Parallel Algorithms and Architectures pp 176-183 | Cite as

# Computing a rectilinear steiner minimal tree in \(n^{O(\sqrt n )}\)time

Invited Papers

First Online:

## Abstract

We propose an algorithm for computing a rectilinear Steiner minimal tree on *n* points in \(2^{O(\sqrt n \log n)}\)time and *O*(*n*^{2}) space. This is an asymptotic improvement on the 2^{O(n)} time required by current algorithms. If the points are distributed uniformly at random on the unit square, the Steiner tree calculated by our algorithm is minimal with high probability. The “constant factors” of our algorithm are such that it should be feasible to obtain exact solutions for *n*-point problems whenever *n*≤25. Previously, only problems of size *n*≤20 were feasible.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. V. Aho, M. R. Garey, and F. K. Hwang. Rectilinear Steiner trees: efficient special-case algorithms.
*Networks*, 7:37–58, 1977.Google Scholar - [2]Y. P. Aneja. An integer linear programming approach to the Steiner problem in graphs.
*Networks*, 10:167–178, 1980.Google Scholar - [3]Marshall W. Bern. A more general special case of the Steiner tree problem. 1986. Manuscript submitted to STOC-19.Google Scholar
- [4]Marshall W. Bern. Two probabilistic results on rectilinear Steiner trees. In
*Proceedings of the 18th Annual ACM Symposium on the Theory of Computing*, pages 433–441, May 1986.Google Scholar - [5]U. Bertele and F. Briosche.
*Nonserial Dynamic Programming*. Academic Press, New York, 1972.Google Scholar - [6]S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs.
*Networks*, 1:195–207, 1972.Google Scholar - [7]M. R. Garey, R. L. Graham, and D. S. Johnson. The complexity of computing Steiner minimal trees.
*SIAM Journal of Applied Mathematics*, 32(4):835–859, 1977.CrossRefGoogle Scholar - [8]M. R. Garey and D. S. Johnson. The rectilinear Steiner tree problem is NP-complete.
*SIAM Journal of Applied Mathematics*, 32(4):826–834, June 1977.Google Scholar - [9]M. Hanan. On Steiner's problem with rectilinear distance.
*SIAM Journal of Applied Mathematics*, 14(2):255–265, March 1966.Google Scholar - [10]F. K. Kwang. An
*O*(*n*log*n*) algorithm for suboptimal rectilinear Steiner trees.*IEEE Transactions on Circuits and Systems*, CAS-26(1):75–77, January 1979.Google Scholar - [11]F. K. Hwang. The rectilinear Steiner problem.
*Design Automation and Fault Tolerant Computing*, 2:303–310, 1978.Google Scholar - [12]Alfred Iwainsky, Enrico Canuto, Oleg Taraszow, and Agostino Villa. Network decomposition for the optimization of connection structures.
*Networks*, 16:205–235, 1986.Google Scholar - [13]Janos Komlos and M. T. Shing. Probabilistic partitioning algorithms for the rectilinear Steiner tree problem.
*Networks*, 15:413–423, 1985.Google Scholar - [14]Harold W. Kuhn. “Steiner's” problem revisited. In G. B. Dantzig and B. C. Eaves, editors,
*Studies in Mathematics, Vol. 10: Studies in Optimization*, pages 52–70, Mathematical Association of America, 1974.Google Scholar - [15]E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, editors.
*The Traveling Salesman Problem: A Guided Tour of Optimization*. John Wiley and Sons Ltd., 1985.Google Scholar - [16]Prabhakar Raghavan.
*Randomized Rounding and Discrete Ham-Sandwich Theorems: Provably Good Algorithms for Routing and Packing Problems*. PhD thesis, U. C. Berkeley, Computer Science Division, August 1986.Google Scholar - [17]Prabhakar Raghavan and Clark D. Thompson. Provably good routing in graphs: regular arrays. In
*Proceedings of the 17th Annual ACM Symposium on the Theory of Computing*, pages 79–87, ACM, May 1985.Google Scholar - [18]Prabhakar Raghavan and Clark D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs.
*Combinatorica*, to appear, 1987.Google Scholar - [19]Michal Servit. Heuristic algorithms for rectilinear Steiner trees.
*Digital Processes*, 7:21–32, 1981.Google Scholar - [20]J. MacGregor Smith, D. T. Lee, and Judith S. Liebman. An
*O*(*n*log*n*) heuristic algorithm for the rectilinear Steiner minimal tree problem.*Engineering Optimization*, 4:179–192, 1980.Google Scholar - [21]David I. Steinberg. The fixed charge problem.
*Naval Research Logistics Quarterly*, 17:217–225, 1970.Google Scholar - [22]V. A. Trubin. Subclass of the Steiner problems on a plane with rectilinear metric.
*Cybernetics (U.S.A.)*, 21(3):320–325, 1985. translated from*Kibernetika 21:3*, 37–40, May–June 1985.Google Scholar - [23]Y. Y. Yang and Omar Wing. Optimal and suboptimal solution algorithms for the wiring problem. In
*IEEE International Symposium on Circuit Theory*, pages 154–158, 1972.Google Scholar - [24]Y. Y. Yang and Omar Wing. Suboptimal algorithm for a wire routing problem.
*IEEE Transactions on Circuit Theory*, 508–510, September 1972.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1987