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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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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...
Geometric and Statistical Analysis of Porous Media
, 2000
"... of the Dissertation Geometric and Statistical Analysis of Porous Media by Arun Bhaskar Venkatarangan Doctor of Philosophy in Applied Mathematics and Statistics State University of New York at Stony Brook 2000 The threedimensional geometry and connectivity of pore space controls the hydra ..."
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of the Dissertation Geometric and Statistical Analysis of Porous Media by Arun Bhaskar Venkatarangan Doctor of Philosophy in Applied Mathematics and Statistics State University of New York at Stony Brook 2000 The threedimensional geometry and connectivity of pore space controls the hydraulic transport behavior of crustal rocks. Direct measurement of owrelevant geometrical properties of the void space in a suite of 4 samples of Fontainebleau sandstone ranging from 7.5% to 22% porosity and 3 samples of Berea sandstone of 8% to 21% porosity is studied. The measurements are obtained from computer analysis of three dimensional, synchrotron Xray computed microtomographic images. Measured distributions of coordination number, channel length, throat size and pore volume, iii and of correlations between throatsize/porevolume and nearest neighbor porevolume/pore volume determined for these samples are presented. Quantitative characterisation of the distributions mea...
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells.
IEEL TRANSACTIONS ON COMPUTERAIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 13, NO 11. NOVEMBER 1994 Closing the Gap: NearOptimal Steiner Trees in Polynomial Time
"... AbstractThe 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 (IIS) method recently proposed by Kahng and Ro ..."
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AbstractThe 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 (IIS) method recently proposed by Kahng and Robins. In this paper, we develop a straightforward, efficient implementation of IIS, 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 0.257 of optimal, and our methods produce optimal solutions in up to 90t/ ' of the cases for typical nets. We generalize 11S 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 algorithms, we prove that any pointset in the Manhattan plane has a minimum spanning tree (MST) with maximum degree 4, and that in threedimensional Manhattan space every pointset has an MST with maximum degree of 14 (the best previous upper bounds on the maximum MST degree in two and three dimensions are 6 and 26, respectively); these results are of independent theoretical interest and also settle an open problem in complexity theory. I.
Closing the Gap: NearOptimal Steiner Trees in Polynomial Time
"... We propose several efficient enhancements to the Iterated 1Steiner (I1S) heuristic of Kahng and Robins [17] for the minimum rectilinear Steiner tree problem. For typical nets, our methods obtain average performance of less than 0.25% from optimal, and produce optimal solutions up to 90% of the t ..."
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We propose several efficient enhancements to the Iterated 1Steiner (I1S) heuristic of Kahng and Robins [17] for the minimum rectilinear Steiner tree problem. For typical nets, our methods obtain average performance of less than 0.25% from optimal, and produce optimal solutions up to 90% of the time. 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. Our algorithms are highly parallelizable, and extend to arbitrary weighted graphs, and thus, our methods are applicable in practical routing regimes. We prove that given a pointset in the Manhattan plane, the minimum spanning tree (MST) degree of any specific point can be made to be 4 or less; similarly, we show that in three dimensions, the MST degree of any specific point can be made 14 or less. Using a perturbative argument, these results have been recently extended to show that for every pointset in the Manhatt...
ABSTRACT
"... The treatise proposes chiasmus is a dominant instrument that conducts processes and products of human thought. The proposition grows out of work in cognitive semantics and cognitive rhetoric. These disciplines establish that conceptualization traces to embodied image schematic knowledge. The Introdu ..."
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The treatise proposes chiasmus is a dominant instrument that conducts processes and products of human thought. The proposition grows out of work in cognitive semantics and cognitive rhetoric. These disciplines establish that conceptualization traces to embodied image schematic knowledge. The Introduction sets out how this knowledge gathers from perceptions, experiences, and memories of the body’s commonplace engagements in space. With these ideas as suppositional foundation, the treatise contends that chiastic instrumentation is a function of a corporeal mind steeped in elementary, nonverbal spatial forms or gestalts. It shows that chiasmus is a space
Randomness in topological models
"... There are two aspects of randomness in topological models. In the first one, topological idealization of random patterns found in the Nature can be regarded as planar representations of threedimensional lattices and thus reconstructed in the space. Another aspect of randomness is related to graphs ..."
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There are two aspects of randomness in topological models. In the first one, topological idealization of random patterns found in the Nature can be regarded as planar representations of threedimensional lattices and thus reconstructed in the space. Another aspect of randomness is related to graphs in which some properties are determined in a random way. For example, combinatorial properties of graphs: number of vertices, number of edges, and connections between them can be regarded as events in the defined probability space. Randomgraph theory deals with a question: at what connection probability a particular property reveals. Combination of probabilistic description of planar graphs and their spatial reconstruction creates new opportunities in structural formfinding, especially in the inceptive, the most creative, stage.
Links of Interest From Joseph Kerski: Why Geography Education Matters
, 2013
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