Summary
We present an algorithm for finding a Steiner tree for a connected, undirected distance graph with a specified subset S of the set of vertices V. The set V-S is traditionally denoted as Steiner vertices. The total distance on all edges of this Steiner tree is at most 2(1−1/l) times that of a Steiner minimal tree, where l is the minimum number of leaves in any Steiner minimal tree for the given graph. The algorithm runs in O(¦E¦log¦V¦) time in the worst case, where E is the set of all edges and V the set of all vertices in the graph. It improves dramatically on the best previously known bound of O(¦S¦¦V¦2), unless the graph is very dense and most vertices are Steiner vertices. The essence of our algorithm is to find a generalized minimum spanning tree of a graph in one coherent phase as opposed to the previous multiple steps approach.
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The work of this author was partially supported by the National Science Foundation under Grants MCS 8342682 and ECS 8340031. This work was performed while this author was a summer visitor at the IBM T.J. Watson Research Center.
On leave from: Institut für Angewandte Informatik und Formale Beschreibungsverfahren, Universität Karlsruhe, Postfach 6380, D-7500 Karlsruhe, Federal Republic of Germany
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Wu, Y.F., Widmayer, P. & Wong, C.K. A faster approximation algorithm for the Steiner problem in graphs. Acta Informatica 23, 223–229 (1986). https://doi.org/10.1007/BF00289500
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DOI: https://doi.org/10.1007/BF00289500