Skip to main content

A fast algorithm for Steiner trees

Summary

Given an undirected distance graph G=(V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S⊑V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦ 2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1−1/l) times that of the optimal tree, where l is the number of leaves in the optimal tree.

This is a preview of subscription content, access via your institution.

References

  1. Gilbert, E.N., Pollak, H.O.: Steiner minimal tree. SIAM J. Appl. Math. 16, 1–29 (1968)

    Google Scholar 

  2. Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. Proc. of the 8th Annual ACM Symposium on Theory of Computing, pp. 10–22 1976

  3. Dijkstra, E.N.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959)

    Google Scholar 

  4. Floyd, R.W.: Algorithm 97: shortest path, CACM 5, 345 (1962)

    Google Scholar 

  5. Tabourier, Y.: All shortest distances in a graph: an improvement to Dantzig's inductive algorithm, Discrete Math. 4, 83–87 (1973)

    Google Scholar 

  6. Yao, A.C.C.: An OE¦ loglog¦V¦) algorithm for finding minimal spanning tree. Information Processing Lett. 4, 21–23 (1975)

    Google Scholar 

  7. Cheriton, D., Tarjan, R.E.: Finding minimal spanning tree, SIAM J. Comput. 5, 724–742 (1976)

    Google Scholar 

  8. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of computer computations (R.E. Miller, J.W. Thatcher eds.), pp. 85–104. New York: Plenum Press 1972

    Google Scholar 

  9. Hwang, F.K.: On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math. 30, 104–114 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kou, L., Markowsky, G. & Berman, L. A fast algorithm for Steiner trees. Acta Informatica 15, 141–145 (1981). https://doi.org/10.1007/BF00288961

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00288961

Keywords

  • Computational Mathematic
  • System Organization
  • Time Complexity
  • Distance Function
  • Optimal Tree