resolving the main open questions of [DunaganVempala, RANDOM 2001] and [KrauthgamerLinialMagen, Discrete Comput. Geom. 2004]. The best previous bound was O(log n), and our bound is nearly tight, as even the 2dimensional volume distortion of an nvertex path is Ω ( √ log n).
A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics
"... We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of ..."
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Cited by 4 (0 self)
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We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of
Interchanging distance and capacity in probabilistic mappings
 CoRR
"... Harald Räcke [STOC 2008] described a new method to obtain hierarchical decompositions of networks in a way that minimizes the congestion. Räcke’s approach is based on an equivalence that he discovered between minimizing congestion and minimizing stretch (in a certain setting). Here we present Räcke’ ..."
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Cited by 3 (0 self)
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Harald Räcke [STOC 2008] described a new method to obtain hierarchical decompositions of networks in a way that minimizes the congestion. Räcke’s approach is based on an equivalence that he discovered between minimizing congestion and minimizing stretch (in a certain setting). Here we present Räcke’s equivalence in an abstract setting that is more general than the one described in Räcke’s work, and clarifies the power of Räcke’s result. In addition, we present a related (but different) equivalence that was developed by Yuval Emek [ESA 2009] and is only known to apply to planar graphs. 1
A PolylogarithmicCompetitive Algorithm for the kServer Problem
"... We give the first polylogarithmiccompetitive randomized online algorithm for the kserver problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log 2 k) for any metric space on n points. Our algorithm improves upon the deterministic (2k − ..."
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Cited by 3 (0 self)
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We give the first polylogarithmiccompetitive randomized online algorithm for the kserver problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log 2 k) for any metric space on n points. Our algorithm improves upon the deterministic (2k − 1)competitive algorithm of Koutsoupias and Papadimitriou [23] whenever n is subexponential in k.
An Improved Approximation Ratio for the Covering Steiner Problem. On the Covering Steiner problem
 Theory of Computing
, 2006
"... Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a nonnegative integer value (called its requirement); the problem is to find a minimumcost tree which spans at least the requi ..."
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Cited by 2 (1 self)
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Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a nonnegative integer value (called its requirement); the problem is to find a minimumcost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the kMST and Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the kMST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem
UltraLowDimensional Embeddings for Doubling Metrics
"... We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argumen ..."
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We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argument shows that this bound is nearly tight: the uniform metric on n points requires Ω(log n / log log n) dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to lowdimensional lowdistortion embeddings. Do doubling metrics, which do not have large uniform submetrics, embed in low dimensional Euclidean spaces with small distortion? In this paper, we answer the question positively and show that any doubling metric embeds into O(log log n) dimensions with o(log n) distortion. In fact, we give a suite of embeddings with a smooth tradeoff between distortion and dimension: given an npoint metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we embed the metric into Euclidean space R T with O(log n � dimD /T) distortion.
Fast ckr partitions of sparse graphs
 Chicago Journal of Theoretical Computer Science
"... We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1 ..."
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We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1
The Online Transportation Problem: On the Exponential Boost of One Extra
"... We present a polylogcompetitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a polylogcompetitive randomized on ..."
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We present a polylogcompetitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a polylogcompetitive randomized online algorithm for the online transportation on an arbitrary metric space when the online algorithm has one extra server per site. 1
New Length Bounds for Cycle Bases ∗
, 2007
"... Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n 2) for any unweighted graph, whence proving the conjecture of Deo et al. (1982). For weighted graphs, we construct cycle bases of length O(W ·log n log log n), where W denotes t ..."
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Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n 2) for any unweighted graph, whence proving the conjecture of Deo et al. (1982). For weighted graphs, we construct cycle bases of length O(W ·log n log log n), where W denotes the sum of the weights of the edges. This improves the upper bound that follows from the result of Elkin et al. (2005) by a logarithmic factor and, for comparison from below, some natural classes of large girth graphs are known to exhibit minimum cycle bases of length Ω(W · log n). We achieve this bound for weighted graphs by not restricting ourselves to strictly fundamental cycle bases—as it is inherent to the approach of Elkin et al.—but rather also considering weakly fundamental cycle bases in our construction. This way we profit from some nice properties of Hierarchically WellSeparated Trees that were introduced by Bartal (1998). 1