Abstract
We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in ℝ2. The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an O(logn)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple O(logd + 1 n)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(n ε)-ratio for MMN in d dimensions [ESA’11]. En route, we show that an existing O(logn)-approximation algorithm for 2D-RSA generalizes to higher dimensions.
This work was supported by the ESF EuroGIGA project GraDR (DFG grant Wo 758/5-1).
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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
Arora, S.: Approximation schemes for NP-hard geometric optimization problems: A survey. Math. Program. 97(1-2), 43–69 (2003)
Chepoi, V., Nouioua, K., Vaxès, Y.: A rounding algorithm for approximating minimum Manhattan networks. Theor. Comput. Sci. 390(1), 56–69 (2008)
Chin, F., Guo, Z., Sun, H.: Minimum Manhattan network is NP-complete. Discrete Comput. Geom. 45, 701–722 (2011)
Cong, J., Leung, K.S., Zhou, D.: Performance-driven interconnect design based on distributed RC delay model. In: 30th IEEE Conf. Design Automation (DAC 1993), pp. 606–611. IEEE Press, New York (1993)
Das, A., Fleszar, K., Kobourov, S.G., Spoerhase, J., Veeramoni, S., Wolff, A.: Approximating the generalized minimum Manhattan network problem. Arxiv report (2012), http://arxiv.org/abs/1203.6481
Das, A., Gansner, E.R., Kaufmann, M., Kobourov, S., Spoerhase, J., Wolff, A.: Approximating minimum Manhattan networks in higher dimensions. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 49–60. Springer, Heidelberg (2011), to appear in Algorithmica, http://dx.doi.org/10.1007/s00453-013-9778-z
Gudmundsson, J., Klein, O., Knauer, C., Smid, M.: Small Manhattan networks and algorithmic applications for the Earth Mover’s Distance. In: 23rd Europ. Workshop Comput. Geom (EuroCG 2007), Graz, Austria, pp. 174–177 (2007)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Approximating a minimum Manhattan network. Nordic J. Comput. 8, 219–232 (2001)
Gudmundsson, J., Narasimhan, G., Smid, M.: Applications of geometric spanner networks. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 1–99. Springer (2008)
Guo, Z., Sun, H., Zhu, H.: Greedy construction of 2-approximate minimum Manhattan networks. Int. J. Comput. Geom. Appl. 21(3), 331–350 (2011)
Knauer, C., Spillner, A.: A fixed-parameter algorithm for the minimum Manhattan network problem. J. Comput. Geom. 2(1), 189–204 (2011)
Lu, B., Ruan, L.: Polynomial time approximation scheme for the rectilinear Steiner arborescence problem. J. Comb. Optim. 4(3), 357–363 (2000)
Muñoz, X., Seibert, S., Unger, W.: The minimal Manhattan network problem in three dimensions. In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 369–380. Springer, Heidelberg (2009)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press (2007)
Nastansky, L., Selkow, S.M., Stewart, N.F.: Cost-minimal trees in directed acyclic graphs. Zeitschrift Oper. Res. 18(1), 59–67 (1974)
Nouioua, K.: Enveloppes de Pareto et Réseaux de Manhattan: Caractérisations et Algorithmes. PhD thesis, Université de la Méditerranée (2005), http://www.lif-sud.univ-mrs.fr/~karim/download/THESE_NOUIOUA.pdf
Rao, S., Sadayappan, P., Hwang, F., Shor, P.: The rectilinear Steiner arborescence problem. Algorithmica 7, 277–288 (1992)
Shi, W., Su, C.: The rectilinear Steiner arborescence problem is NP-complete. SIAM J. Comput. 35(3), 729–740 (2005)
Zachariasen, M.: On the approximation of the rectilinear Steiner arborescence problem in the plane (2000) (Manuscript), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.4529
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Das, A., Fleszar, K., Kobourov, S., Spoerhase, J., Veeramoni, S., Wolff, A. (2013). Approximating the Generalized Minimum Manhattan Network Problem. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_67
Download citation
DOI: https://doi.org/10.1007/978-3-642-45030-3_67
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45029-7
Online ISBN: 978-3-642-45030-3
eBook Packages: Computer ScienceComputer Science (R0)