Abstract
In this paper, motivated by many practical applications, we address the 1-line minimum rectilinear Steiner tree (1L-MRStT) problem, which is a variation of the Euclidean minimum rectilinear Steiner tree problem. More specifically, given n points in the Euclidean plane \({\mathbb {R}}^2\), it is asked to find the location of a line l and a Steiner tree T(l), consisting only of vertical and horizontal line segments plus several successive segments located on this line l, to interconnect these n points and at least one point located on the line l, the objective is to minimize total weight of this Steiner tree T(l), i.e., \(\min \{\sum _{uv\in T(l)} w(u,v)\) | T(l) is a Steiner tree mentioned-above\(\}\), where we define a weight \(w(u,v)=0\) if the two endpoints u and v of that edge \(uv \in T(l)\) is located on the line l and otherwise we define a weight w(u, v) as the rectilinear distance between the two endpoints u and v of that edge \(uv \in T(l)\). Given a line l as an input in \({\mathbb {R}}^2\), we denote this problem as the 1-line-fixed minimum rectilinear Steiner tree (1LF-MRStT) problem; Furthermore, when the Steiner points of T(l) are all located on the fixed line l, we recall this problem as the 1-line-fixed-constrained minimum rectilinear Steiner tree (1LFC-MRStT) problem. We provide three following main contributions. (1) We design an algorithm \({{\mathcal {A}}}_{C}\) to optimally solve the 1LFC-MRStT problem, where the algorithm \({{\mathcal {A}}}_{C}\) runs in time \(O(n\log n)\); (2) We prove that this algorithm \({{\mathcal {A}}}_{C}\) is a 1.5-approximation algorithm to solve the 1LF-MRStT problem; (3) Combining the algorithm \({{\mathcal {A}}}_{C}\) for many times and a key lemma proved by some techniques of computational geometry, we present a 1.5-approximation algorithm to solve the 1L-MRStT problem, where this algorithm runs in time \(O(n^3\log n)\), and we finally provide another approximation algorithm to solve a special version of the 1L-MRStT problem, where that new algorithm runs in lower time \(O(n^2\log n)\).
Similar content being viewed by others
References
Aazami A, Cheriyan J, Jampani KR (2012) Approximation algorithms and hardness results for packing element-disjoint Steiner trees in planar graphs. Algorithmica 63(1–2):425–456
Arora S (1998) Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J ACM 45(5):753–782
Baratz A (1983) Algorithms for integrated circuit signal routing. PhD thesis, Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge
Berg M, Cheong O, Kreveld M, Overmars M (2008) Computational geometry: algorithms and applications. Springer, New York
Bern M, Plassmann P (1989) The Steiner problem with edge lengths 1 and 2. Inf Process Lett 32(4):171–176
Byrka J, Grandoni F, Rothvoß T, Sanità L (2010) An improved LP-based approximation for Steiner tree. In: Proceedings of the 2010 ACM International Symposium on Theory of Computing, pp 583–592
Chen G, Zhang G (2000) A constrained minimum spanning tree problem. Comput Oper Res 27(9):867–875
Cheng X, Du D (2001) Steiner Trees in Industry. Combinatorial Optimization 11, Springer US
Chung F, Graham R (1982) A new bound for Euclidean Steiner minimal trees. Ann New York Acad Sci 440(1):328–346
Cieslik D (1998) Steiner Minimal Trees. Kluwer Academic Publishers, Amsterdam
Garey M, Graham R, Johnson D (1977) The complexity of computing Steiner minimal trees. SIAM J Appl Math 32(4):835–859
Georgakopoulos G, Papadimitriou CH (1987) The 1-Steiner tree problem. J Alg 8(1):122–130
Gilbert EN, Pollak HO (1968) Steiner minimal trees. SIAM J Appl Math 16(1):1–29
Holby J (2017) Variations on the Euclidean Steiner tree problem and algorithms. Rose-Hulman Undergrad Math J 18(1):123–155
Hwang FK (1976) On Steiner minimal trees with rectilinear distance. SIAM J Appl Math 30(1):104–114
Hwang FK (1979) An \(O ( n \log n )\) algorithm for rectilinear minimal spanning trees. J ACM 26(2):177–182
Hwang FK, Richards DS (1992) Steiner tree problems. Networks 22(1):55–89
Imase M, Waxman BM (1991) Dynamic Steiner tree problem. SIAM J Dis Math 4(3):369–384
Johnson MR, Garey DS (1977) The rectilinear Steiner tree problem is NP-complete. SIAM J Appl Math 32(4):826–834
Korte B, Vygen J (2008) Combinatorial optimization: theory and algorithms. Springer, Berlin
Kruskal JB (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proceed Am Math Soc 7(1):48–50
Li JP, Liu SD, Lichen JR, Wang WC, Zheng YJ (2020) Approximation algorithms for solving the 1-line Euclidean minimum Steiner tree problem. J Comb Opt 39(2):492–508
Morris JG, Norback JP (1980) A simple approach to linear facility location. Transp Sci 14(1):1–8
Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36(6):1389–1401
Richards DS (1989) Fast heuristic algorithms for rectilinear Steiner trees. Algorithmica 4(1–4):191–207
Salowe JS (1992) A simple proof of the planar rectilinear Steiner ratio. Oper Res Lett 12(4):271–274
Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin
Shamos MI, Hoey D (1975) Closest-point problems. In: The 16th Annual Symposium on Foundations of Computer Science. IEEE Computer Society, pp 151–162
Williamson DP, Shmoys DB (2011) The design of approximation algorithms. Cambridge University Press, Cambridge
Yao AC (1982) On constructing minimum spanning trees in \(k\)-dimensional spaces and related problems. SIAM J Comput 11(4):721–736
Zhou H, Shenoy N, Nicholls W (2002) Efficient minimum spanning tree construction without Delaynay triangulation. Inf Proc Lett 81(5):271–276
Acknowledgements
The authors are indeed grateful to an anonymous editor and two anonymous reviewers whose kind suggestions and comments have led to a substantially improved presentation for this manuscript.
This paper is supported by Project of the National Natural Science Foundation of China [Nos.11861075, 12101593], Project for Innovation Team (Cultivation) of Yunnan Province, Joint Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No.018FY001014] and Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province [C176240111009]. Jianping Li is also supported by Project of Yunling Scholars Training of Yunnan Province, and J.R. Lichen is also supported Project of Doctorial Fellow Award of Yunnan Province [No. 2018010514].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, J., Lichen, J., Wang, W. et al. 1-line minimum rectilinear steiner trees and related problems. J Comb Optim 44, 2832–2852 (2022). https://doi.org/10.1007/s10878-021-00796-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-021-00796-0