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Closing the Gap: NearOptimal Steiner Trees in Polynomial Time
 IEEE Trans. ComputerAided Design
, 1994
"... The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NPhard, and the best performing MRST heuristic to date is the Iterated 1Steiner (I1S) method recently proposed by Kahng and Robins. In ..."
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Cited by 43 (13 self)
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The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NPhard, and the best performing MRST heuristic to date is the Iterated 1Steiner (I1S) method recently proposed by Kahng and Robins. In this paper we develop a straightforward, efficient implementation of I1S, achieving a speedup factor of three orders of magnitude over previous implementations. We also give a parallel implementation that achieves nearlinear speedup on multiple processors. Several performanceimproving enhancements enable us to obtain Steiner trees with average cost within 0.25% of optimal, and our methods produce optimal solutions in up to 90% of the cases for typical nets. We generalize I1S and its variants to three dimensions, as well as to the case where all the pins lie on k parallel planes, which arises in, e.g., multilayer routing. Motivated by the goal of reducing the running times of our algorith...
Low Degree Spanning Trees Of Small Weight
, 1996
"... . Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n ..."
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Cited by 33 (3 self)
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. Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree three whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree four whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in O(n) time. The results are generalized to points in higher dimensions. It is shown that for any d 3, an arbitrary collection of points in ! d contains a spanning tree of degree three, whose weight is at most 5/3 times the weight of a minimum spanning tre...
LowDegree Minimum Spanning Trees
 Discrete Comput. Geom
, 1999
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where ..."
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Cited by 23 (1 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary Lp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane there exists an MST with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14. 1 Introduction Minimum spanning tree (MST) construction is a classic optimization problem for which several efficient algorithms are known [9] [15] [19]. Solutions of many other problems hinge on the construction of an MST as an intermediary step [4], with th...
Forming connected topologies in Bluetooth adhoc networks
, 2002
"... This paper represents a first step in exploring the formation of connected topologies in adhoc networks built on the Bluetooth technology. Connectivity is the most basic requirement for any system aimed at allowing devices to communicate with each other and in this paper we illustrate that this see ..."
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Cited by 16 (2 self)
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This paper represents a first step in exploring the formation of connected topologies in adhoc networks built on the Bluetooth technology. Connectivity is the most basic requirement for any system aimed at allowing devices to communicate with each other and in this paper we illustrate that this seemingly innocuous goal gives rise to many significant challenges in the context of the Bluetooth technology. We start with a brief overview of Bluetooth and its operation and then identify some of the major problems the technology faces when used to build adhoc networks. The paper's contributions are in introducing basic algorithmic problems associated with building connected Bluetooth networks and in developing several possible solutions capable
Can Bluetooth Succeed as a LargeScale Ad Hoc Networking Technology
 IEEE Journal on Selected Areas in Communications
"... endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution m ..."
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Cited by 10 (0 self)
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endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubspermissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.
A ptas for nodeweighted steiner tree in unit disk graphs
 in COCOA, 2009
"... Abstract. The nodeweighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G =(V,E)withnode weight function C: V → R + and a subset X of V, the nodeweighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minim ..."
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Abstract. The nodeweighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G =(V,E)withnode weight function C: V → R + and a subset X of V, the nodeweighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minimum. In this paper, we study this problem in unit disk graphs and present a (1+ε)approximation algorithm for any ε>0, when the given set of vertices is clocal. As an application, we use nodeweighted Steiner tree to solve the nodeweighted connected dominating set problem in unit disk graphs and obtain a (5+ε)approximation algorithm.
WorstCase Ratios for DegreeConstrained Trees
, 1995
"... We discuss problems of minimum degreeconstrained trees T k , where each vertex of a complete graph Kn with a metric satifying triangle inequality is restricted to at most k neighbors. We show that for any k and m, the ratio w(Tk ) w(Tk+m ) can be arbitrarily close to ae k;m = 1 + m m+k\Gamma2 ..."
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We discuss problems of minimum degreeconstrained trees T k , where each vertex of a complete graph Kn with a metric satifying triangle inequality is restricted to at most k neighbors. We show that for any k and m, the ratio w(Tk ) w(Tk+m ) can be arbitrarily close to ae k;m = 1 + m m+k\Gamma2 and give an O(n log(k + m))) algorithm that converts a T k+m into a T k such that w(T k ) ! ae k;m T k+m . For the special case of a planar point set with L 1 distances, this implies that we can find a T 3 with w(T3 ) w(Tmin ) 3 2 .
Extreme Distances in Multicolored Point Sets
"... Given a set of n colored points in some ddimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of dierent colors. We present ecient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the ..."
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Given a set of n colored points in some ddimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of dierent colors. We present ecient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are xed but they dynamically change color. We do this by solving the more general problem of maintaining a bichromatic edge of minimum (resp. maximum) weight in an undirected weighted graph with colored vertices, when vertices dynamically change color.
Abstract Spanning Trees of Small Weight
"... Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing low weight degreeK spanning trees for K>2. It is shown that for an arbitrary collection of ..."
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Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing low weight degreeK spanning trees for K>2. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree three whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree four whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in O(n) time. The results are generalized to points in higher dimensions. It is shown that for any d 23, an arbitrary collection of points in R ~ contains a spanning tree of degree three, whose weight is at most 5/3 times the weight of a minimum spanning tree. This is the first paper that achieves factors better than two for these problems.
Contents lists available at ScienceDirect
"... journal homepage: www.elsevier.com/locate/tcs New approximations for minimumweighted dominating sets and ..."
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journal homepage: www.elsevier.com/locate/tcs New approximations for minimumweighted dominating sets and