Abstract
In this paper we propose and study a new complexity model for approximation algorithms. The main motivation are practical problems over large data sets that need to be solved many times for different scenarios, e.g., many multicast trees that need to be constructed for different groups of users. In our model we allow a preprocessing phase, when some information of the input graph G = (V,E) is stored in a limited size data structure. Next, the data structure enables processing queries of the form “solve problem A for an input S ⊆ V”. We consider problems like Steiner Forest, Facility Location, k -Median, k -Center and TSP in the case when the graph induces a doubling metric. Our main results are data structures of near-linear size that are able to answer queries in time close to linear in |S|. This improves over typical worst case reuniting time of approximation algorithms in the classical setting which is Ω(|E|) independently of the query size. In most cases, our approximation guarantees are arbitrarily close to those in the classical setting. Additionally, we present the first fully dynamic algorithm for the Steiner tree problem.
This work was partially supported by the Polish Ministry of Science grant N206 355636.
This is a preview of subscription content, log in via an institution to check access.
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
Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)
Bilò, D., Böckenhauer, H.-J., Hromkovič, J., Královič, R., Mömke, T., Widmayer, P., Zych, A.: Reoptimization of steiner trees. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 258–269. Springer, Heidelberg (2008)
Böckenhauer, H.-J., Hromkovič, J., Královič, R., Mömke, T., Rossmanith, P.: Reoptimization of steiner trees: Changing the terminal set. Theor. Comput. Sci. 410(36), 3428–3435 (2009)
Böckenhauer, H.-J., Hromkovič, J., Mömke, T., Widmayer, P.: On the hardness of reoptimization. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 50–65. Springer, Heidelberg (2008)
Chan, H.T.-H., Gupta, A., Maggs, B.M., Zhou, S.: On hierarchical routing in doubling metrics. In: Proc. SODA 2005, pp. 762–771 (2005)
Clarkson, K.L.: Nearest neighbor queries in metric spaces. Discrete & Computational Geometry 22(1), 63–93 (1999)
Cole, R., Hariharan, R., Lewenstein, M., Porat, E.: A faster implementation of the Goemans-Williamson clustering algorithm. In: Proc. SODA 2001, pp. 17–25 (2001)
Escoffier, B., Milanic, M., Paschos, V.T.: Simple and fast reoptimizations for the Steiner tree problem. Algorithmic Operations Research 4(2), 86–94 (2009)
Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. In: Proc. SCG 2005, pp. 150–158 (2005)
Jia, L., Lin, G., Noubir, G., Rajaraman, R., Sundaram, R.: Universal aproximations for TSP, Steiner Tree and Set Cover. In: STOC 2005, pp. 1234–5415 (2005)
Krauthgamer, R., Lee, J.R.: Navigating nets: simple algorithms for proximity search. In: Proc. SODA 2004, pp. 798–807 (2004)
Mahdian, M., Ye, Y., Zhang, J.: Approximation algorithms for metric facility location problems. SIAM Journal on Computing 36(2), 411–432 (2006)
Salama, H.F., Reeves, D.S., Viniotis, Y., Sheu, T.-L.: Evaluation of multicast routing algorithms for real-time communication on high-speed networks. In: Proceedings of the IFIP Sixth International Conference on High Performance Networking VI, pp. 27–42 (1995)
Thorup, M.: Quick and good facility location. In: Proc. SODA 2003, pp. 178–185 (2003)
Thorup, M.: Quick k-median, k-center, and facility location for sparse graphs. SIAM Journal on Computing 34(2), 405–432 (2005)
Winter, P.: Steiner problem in networks: A survey. Networks 17(2), 129–167 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cygan, M., Kowalik, L., Mucha, M., Pilipczuk, M., Sankowski, P. (2010). Fast Approximation in Subspaces by Doubling Metric Decomposition . In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-15775-2_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15774-5
Online ISBN: 978-3-642-15775-2
eBook Packages: Computer ScienceComputer Science (R0)