Abstract
We study the problem of finding non-crossing minimum-link \(\mathcal{C}\)-oriented paths that are homotopic to a set of input paths in an environment with \(\mathcal{C}\)-oriented obstacles. We introduce a special type of \(\mathcal{C}\)-oriented paths—smooth paths—and present a 2-approximation algorithm that runs in O(n 2 (n + logκ) + k in logn) time, where n is the total number of paths and obstacle vertices, k in is the total number of links in the input, and \(\kappa = |\mathcal{C}|\). The algorithm also computes an O(κ)-approximation for general \(\mathcal{C}\)-oriented paths. As a related result we show that, given a set of \(\mathcal{C}\)-oriented paths with L links in total, non-crossing \(\mathcal{C}\)-oriented paths homotopic to the input paths can require a total of Ω(L logκ) links.
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
Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. J. Alg. 49(2), 284–303 (2003)
Cabello, S., de Berg, M., van Kreveld, M.: Schematization of networks. Comp. Geom.: Theory and Appl. 30(3), 223–238 (2005)
Cabello, S., Liu, Y., Mantler, A., Snoeyink, J.: Testing homotopy for paths in the plane. Discrete & Computational Geometry 31, 61–81 (2004)
Cole, R., Siegel, A.: River routing every which way, but loose. In: Proc. 25th Symp. Found. Comp. Sci., pp. 65–73 (1984)
Duncan, C.A., Efrat, A., Kobourov, S.G., Wenk, C.: Drawing with fat edges. Intern. J. Found. Comp. Sci. 17(5), 1143–1163 (2006)
Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. Comp. Geom.: Theory and Appl. 35(3), 162–172 (2006)
Gao, S., Jerrum, M., Kaufmann, M., Mehlhorn, K., Rülling, W.: On continuous homotopic one layer routing. In: Proc. 4th Symp. Comp. Geom., pp. 392–402 (1988)
Gupta, H., Wenger, R.: Constructing pairwise disjoint paths with few links. ACM Trans. Alg. 3(3), 26 (2007)
Leiserson, C.E., Maley, F.M.: Algorithms for routing and testing routability of planar VLSI layouts. In: Proc. 17th Symp. Theory. Comp., pp. 69–78 (1985)
Maley, F.M.: Single-layer wire routing and compaction. MIT Press, Cambridge (1990)
Merrick, D., Gudmundsson, J.: Path Simplification for Metro Map Layout. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 258–269. Springer, Heidelberg (2007)
Neyer, G.: Line Simplification with Restricted Orientations. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 13–24. Springer, Heidelberg (1999)
Nöllenburg, M., Wolff, A.: Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE Transactions on Visualization and Computer Graphics 17(5), 626–641 (2011)
Speckmann, B., Verbeek, K.: Homotopic Rectilinear Routing with Few Links and Thick Edges. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 468–479. Springer, Heidelberg (2010)
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
Verbeek, K. (2013). Homotopic \(\mathcal{C}\)-Oriented Routing. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-36763-2_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36762-5
Online ISBN: 978-3-642-36763-2
eBook Packages: Computer ScienceComputer Science (R0)