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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 42 (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 31 (2 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 22 (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  An Algorithmic Perspective
 In Proc. International Teletraffic Congress (ITC
, 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 11 (1 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 of generating "good" connected topologies
Can Bluetooth succeed as a largescale ad hoc networking technology?
 IEEE JOURNAL ON SPECIAL AREAS IN COMMUNICATIONS – WIRELESS SERIES
, 2005
"... We investigate issues that Bluetooth may face in evolving from a simple wire replacement to a largescale ad hoc networking technology. We do so by examining the efficacy of Bluetooth in establishing a connected topology, which is a basic requirement of any networking technology. We demonstrate that ..."
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Cited by 6 (0 self)
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We investigate issues that Bluetooth may face in evolving from a simple wire replacement to a largescale ad hoc networking technology. We do so by examining the efficacy of Bluetooth in establishing a connected topology, which is a basic requirement of any networking technology. We demonstrate that Bluetooth experiences some fundamental algorithmic challenges in accomplishing this seemingly simple task. Specifically, deciding whether there exists at least one connected topology that satisfies the Bluetooth constraints is NPhard. Several implementation problems also arise due to the internal structure of the Bluetooth protocol stack. All these together degrade the performance of the network, or increase the complexity of operation. Given the availability of efficient substitute technologies, Bluetooth’s use may endup being limited to small ad hoc networks.
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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Cited by 1 (0 self)
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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.
Practical Delaunay triangulation algorithms for surface reconstruction and related problems
, 2003
"... ..."
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.
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 .
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