Summary
This paper studies the design and implementation of an approximation algorithm for the Steiner tree problem. Given any undirected distance graph G and a set of Steiner points S, the algorithm produces a Steiner tree with total weight on its edges no more than 2(1−1/L) times the total weight on the optimal Steiner tree, where L is the number of leaves in the optimal Steiner tree. Our implementation of the algorithm, in the worst case, makes it run in 0(¦E g¦+¦V g−S¦log¦V g−S¦+¦S¦log ¦S¦) time for general graph G and in 0(¦S¦ log¦S¦+M log β(M,¦V g−S¦)) time for sparse graph G, where E g is the set of edges in G, Vg is the set of vertices in G, M = min {¦E g, (¦V g−S¦−1)2/2} and β(x,y) = min {i¦log(i) y ≦ x/y}.The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦V g¦ vertices and ¦E g¦ edges and the minimum spanning tree of a graph with ¦V g−S¦ vertices and M edges.
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Kou, L.T. On efficient implementation of an approximation algorithm for the Steiner tree problem. Acta Informatica 27, 369–380 (1990). https://doi.org/10.1007/BF00264613
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DOI: https://doi.org/10.1007/BF00264613