## 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

Genetic Algorithm Approaches to Solve Various | 30 |

Solving MStTG Problem by Genetic Algorithm | 39 |

Solving CMStTG Problem by Genetic Algorithm | 49 |

Solving MEStT Problem by Genetic Algorithm | 55 |

29 | 66 |

Neural Network Approaches to Solve Various | 71 |

vi | 73 |

Steiner Tree Problems in VLSI Layout Designs | 101 |

Approximation Algorithms for the Steiner | 235 |

A Proposed Experiment on Soap Film Solutions | 280 |

Steiner Tree Based Distributed Multicast Routing | 326 |

On Cost Allocation in Steiner Tree Networks | 353 |

Steiner Trees and the Dynamic Quadratic | 376 |

Polynomial Time Algorithms for the Rectilinear | 405 |

Minimum Networks for Separating | 427 |

A First Level Scatter Search Implementation for Solving | 440 |

Polyhedral Approaches for the Steiner Tree | 174 |

The Perfect Phylogeny Problem | 203 |

A Tutorial | 467 |

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### Common terms and phrases

A-Tree apply approach approximation algorithm binary buffer chromosome cladistic cladistic characters clock skew clock tree cluster Computer connected consider constructed crossover degree delay bound denote distance distribution edge weights Figure follows formulation full components genetic algorithm given graph G greedy algorithm grid heuristic IEEE inequality instances integer iteration Lemma length linear lower bound LP-relaxations Min-Cost minimal minimum spanning tree MRDPT MRST MStTG multicast multicast routing neural network neurons node weights NP-complete NP-hard number of Steiner optimal solution optimum orthogonal perfect phylogeny performance ratio plane polynomial problem in graphs Proof proposed quasi-bipartite rectilinear Steiner tree selected shortest path shown sink solve Steiner minimum tree Steiner nodes Steiner point Steiner problem Steiner ratio Steiner tree problem Steiner vertices step subset subtree taxa Theorem topology undirected variables vertex VLSI wirelength X-tree Zelikovsky