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29
Bypassing the embedding: Algorithms for lowdimensional metrics
 In Proceedings of the 36th ACM Symposium on the Theory of Computing (STOC
, 2004
"... The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2 k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into l ..."
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Cited by 64 (4 self)
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The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2 k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into low dimensional Euclidean spaces, they would inherit several algorithmic and structural properties of the Euclidean spaces. Unfortunately however, such a restriction on dimension does not suffice to guarantee embeddibility in a normed space. In this paper we explore the option of bypassing the embedding. In particular we show the following for low dimensional metrics: • Quasipolynomial time (1+ɛ)approximation algorithm for various optimization problems such as TSP, kmedian and facility location. • (1 + ɛ)approximate distance labeling scheme with optimal label length. • (1+ɛ)stretch polylogarithmic storage routing scheme.
A constantfactor approximation algorithm for the kMST problem
 In Proc. of ACM symposium on Theory of computing (STOC ’96
, 1996
"... In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constantfactor approximation algorithm for TSPN on an arbitrary set of disjoint, connected neigh ..."
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Cited by 47 (5 self)
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In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constantfactor approximation algorithm for TSPN on an arbitrary set of disjoint, connected neighborhoods in the plane. Prior approximation bounds were O(log n), except in special cases. Our approximation algorithm applies to arbitrary connected neighborhoods of any size or shape. 1
On trip planning queries in spatial databases
 In SSTD
, 2005
"... In this paper we discuss a new type of query in Spatial Databases, called the Trip Planning Query (TPQ). Given a set of points of interest P in space, where each point belongs to a specific category, a starting point S and a destination E, TPQ retrieves the best trip that starts at S, passes through ..."
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Cited by 28 (1 self)
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In this paper we discuss a new type of query in Spatial Databases, called the Trip Planning Query (TPQ). Given a set of points of interest P in space, where each point belongs to a specific category, a starting point S and a destination E, TPQ retrieves the best trip that starts at S, passes through at least one point from each category, and ends at E. For example, a driver traveling from Boston to Providence might want to stop to a gas station, a bank and a post office on his way, and the goal is to provide him with the best possible route (in terms of distance, traffic, road conditions, etc.). The difficulty of this query lies in the existence of multiple choices per category. In this paper, we study fast approximation algorithms for TPQ in a metric space. We provide a number of approximation algorithms with approximation ratios that depend on either the number of categories, the maximum number of points
Networkaware overlays with network coordinates
 In Proc. of International Workshop on Dynamic Distributed Systems
, 2006
"... Network coordinates, which embed network distance measurements in a coordinate system, were introduced as a method for determining the proximity of nodes for routing table updates in overlay networks. Their power has far broader reach: due to their low overhead and automatic adaptation to changes in ..."
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Cited by 22 (5 self)
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Network coordinates, which embed network distance measurements in a coordinate system, were introduced as a method for determining the proximity of nodes for routing table updates in overlay networks. Their power has far broader reach: due to their low overhead and automatic adaptation to changes in the network, network coordinates provide a new paradigm for managing dynamic overlay networks. We compare network coordinates to other proposals for networkaware overlays and show how they permit the lucid expression of a range of distributed systems problems in wellunderstood geometric terms. 1.
Euclidean BoundedDegree Spanning Tree Ratios
, 2003
"... Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that ..."
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Cited by 16 (0 self)
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Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that
Approximation schemes for degreerestricted MST and redblue separation problem
 Algorithmica
, 2003
"... Abstract We develop a quasipolynomial time approximation scheme for the Euclidean version of the Degreerestricted MST problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ffl? 0, the scheme finds an approximation with cost with ..."
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Cited by 13 (1 self)
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Abstract We develop a quasipolynomial time approximation scheme for the Euclidean version of the Degreerestricted MST problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the lowest cost spanning tree with the property that all nodes have degree at most d. We also develop a polynomial time approximation scheme for the Euclidean version of the RedBlue Separation Problem, again extending Arora's techniques. Given ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the cost of the optimum separating polygon of the input nodes, in nearly linear time.
Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
 In Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science
, 2009
"... Abstract. We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on b ..."
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Cited by 11 (2 self)
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Abstract. We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs. 1.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 8 (1 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
A QuasiPolynomial Time Approximation Scheme for Minimum Weight Triangulation
 Proceedings of the 38th ACM Symposium on Theory of Computing
, 2006
"... The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1 ..."
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Cited by 7 (0 self)
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The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1979. Very recently the problem was shown to be NPhard by Mulzer and Rote. In this paper, we present a quasipolynomial time approximation scheme for Minimum Weight Triangulation.
A polynomialtime approximation scheme for euclidean steiner forest
 Foundations of Computer Science, Annual IEEE Symposium on
"... We give a randomized O(n polylog n)time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed ǫ> 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1 + ǫ)approximation to the minimumlength forest ..."
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Cited by 6 (1 self)
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We give a randomized O(n polylog n)time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed ǫ> 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1 + ǫ)approximation to the minimumlength forest that connects every requested pair of terminals. 1