Minimum Tree Supports for Hypergraphs and Low-Concurrency Euler Diagrams

  • Boris Klemz
  • Tamara Mchedlidze
  • Martin Nöllenburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)

Abstract

In this paper we present an O(n2(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(n4m2). 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Angluin, D., Aspnes, J., Reyzin, L.: Network construction with subgraph connectivity constraints. J. Comb. Optim. (2013)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Chow, S.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. PhD thesis, University of Victoria (2007)Google Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Du, D.-Z., Kelley, D.F.: On complexity of subset interconnection designs. J. Global Optim. 6, 193–205 (1995)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Flower, J., Fish, A., Howse, J.: Euler diagram generation. J. Visual Languages and Computing 19(6), 675–694 (2008)CrossRefGoogle Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. J. Graph Theory 11(3), 309–325 (1987)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Mathematical Programming 98(1-3), 385–414 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Korach, E., Stern, M.: The complete optimal stars-clustering-tree problem. Discrete Applied Mathematics 156, 444–450 (2008)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    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)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Boris Klemz
    • 1
  • Tamara Mchedlidze
    • 1
  • Martin Nöllenburg
    • 1
  1. 1.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)Germany

Personalised recommendations