### Abstract

Given n regular points in the Euclidean plane, the node-weighted Steiner minimum tree (NWSMT) is a straight line network interconnecting these n regular points and some Steiner points with a minimum cost, where the cost of the network is the sum of the edge lengths plus the total cost of the Steiner points. In 1995, [11] proved that a tight upper bound on the maximum degree of Steiner points in a NWSMT is 4. In 1996, [14] used this result to propose a modified Melzak procedure for computing a NWSMT. However, that procedure requires exponential time to compute a minimum cost network under a given topology. In this paper, we prove that there exists a NWSMT in which the maximum degree of regular points is no more than 5 and that this upper bound is tight. For a given topology interconnecting n regular points, we show that the Xue-Ye algorithm [15] for minimizing a sum of Euclidean norms can be used to compute an (1 + ε)-approximation of the minimum cost network in n^{1.5}(log n + log 1/ε time for any positive ε. These results enable an algorithm that computes a NWSMT by enumerating all the possible Steiner topologies. We prove a bounding theorem that can be used in a branch-and-bound algorithm and present preliminary computational experience.

Original language | English (US) |
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Title of host publication | Computing and Combinatorics - 3rd Annual International Conference COCOON 1997, Proceedings |

Editors | Tao Jiang, D.T. Lee |

Publisher | Springer Verlag |

Pages | 383-392 |

Number of pages | 10 |

ISBN (Print) | 354063357X, 9783540633570 |

DOIs | |

State | Published - 1997 |

Externally published | Yes |

Event | 3rd Annual International Computing and Combinatorics Conference, COCOON 1997 - Shanghai, China Duration: Aug 20 1997 → Aug 22 1997 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1276 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 3rd Annual International Computing and Combinatorics Conference, COCOON 1997 |
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Country | China |

City | Shanghai |

Period | 8/20/97 → 8/22/97 |

### Keywords

- Branch-and-bound
- Maximum node degrees
- Minimum cost network under a given topology
- Node weighted Steiner minimum trees

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Computing and Combinatorics - 3rd Annual International Conference COCOON 1997, Proceedings*(pp. 383-392). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1276). Springer Verlag. https://doi.org/10.1007/bfb0045105