A branch-and-bound algorithm for computing node weighted steiner minimum trees

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 n1.5(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 languageEnglish (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages383-392
Number of pages10
Volume1276
ISBN (Print)354063357X, 9783540633570
StatePublished - 1997
Externally publishedYes
Event3rd Annual International Computing and Combinatorics Conference, COCOON 1997 - Shanghai, China
Duration: Aug 20 1997Aug 22 1997

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1276
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other3rd Annual International Computing and Combinatorics Conference, COCOON 1997
CountryChina
CityShanghai
Period8/20/978/22/97

Fingerprint

Branch and Bound Algorithm
Computing
Vertex of a graph
Steiner Point
Topology
Costs
Maximum Degree
Upper bound
Euclidean norm
Euclidean plane
Exponential time
Straight Line
Approximation
Theorem

Keywords

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

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Xue, G. (1997). A branch-and-bound algorithm for computing node weighted steiner minimum trees. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1276, pp. 383-392). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1276). Springer Verlag.

A branch-and-bound algorithm for computing node weighted steiner minimum trees. / Xue, Guoliang.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1276 Springer Verlag, 1997. p. 383-392 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1276).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Xue, G 1997, A branch-and-bound algorithm for computing node weighted steiner minimum trees. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 1276, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1276, Springer Verlag, pp. 383-392, 3rd Annual International Computing and Combinatorics Conference, COCOON 1997, Shanghai, China, 8/20/97.
Xue G. A branch-and-bound algorithm for computing node weighted steiner minimum trees. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1276. Springer Verlag. 1997. p. 383-392. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Xue, Guoliang. / A branch-and-bound algorithm for computing node weighted steiner minimum trees. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1276 Springer Verlag, 1997. pp. 383-392 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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