论文信息 - Shortest Networks for Smooth Curves

Shortest Networks for Smooth Curves

In this paper we set up a new model of shortest networks that interconnects a set of smooth curves and avoids a set of smoothly bounded obstacles. Using the hexagonal coordinate system we show how the problem of determining a full Steiner tree with a given topology in such a network can be converted to a problem of solving a set of simultaneous equations. Moreover, the number of equations is linearly dependent on the number of curves and obstacles if all curves and all boundaries of obstacles are convex. Hence, any existing numerical methods and computer programs for solving equations can be used to solve this shortest network problem.

J. F. Weng

[1] J. Scott Provan. An Approximation Scheme for Finding Steiner Trees with Obstacles , 1988, SIAM J. Comput..

[2] Dan Trietsch. Interconnecting networks in the plane: The steiner case , 1990, Networks.

[3] Dan Trietsch. Augmenting Euclidean Networks—the Steiner Case , 1985 .

[4] F. Hwang,et al. A linear time algorithm for full steiner trees , 1986 .

[5] H. Pollak,et al. Steiner Minimal Trees , 1968 .

[6] Pawel Winter,et al. Steiner minimal trees for three points with one convex polygonal obstacle , 1991, Ann. Oper. Res..

[7] E. Gilbert. Minimum cost communication networks , 1967 .

[8] Z. A. Melzak,et al. Steiner’s problem for set-terminals , 1968 .

[9] Frank K. Hwang,et al. Hexagonal coordinate systems and steiner minimal trees , 1986, Discret. Math..

[10] E. Cockayne. ON THE EFFICIENCY OF THE ALGORITHM FOR STEINER MINIMAL TREES , 1970 .

[11] J. Weng. Steiner polygons in the Steiner problem , 1994 .

[12] David S. Johnson,et al. The Complexity of Computing Steiner Minimal Trees , 1977 .

[13] Z. A. Melzak. On the Problem of Steiner , 1961, Canadian Mathematical Bulletin.