Abstract
The problem of constructing Steiner minimal trees in the Euclidean plane is NP-hard. When in addition obstacles are present, difficulties of constructing obstacle-avoiding Steiner minimal trees are compounded. This problem, which has many obvious practical applications when designing complex transportation and distribution systems, has received very little attention in the literature. The construction of Steiner minimal trees for three terminal points in the Euclidean plane (without obstacles) has been completely solved (among others by Fermat, Torricelli, Cavallieri, Simpson, Heinen) during the span of the last three centuries. This construction is a cornerstone for both exact algorithms and heuristics for the Euclidean Steiner tree problem with arbitrarily many terminal points. An algorithm for three terminal points in the presence of one polygonal convex obstacle is given. It is shown that this algorithm has the worst-case time complexityO(n), wheren is the number of extreme points on the obstacle. As an extension to the underlying algorithm, if the obstacle is appropriately preprocessed inO(n) time, we can solve any problem instance with three arbitrary terminal points and the preprocessed convex polygonal obstacle inO(logn) time. We believe that the three terminal points algorithm will play a critical role in the development of heuristics for problem instances with arbitrarily many terminal points and obstacles.
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Abbreviations
- S-point:
-
Steiner point
- Z-point:
-
Terminal point
- EMSTO:
-
Euclidean minimum spanning tree with obstacles
- ESTP:
-
Euclidean Steiner tree problem
- ESTPO:
-
Euclidean Steiner tree problem with obstacles
- STP:
-
Steiner tree problem
- STPO:
-
Steiner tree problem with obstacles
References
A. Armillotta and G. Mummolo, A heuristic algorithm for the Steiner problem with obstacles, Technical Report, Dipt. di Pregettazione e Produzione Industriale, Univ. degli Studi di Bari, Italy (1988).
M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).
E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16(1968)1–29.
F.K. Hwang and D. Richards, Steiner tree problems, Networks (1989), to appear.
F.P. Preparata and M.I. Shamos,Computational Geometry: An Introduction (Springer, 1985).
J. Scott Provan, An approximation scheme for finding Steiner trees with obstacles, SIAM J. Comput. 17(1988)920–934.
J. MacGregor Smith and J.S. Liebman, Steiner trees, Steiner circuits, and the interference problem in building design, Eng. Design 4(1979)15–36.
J. MacGregor Smith, D.T. Lee and J.S. Liebman, AnO(n logn) heuristic for the Steiner minimal tree problem on the Euclidean metric, Networks 11(1981)23–29.
J. MacGregor Smith, Steiner minimal trees with obstacles, Paper presented at the ORSA/TIMS Meeting, Detroit, MI (1982).
J. MacGregor Smith, Generalized Steiner network problems in engineering design, in:Design Optimization, ed. J.S. Gero (Academic Press, 1985), ch. 5.
R.E. Tarjan,Data Structures and Network Algorithms (SIAM, Philadelphia, 1983).
P. Winter, An algorithm for the Steiner problem in the Euclidean plane, Networks 15(1985)323–345.
P. Winter, Steiner problem in networks: A survey, Networks 17(1987)129–167.
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Winter, P., MacGregor Smith, J. Steiner minimal trees for three points with one convex polygonal obstacle. Ann Oper Res 33, 577–599 (1991). https://doi.org/10.1007/BF02067243
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DOI: https://doi.org/10.1007/BF02067243