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New Approximation Algorithms for the Steiner Tree Problems
"... The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solution ..."
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Cited by 48 (5 self)
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The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions. We achieve the best up to now approximation ratios of 1.644 in arbitrary metric and 1.267 in rectilinear plane, respectively.
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 (12 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...
Spanning Trees in Hypergraphs with Applications to Steiner Trees
, 1998
"... This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of ..."
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Cited by 25 (1 self)
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This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of finding a minimumweight spanning tree in a hypergraph. The geometric Steiner tree problem is known to be NPcomplete for the rectilinear metric, and NPhard for the Euclidean metric. The fastest exact algorithms (in practice) for these problems use two phases: First a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimal tree is constructed from this set. These phases are called FST generation and FST concatenation, respectively, and an overview of each phase is presented. FST concatenation is almost always the most expensive phase, and has traditionally been accomplished via simple backtrack search or dynamic programming.
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...
A New Exact Algorithm for Rectilinear Steiner Trees
 IN INTERNATIONAL SYMPOSIUM ON MATHEMATICAL PROGRAMMING
, 1997
"... Given a finite set V of points in the plane (called terminals), the rectilinear Steiner minimal tree is a shortest network of horizontal and vertical lines connecting all the terminals of V . The decision form of this problem has been shown to be NPcomplete [8]. A new algorithm is presented that ..."
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Cited by 15 (1 self)
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Given a finite set V of points in the plane (called terminals), the rectilinear Steiner minimal tree is a shortest network of horizontal and vertical lines connecting all the terminals of V . The decision form of this problem has been shown to be NPcomplete [8]. A new algorithm is presented that computes provably optimal Steiner trees using the "FST concatenation" approach. In the "FST generation" phase, extensive geometric processing is used to identify a set of full Steiner trees (FSTs). In the subsequent FST concatenation phase, a Steiner minimal tree is then constructed by a finding a minimal spanning subset of the FSTs. This FST concatenation approach has been more efficient in practice than all other methods currently known. In previous work [19], [20] the author used problem decomposition methods and a "dumb" backtrack search to concatenate FSTs, solving problem instances with up to 65 terminals. Most 45 point instances could be solved within one CPU day on a worksta...
A faster dynamic programming algorithm for exact rectilinear Steiner minimal trees
 In Proceedings of the Fourth Great Lakes Symposium on VLSI
, 1994
"... An exact rectilinear Steiner minimal tree algorithm is presented that improves upon the time and space complexity of previous guarantees and is easy to implement. Experimental evidence is presented that demonstrates that the algorithm also works well an practice.TI 1 ..."
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Cited by 8 (5 self)
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An exact rectilinear Steiner minimal tree algorithm is presented that improves upon the time and space complexity of previous guarantees and is easy to implement. Experimental evidence is presented that demonstrates that the algorithm also works well an practice.TI 1
Optimal Rectilinear Steiner Minimal Trees in O(n²2.62²) Time
, 1994
"... This paper presents an algorithm that computes an optimal rectilinear Steiner minimal tree of n points in at most O(n 2 2:62 n ) time. For instances small enough to solve in practice, this time bound is provably faster than any previous algorithm, and improves the previous best bound for practic ..."
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Cited by 7 (3 self)
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This paper presents an algorithm that computes an optimal rectilinear Steiner minimal tree of n points in at most O(n 2 2:62 n ) time. For instances small enough to solve in practice, this time bound is provably faster than any previous algorithm, and improves the previous best bound for practically solvable instances, which is O(n3 n ). Experimental evidence is also presented that demonstrates that the algorithm is fast in practice as wellmuch faster, in fact, than its worstcase time complexity suggests. As part of the analysis, it is proven that the number of full sets on a set of n terminals is at most O(n1:62 n ). 1 Introduction The rectilinear Steiner minimal tree (RSMT) problem is stated as follows: given a set T of points called terminals in the plane, find a set S of additional points called Steiner points such that the length of a rectilinear minimum spanning tree of T [S is minimized. For example, Figure 1 illustrates an optimal RSMT for a set of 27 terminals. G...
Optimal Rectilinear Steiner Tree Routing in the Presence of Obstacles
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1993
"... This paper presents a new model for VLSI routing in the presence of obstacles, that transforms any routing instance from a geometric problem into a graph problem. It is the first model that allows computation of optimal obstacleavoiding rectilinear Steiner trees in time corresponding to the instanc ..."
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Cited by 3 (0 self)
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This paper presents a new model for VLSI routing in the presence of obstacles, that transforms any routing instance from a geometric problem into a graph problem. It is the first model that allows computation of optimal obstacleavoiding rectilinear Steiner trees in time corresponding to the instance size (the number of terminals and obstacle border segments) rather than the size of the routing area. For the most common multiterminal critical netsthose with three or four terminalswe observe that optimal trees can be computed as efficiently as good heuristic trees, and present algorithms that do so. For nets with five or more terminals, we present algorithms that heuristically compute obstacleavoiding Steiner trees. Analysis and experimental results demonstrate that the model and algorithms work well in both theory and practice. Also presented are several theoretical results: a derivation of the Steiner ratio for obstacleavoiding rectilinear Steiner trees, and complexity results...
Minimum steiner tree construction
 IN ALPERT, C.J., MEHTA, D.P. AND SAPATNEKAR, S.S. (EDS), THE HANDBOOK OF ALGORITHMS FOR VLSI PHYSICAL DESIGN AUTOMATION
, 2009
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On the Maximum Degree of Minimum Spanning Trees (Extended Abstract)
 IN PROC. ACM SYMP. COMPUTATIONAL GEOMETRY, STONY
, 1994
"... Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximumdegree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum verte ..."
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Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximumdegree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum vertex degree in a minimumdegree L p MST; towards this end, we define the MST number, which is closely related to the Hadwiger number. We bound Hadwiger and MST numbers for arbitrary L p metrics, and focus on the L 1 metric, where little was known. We show that the MST number of a diamond is 4, and that for the octahedron the Hadwiger number is 18 and the MST number is either 13 or 14. We also give an exponential lower bound on the MST number for an L p unit ball. Implications to L p minimum spanning trees and related problems are explored.