This Article 
 Bibliographic References 
 Add to: 
Bottleneck Steiner Trees in the Plane
March 1992 (vol. 41 no. 3)
pp. 370-374

A Steiner tree with maximum-weight edge minimized is called a bottleneck Steiner tree (BST). The authors propose a Theta ( mod rho mod log mod rho mod ) time algorithm for constructing a BST on a point set rho , with points labeled as Steiner or demand; a lower bound, in the linear decision tree model, is also established. It is shown that if it is desired to minimize further the number of used Steiner points, then the problem becomes NP-complete. It is shown that when locations of Steiner points are not fixed the problem remains NP-complete; however, if the topology of the final tree is given, then the problem can be solved in Theta ( mod rho mod log mod rho mod ) time. The BST problem can be used, for example, in VLSI layout, communication network design, and (facility) location problems.

[1] L. P. Chew and R. L. Drysdale, "Voronoi diagrams based on convex distance functions," inProc. ACM Symp. Computat. Geometry, 1985, pp. 235-244.
[2] C. Chiang, M. Sarrafzadeh, and C. K. Wong, "A powerful global router: Based on Steiner min-max tree," inProc. ICCAD-89, alsoIEEE Trans. Comput.-Aided Design, vol. 19, no. 2, pp. 1318-1325, Dec. 1990.
[3] D. Cheriton and R. E. Tarjan, "Finding minimum spanning trees,"SIAM J. Comput., vol. 5, no. 4, pp. 724-742, Dec. 1976.
[4] Z. Drezner, "Competitive location strategy for two facilities,"Regional Sci. and Urban Econom., vol. 12, 1982, North-Holland, pp. 485-493.
[5] E. N. Gilbert, "Minimum cost communication networks,"Bell Syst. Tech. J., vol. 9, pp. 2209-2227, 1967.
[6] M. R. Garey, R. L. Graham, and D. S. Johnson, "The complexity of computing Steiner minimal tree,"SIAM J. Appl. Mathemat., vol. 32, no. 4, 1977.
[7] M. R. Garey and D. S. Johnson, "The rectilinear Steiner tree problem is NP-complete,"SIAM J. Appl. Mathemat., vol. 32, no. 4, 1977.
[8] M. R. Garey and D. S. Johnson,Computers and Intractability. San Francisco, CA: Freeman, 1979.
[9] F. K. Hwang, "On Steiner minimal tree with rectilinear distance,"SIAM J. Appl. Mathemat., vol. 30, no. 1, Jan. 1976.
[10] F. K. Hwang, "AnO(nlogn) algorithm for rectilinear minimal spanning trees,"J. ACM, vol. 26, no. 2, pp. 177-182, Apr. 1979.
[11] R. M. Karp, "Reducibility among combinatorial problems," inComplexity of Computer Computations. New York: Plenum, 1972, pp. 85-163.
[12] O. Kariv and S. L. Hakimi, "An algorithmic approach to network location problem--Part I: The p-centers," unpublished manuscript.
[13] D. Lichtenstein, "Planar formulae and their uses,"SIAM J. Comput., vol. 11, no. 2, pp. 329-343, 1982.
[14] R. J. Lipton and R. E. Tarjan, "A separator theorem for planar graphs,"SIAM J. Appl. Mathemat., vol. 36, no. 2, pp. 177-189, Apr. 1979.
[15] D. T. Lee, private communication.
[16] D. T. Lee, M. Sarrafadeh, and Y. F. Wu, "Minimum cut for circular-arc graphs,"SIAM J. Comput., vol. 9, no. 6, p. 1041-1050, Dec. 1990.
[17] D. T. Lee and C. K. Wong, "Voronoi diagrams inL1(L∞) metric with two-dimensional storage applications,"SIAM J. Comput., vol. 9, pp. 200-211, 1980.
[18] D. T. Lee and Y. F. Wu, "Geometric complexity of some location problem,"Algorithmica, vol. 1, pp. 193-211, 1985.
[19] U. Manber and M. Tompa, "Probabilistic, nondeterministic, and alternating decision trees," TR 82-03-01, Univ. of Washington.
[20] F. P. Preparata and D. E. Muller, "Finding the intersection ofnhalf spaces inO(nlogn) time,"Theoret. Comput. Sci., vol. 8, no. 1, pp. 45-55, Jan. 1979.
[21] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
[22] M. I. Shamos, "Geometric complexity," inProc. Seventh Annu. ACM SIGACT Conf., May 1975, pp. 224-233.
[23] J. Soukup, "On minimum cost networks with nonlinear costs,"SIAM J. Comput., vol. 29, pp. 571-581, 1975.
[24] P. Widmayer, Y. F. Wu, and C. K. Wong, "On some distance problems in fixed orientations,"SIAM J. Comput., vol. 16, no. 4, pp. 728-746, Aug. 1987.

Index Terms:
bottleneck Steiner trees; maximum-weight edge minimized; lower bound; linear decision tree model; NP-complete; VLSI layout; communication network design; location problems; computational complexity; optimisation; trees (mathematics).
M. Sarrafzadeh, C.K. Wong, "Bottleneck Steiner Trees in the Plane," IEEE Transactions on Computers, vol. 41, no. 3, pp. 370-374, March 1992, doi:10.1109/12.127452
Usage of this product signifies your acceptance of the Terms of Use.