Algorithms for Finding Noncrossing Steiner Forests in Plane Graphs
Let G = (V,E) be a plane graph with nonnegative edge lengths, and let N be a family of k vertex sets N 1,N 2,...,N k \( \subseteq \) V, called nets. Then a noncrossing Steiner forest for N in G is a set T of k trees T 1, T 2,...,T k in G such that each tree T i ∈ T connects all vertices in N i , any two trees in T do not cross each other, and the sum of edge lengths of all trees is minimum. In this paper we give an algorithm to find a noncrossing Steiner forest in a plane graph G for the case where all vertices in nets lie on two of the face boundaries of G. The algorithm takes time O(n log n) if G has n vertices.
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- 1.M. Bern: “Faster exact algorithm for Steiner trees in planar networks,” Networks, 20, pp. 109–120 (1990).Google Scholar
- 2.S. E. Dreyfus and R. A. Wagner: “The Steiner problem in graphs,” Networks, 1, pp. 195–208 (1972).Google Scholar
- 3.R. E. Erickson, C. L. Monma, and A. F. Veinott: “Send-and-split method for minimum-concave-cost network flows,” Math. Oper. Res., 12, pp. 634–664 (1987).Google Scholar
- 4.M. R. Garey and D. S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman (1979).Google Scholar
- 5.M. R. Kramer and J. V. Leeuwen: “Wire-routing is NP-complete,” RUU-CS-82-4, Department of Computer Science, University of Utrecht, Utrecht, the Netherlands (1982).Google Scholar
- 6.P. Klein, S. Rao, M. Rauch, and S. Subramanian: “Faster shortest-path algorithms for planar graphs,” Proc. of 26th Annual Symp. on Theory of Computing, pp. 27–37 (1994).Google Scholar
- 7.T. Lengauer: Combinatorial Algorithms for Integrated Circuit Layout, John Wiley & Sons, Chichester, England (1990).Google Scholar
- 8.J. F. Lynch: “The equivalence of theorem proving and the interconnection problem,” ACM SIGDA Newsletter, 5, 3, pp. 31–36 (1975).Google Scholar
- 9.J. Takahashi, H. Suzuki and T. Nishizeki: “Shortest noncrossing paths in plane graphs,” Algorithmica, 16, pp. 339–357 (1996).Google Scholar
- 10.M. Thorup: “Undirected single source shortest paths in linear time,” Proc. of 38th Annual Symp. on Foundations of Computer Science, pp. 12–21 (1997).Google Scholar
- 11.P. Winter: “Steiner problem in networks: A survey,” Networks, 17, pp. 129–167 (1987).Google Scholar