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.
Unable to display preview. Download preview PDF.
- 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 1976Google Scholar
- 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 O(¦E¦ 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 1972Google Scholar
- 9.Hwang, F.K.: On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math. 30, 104–114 (1976)Google Scholar