# Shortest Networks on Surfaces

• J. F. Weng
Chapter

## Abstract

Suppose A = {a l, a 2, ... , a n } is a point set in a metric space M. The shortest network problem asks for a minimum length network S(A) that interconnects all points of A (called terminals), possibly with some additional points to shorten the network. S(A) must be a tree since it cannot contain any cycle for minimality. In the literature this problem is called the Steiner tree problem, and S(A) is called a Steiner minimal tree for A [9]. If no additional points are added, then the network, denoted by T(A), is called a minimal spanning tree on A. Sometimes these networks are simply denoted by S and T if no confusion is caused.

## Keywords

Minimal Span Tree Steiner Tree Euclidean Plane Steiner Point Steiner Tree Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng and N.C. Wormald, Shortest Networks on Spheres, Proceedings of the DIMACS Workshop on Network Design, to appear.Google Scholar
2. [2]
J. Cheeger and D.G. Ebin, Comparison theorems in Riemannian geometry, ( North-Holland Publishing Co., Amsterdam, 1975 ).
3. [3]
E.J. Cockayne, On Fermat’s problem on the surface of a sphere, Math. Magazine, Vol. 45 (1972), pp. 216–219.
4. [4]
J. Dolan, R. Weiss and J. MacGregor Smith, Minimal length tree networks on the unit sphere, Annals of Oper. Res., Vol. 33 (1991), pp. 503–535.
5. [5]
D.Z. Du and F.K. Hwang, A proof of the Gilbert-Pollak Conjecture on the Steiner ratio, Algorithmica, Vol. 7 (1992), pp. 121–135.
6. [6]
D.Z. Du, F.K. Hwang and J.F. Weng, Steiner minimal trees for regular polygons, Discrete Comput. Geometry, Vol. 2 (1987), pp. 65–87.
7. [7]
M.R. Carey, R.L. Graham and D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math., Vol. 32 (1977), pp. 835–859.
8. [8]
F.K. Hwang and J.F. Weng, Hexagonal coordinate systems and Steiner minimal trees, Discrete Math., Vol. 62 (1986), pp. 49–57.
9. [9]
F.K. Hwang, D.S. Richard and P. Winter, The Steiner tree problem, ( North-Holland Publishing Co., Amsterdam, 1992 ).
10. [10]
J.H. Rubinstein and D.A. Thomas, A variational approach to the Steiner network problem, Ann. Oper. Res., Vol. 33 (1991), pp. 481–499.
11. [11]
J.H. Rubinstein and D.A. Thomas, Graham’s problem on shortest networks for points on a circle, Algorithmica, Vol. 7 (1992), pp. 193–218.
12. [12]
J.H. Rubinstein and J.F. Weng, Compression theorems and Steiner ratios on spheres, J. Combin. Optimization, Vol. 1 (1997) pp. 67–78.
13. [13]
I. Todhunter and J.G. Leathem, Spherical trigonometry, ( Macmillan and CO., London, 1901 ).Google Scholar
14. [14]
J.F. Weng, Steiner minimal trees on vertices of regular polygons (Chinese. English summary), Acta Math. Appl. Sinica, Vol. 8 (1985), pp. 129–141.
15. [15]
J.F. Weng, Generalized Steiner problem and hexagonal coordinate system ( Chinese. English summary), Acta. Math. Appl. Sinica, Vol. 8 (1985), pp. 383–397.
16. [16]
J.F. Weng, Steiner trees on curved surfaces, preprint, 1997.Google Scholar
17. [17]
J.F. Weng, Shortest networks for cocircular points on spheres, J. Combin. Optimization, to appear.Google Scholar
18. [18]
J.F. Weng, Finding Steiner points on spheres, preprint, 1997.Google Scholar
19. [19]
J.F. Weng and J.H. Rubinstein, A note on the compression theorem for convex surfaces, preprint, 1996.Google Scholar