Abstract
Let P be a set of n points in a metric space. A Steiner Minimal Tree (SMT) on P is a shortest network interconnecting P while a Minimum Spanning Tree (MST) is a shortest network interconnecting P with all edges between points of P. The Steiner ratio is the infimum over P of ratio of the length of SMT over that of MST. Steiner ratio problem is to determine the value of the ratio. In this paper we consider the Steiner ratio problem in uniform orientation metrics, which find important applications in VLSI design. Our study is based on the fact that lengths of MSTs and SMTs could be reduced through properly rotating coordinate systems without increasing the number of orientation directions. We obtain the Steiner ratios with |P| = 3 when rotation is allowed and some bounds of Steiner ratios for general case.
Supported in part by NSF of China under Grant No. 70221001, 10401038, 10531070, and the Key Project of Chinese Ministry of Education under grant No. 106008.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S.: Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)
Brazil, M., Nielsen, B.K., Winter, P., Zachariasen, M.: Rotationally Optimal Spanning and Steiner Trees in Uniform Orientation Metrics. Computational Geometry: Theory and Applications 29(3), 251–263 (2004)
Brazil, M., Thomas, D.A., Wang, J.F.: Minimum Networks in Uniform Orientation Metrics. SIAM Journal on Computing 30, 1579–1593 (2000)
Cieslik, D.: The Steiner Ratio. Kluwer Academic Publishers, Boston (2001)
Du, D.Z., Hwang, F.K.: A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratio. Algorithmica 7(2-3), 121–135 (1992)
Garey, M.R., Johnson, D.S.: The Rectilinear Steiner Tree Problem is NP-Complete. SIAM Journal of Applied Mathematics 32(4), 826–834 (1977)
Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-completeness. Freeman, San Francisico (1979)
Hanan, M.: On Steiner’s Problem with Rectilinear Distance. SIAM J. Appl. Math. 14(2), 255–265 (1966)
Hwang, F.K.: On Steiner Minimal Trees with Rectilinear Distance. SIAM Journal of Applied Mathematics 30(1), 104–114 (1976)
Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. North Holland, Springer Verlag, Berlin Heidelberg New York (1994)
Lee, D.T., Shen, C.F.: The Steiner Minimal Tree Problem in the λ-Geormetry Plane. In: Nagamochi, H., Suri, S., Igarashi, Y., Miyano, S., Asano, T. (eds.) ISAAC 1996. LNCS, vol. 1178, pp. 247–255. Springer, Berlin Heidelberg New York (1996)
Lee, D.T., Shen, C.F., Ding, C.-L.: On Steiner Tree Problem with 45° Routing. In: Proc. IEEE Int. Symp. on Circuit and Systems, pp. 1680–1682 (1995)
Teig, S.L.: The X-Architecture: Not Your Father’s Diagonal Wiring. In: Proc. ACM International Workshop on System -Level Interconnect Prediction, pp. 33-37 (2002)
Widmayer, P., Wu, Y.F., Wong, C.K.: On Some Distance Problem in Orientation Metrics. Journal on Computing 16, 728–746 (1987)
X Initiative Homepage, www.xinitiative.org (March 2003)
Zhu, Q., Zhou, H., Jing, T., Hong, X.-L., Yang, Y.: Spanning Graph-Based Nonrectilinear Steiner Tree Algorithms. IEEE Trans. on CAD 24(7), 1066–1075 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Shang, S., Hu, X., Jing, T. (2007). Rotational Steiner Ratio Problem Under Uniform Orientation Metrics. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds) Discrete Geometry, Combinatorics and Graph Theory. CJCDGCGT 2005. Lecture Notes in Computer Science, vol 4381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70666-3_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-70666-3_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70665-6
Online ISBN: 978-3-540-70666-3
eBook Packages: Computer ScienceComputer Science (R0)