Abstract
In this paper we present an O(n 2(m + logn))-time algorithm for computing a minimum-weight tree support (if one exists) of a hypergraph H = (V,S) with n vertices and m hyperedges. This improves the previously best known algorithm with running time O(n 4 m 2). A support of H is a graph G on V such that each hyperedge in S induces a connected subgraph in G. If G is a tree, it is called a tree support and it is a minimum tree support if its edge weight is minimum for a given edge weight function. Tree supports of hypergraphs have several applications, from social network analysis and network design problems to the visualization of hypergraphs and Euler diagrams. We show in particular how a minimum-weight tree support can be used to generate an area-proportional Euler diagram that satisfies typical well-formedness conditions and additionally minimizes the number of concurrent curves of the set boundaries in the Euler diagram.
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
Angluin, D., Aspnes, J., Reyzin, L.: Inferring social networks from outbreaks. In: Hutter, M., Stephan, F., Vovk, V., Zeugmann, T. (eds.) Algorithmic Learning Theory. LNCS, vol. 6331, pp. 104–118. Springer, Heidelberg (2010)
Angluin, D., Aspnes, J., Reyzin, L.: Network construction with subgraph connectivity constraints. J. Comb. Optim. (2013)
Brandes, U., Cornelsen, S., Pampel, B., Sallaberry, A.: Blocks of hypergraphs applied to hypergraphs and outerplanarity. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 201–211. Springer, Heidelberg (2011)
Buchin, K., van Kreveld, M., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. Technical Report UU-CS-2009-035, Utrecht University (2009)
Buchin, K., van Kreveld, M., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 345–356. Springer, Heidelberg (2010)
Chen, J., Komusiewicz, C., Niedermeier, R., Sorge, M., Suchý, O., Weller, M.: Effective and efficient data reduction for the subset interconnection design problem. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 361–371. Springer, Heidelberg (2013)
Chockler, G., Melamed, R., Tock, Y., Vitenberg, R.: Constructing scalable overlays for pub-sub with many topics. In: Principles of Distributed Computing (PODC 2007), pp. 109–118 (2007)
Chow, S.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. PhD thesis, University of Victoria (2007)
Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 466–477. Springer, Heidelberg (2004)
Du, D.-Z., Kelley, D.F.: On complexity of subset interconnection designs. J. Global Optim. 6, 193–205 (1995)
Fan, H., Hundt, C., Wu, Y.-L., Ernst, J.: Algorithms and implementation for interconnection graph problem. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 201–210. Springer, Heidelberg (2008)
Flower, J., Fish, A., Howse, J.: Euler diagram generation. J. Visual Languages and Computing 19(6), 675–694 (2008)
Hosoda, J., Hromkovič, J., Izumi, T., Ono, H., Steinová, M., Wada, K.: On the approximability and hardness of minimum topic connected overlay and its special instances. Theoretical Computer Science 429, 144–154 (2012)
Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. J. Graph Theory 11(3), 309–325 (1987)
Kaufmann, M., van Kreveld, M., Speckmann, B.: Subdivision drawings of hypergraphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 396–407. Springer, Heidelberg (2009)
Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Mathematical Programming 98(1-3), 385–414 (2003)
Korach, E., Stern, M.: The complete optimal stars-clustering-tree problem. Discrete Applied Mathematics 156, 444–450 (2008)
Rodgers, P.J., Zhang, L., Fish, A.: General Euler diagram generation. In: Stapleton, G., Howse, J., Lee, J. (eds.) Diagrams 2008. LNCS (LNAI), vol. 5223, pp. 13–27. Springer, Heidelberg (2008)
Stapleton, G., Rodgers, P., Howse, J.: A general method for drawing area-proportional Euler diagrams. J. Visual Languages and Computing 22(6), 426–442 (2011)
Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Klemz, B., Mchedlidze, T., Nöllenburg, M. (2014). Minimum Tree Supports for Hypergraphs and Low-Concurrency Euler Diagrams. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-08404-6_23
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08403-9
Online ISBN: 978-3-319-08404-6
eBook Packages: Computer ScienceComputer Science (R0)