Full minimal Steiner trees on lattice sets. (English) Zbl 0874.05018

Given a finite set of points in Euclidean plane, the Steiner tree problem is to find a shortest network interconnecting the points in the set. The authors determine the Steiner minimum tree for \(n\times n\) integer lattice points. This solves an open problem posed by F. Chung, M. Gardner and R. Graham [Math. Mag. 62, No. 2, 83-96 (1989; Zbl 0681.05018)].


05C05 Trees


Zbl 0681.05018
Full Text: DOI


[1] Brazil, M.; Cole, T.; Rubinstein, J. H.; Thomas, D. A.; Weng, J. F.; Wormald, N. C., Minimal Steiner trees for \(2^k\)×\(2^k\) square lattices, J. Combin. Theory Ser. A, 73, 91-110 (1996) · Zbl 0844.05036
[3] Chung, F. R.K.; Gardner, M.; Graham, R. L., Steiner trees on a checkerboard, Math. Magazine, 62, 83-96 (1989) · Zbl 0681.05018
[4] Chung, F. R.K.; Graham, R. L., Steiner trees for ladders, Ann. Distcrete Math., 2, 173-200 (1978) · Zbl 0384.05030
[5] Cockayne, E. J., On the efficiency of the algorithm for Steiner minimal trees, SIAM J. Appl. Math., 18, 150-159 (1970) · Zbl 0218.90064
[6] Du, D. Z.; Hwang, F. K.; Song, G. D.; Ting, G. Y., Steiner minimal trees on sets of four points, Discrete Comput. Geom., 2, 401-414 (1987) · Zbl 0623.05013
[7] Garey, M. R.; Graham, R. L.; Johnson, D. S., The complexity of computing Stiener minimal trees, SIAM J. Appl. Math., 32, 835-859 (1977) · Zbl 0399.05023
[8] Gilbert, E. N.; Pollak, H. O., Steiner minimal trees, SIAM J. Appl. Math., 16, 1-29 (1968) · Zbl 0159.22001
[9] Kuhn, F. W., Steiner’s problem revisited, Studies in Optimization. Studies in Optimization, Studies in Mathematics, 10 (1975), Math. Assoc. Amer: Math. Assoc. Amer Washington · Zbl 0347.90054
[10] Melzak, Z. A., On the problem of Steiner, Can. Math. Bull., 4, 143-148 (1961) · Zbl 0101.13201
[11] Pollak, H. O., Some remarks on the Steiner Problem, J. Combin. Theor. Ser. A, 24, 278-295 (1978) · Zbl 0392.05021
[12] Rubinstein, J. H.; Thomas, D. A., A variational approach to the Steiner network problem, Ann. Oper. Res., 33, 481-499 (1991) · Zbl 0734.05040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.