## Steiner Trees in IndustryThis book is a collection of articles studying various Steiner tree prob lems with applications in industries, such as the design of electronic cir cuits, computer networking, telecommunication, and perfect phylogeny. The Steiner tree problem was initiated in the Euclidean plane. Given a set of points in the Euclidean plane, the shortest network interconnect ing the points in the set is called the Steiner minimum tree. The Steiner minimum tree may contain some vertices which are not the given points. Those vertices are called Steiner points while the given points are called terminals. The shortest network for three terminals was first studied by Fermat (1601-1665). Fermat proposed the problem of finding a point to minimize the total distance from it to three terminals in the Euclidean plane. The direct generalization is to find a point to minimize the total distance from it to n terminals, which is still called the Fermat problem today. The Steiner minimum tree problem is an indirect generalization. Schreiber in 1986 found that this generalization (i.e., the Steiner mini mum tree) was first proposed by Gauss. |

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### Contents

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A Proposed Experiment on Soap Film Solutions of Planar Euclidean Steiner Trees | 281 |

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apply approach approximation algorithm binary buffer characters chromosome clock tree Computer connected consider constraint constructed core corner point cost allocation crossover defined degree denote distance distributed Euclidean Steiner tree evaluation exists Figure follows formulation full components genetic algorithm given graph G grid graph heuristic IEEE inequality integer iteration Lemma length linear lower bound LP-relaxations member node method minimal minimum spanning tree MRDPT multicast multicast routing neural network neurons node weights number of Steiner optimal solution optimum perfect phylogeny performance ratio plane polynomial problem in graphs programming proof rectilinear Steiner tree RSMT segments selected shortest path shown sink skew solve Steiner minimum A-tree Steiner minimum tree Steiner nodes Steiner points Steiner problem Steiner ratio Steiner tree problem step subset subtree tabu search terminal set Theorem topology undirected vertex VLSI wirelength Zelikovsky

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Page 405 - Australian Photonics Cooperative Research Centre, Photonics Research Laboratory, Department of Electrical and Electronic Engineering, The University of Melbourne, VIC 3010, Australia. Tel: +6J -3-83444650: Fax: +61-3-83446678; E-mail: e.wong@ee.mu.oz.au *Redfem Broadband Networks, Australian Technology Park, Eveleigh, NSW 1430, Australia.

Page 405 - Centre for Ultra-Broadband Information Networks Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, VIC 3010, Australia Email: j.guo@ee.mu.oz.au; m.zukerman@ee.mu.oz.au; h.vu@ee.mu.oz.au M.