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.
The research of this author was supported in part by the US Army Research Office grant DAAH04-96-1-0233 and by the National Science Foundation grants ASC-9409285 and OSR-9350540.
Preview
Unable to display preview. Download preview PDF.
References
S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, 37th IEEE FOGS (1996), pp. 2–11.
D.-Z. Du and F.K. Hwang, A proof of Gilbert-Pollak's conjecture on the Steiner ratio, Algorithmica, Vol. 7 (1992), pp. 121–135.
M.R. Garey, R.L. Graham and D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math., Vol. 32 (1977), pp. 835–859.
E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., Vol. 16 (1968), pp. 1–29.
F. Harary, Graph Theory, Addison-Wesley, 1969.
F.K. Hwang, A linear time algorithm for full Steiner trees, Operations Research Letters, Vol. 4 (1986), pp. 235–237.
F.K. Hwang, D.S. Richard and Pawel Winter, The Steiner tree problem, North-Holland, 1992.
F.K. Hwang and J.F. Weng, The shortest network under a given topology, J. of Algorithms, Vol. 13 (1992), pp. 468–488.
E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, Comp. Sci. Press, 1978.
Z.A. Melzak, On the problem of Steiner, Canadian Math. Bull., Vol. 4 (1961), pp. 143–148.
J.H. Rubinstein, D.A. Thomas and J.F. Weng, Degree-five Steiner points cannot reduce network costs for planar sets, Networks, Vol. 22 (1995), pp. 531–537.
W.D. Smith, How to find Steiner minimal trees in Euclidean d-space, Algorithmica, Vol. 7 (1992), pp. 137–177.
D. Trietsch and F.K. Hwang, An improved algorithm for Steiner trees, SIAM J. Appl. Math., Vol. 50 (1990), pp. 244–263.
Alice Underwood, A modified Melzak procedure for computing node-weighted Steiner trees, Networks, Vol. 27 (1996), pp. 73–79.
G.L. Xue and Y.Y. Ye, An efficient algorithm for minimizing a sum of Euclidean norms with applications, manuscript, June 1995. SIAM J. Optimization, accepted for publication.
G.L. Xue and D.Z. Du, An O(n log n) average time algorithm for computing the shortest network under a given topology, manuscript, August 1995. Algorithmica, accepted for publication.
G.L. Xue, T. Lillys and D.E. Dougherty, Computing the minimum cost pipe network interconnecting one sink and many sources, manuscript, December 1996. submitted to SIAM J. Optimization.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xue, G. (1997). A branch-and-bound algorithm for computing node weighted steiner minimum trees. In: Jiang, T., Lee, D.T. (eds) Computing and Combinatorics. COCOON 1997. Lecture Notes in Computer Science, vol 1276. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0045105
Download citation
DOI: https://doi.org/10.1007/BFb0045105
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63357-0
Online ISBN: 978-3-540-69522-6
eBook Packages: Springer Book Archive