TY - GEN
T1 - A branch-and-bound algorithm for computing node weighted steiner minimum trees
AU - Xue, Guoliang
N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1997.
PY - 1997
Y1 - 1997
N2 - 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.
AB - 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.
KW - Branch-and-bound
KW - Maximum node degrees
KW - Minimum cost network under a given topology
KW - Node weighted Steiner minimum trees
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U2 - 10.1007/bfb0045105
DO - 10.1007/bfb0045105
M3 - Conference contribution
AN - SCOPUS:84947798093
SN - 354063357X
SN - 9783540633570
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 383
EP - 392
BT - Computing and Combinatorics - 3rd Annual International Conference COCOON 1997, Proceedings
A2 - Jiang, Tao
A2 - Lee, D.T.
PB - Springer Verlag
T2 - 3rd Annual International Computing and Combinatorics Conference, COCOON 1997
Y2 - 20 August 1997 through 22 August 1997
ER -