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

  • Guoliang Xue
Session 11: Algorithms and Applications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1276)


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.


Node weighted Steiner minimum trees maximum node degrees minimum cost network under a given topology branch-and-bound 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Guoliang Xue
    • 1
  1. 1.University of VermontBurlingtonUSA

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