Advertisement

The Bundled Crossing Number

  • Md. Jawaherul Alam
  • Martin Fink
  • Sergey Pupyrev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all crossings in a drawing of the graph.

We show that the bundled crossing number is closely related to the orientable genus of the graph. If multiple crossings and self-intersections of edges are allowed, the two values are identical; otherwise, the bundled crossing number can be higher than the genus.

We then investigate the problem of minimizing the number of bundled crossings. For circular graph layouts with a fixed order of vertices, we present a constant-factor approximation algorithm. When the circular order is not prescribed, we get a \(\frac{6c}{c-2}\)-approximation for a graph with n vertices having at least cn edges for \(c>2\). For general graph layouts, we develop an algorithm with an approximation factor of \(\frac{6c}{c-3}\) for graphs with at least cn edges for \(c > 3\).

References

  1. 1.
    Ackerman, E., Pinchasi, R.: On the degenerate crossing number. Discret. Comput. Geom. 49(3), 695–702 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. N.-Holl. Math. Stud. 60, 9–12 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alam, M.J., Fink, M., Pupyrev, S.: The bundled crossing number. CoRR, cs.CG/1608.08161 (2016)Google Scholar
  4. 4.
    Baur, M., Brandes, U.: Crossing reduction in circular layouts. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 332–343. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30559-0_28 CrossRefGoogle Scholar
  5. 5.
    Bouts, Q.W., Speckmann, B.: Clustered edge routing. In: PacificVis 2015, pp. 55–62 (2015)Google Scholar
  6. 6.
    Buchheim, C., Chimani, M., Gutwenger, C., Jünger, M., Mutzel, P.: Crossings and planarization. In: Handbook of Graph Drawing and Visualization. CRC Press (2013)Google Scholar
  7. 7.
    Cabello, S.: Hardness of approximation for crossing number. Discret. Comput. Geom. 49(2), 348–358 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chuzhoy, J.: An algorithm for the graph crossing number problem. In: STOC 2011, pp. 303–312 (2011)Google Scholar
  9. 9.
    Cui, W., Zhou, H., Qu, H., Wong, P.C., Li, X.: Geometry-based edge clustering for graph visualization. TVCG 14(6), 1277–1284 (2008)Google Scholar
  10. 10.
    Fraysseix, H., Mendez, P.O.: Stretching of Jordan arc contact systems. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 71–85. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24595-7_7 CrossRefGoogle Scholar
  11. 11.
    Dickerson, M., Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent drawings: visualizing non-planar diagrams in a planar way. JGAA 9(1), 31–52 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eppstein, D., Holten, D., Löffler, M., Nöllenburg, M., Speckmann, B., Verbeek, K.: Strict confluent drawing. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 352–363. Springer, Heidelberg (2013). doi: 10.1007/978-3-319-03841-4_31 CrossRefGoogle Scholar
  13. 13.
    Ersoy, O., Hurter, C., Paulovich, F.V., Cantareiro, G., Telea, A.: Skeleton-based edge bundling for graph visualization. TVCG 17(12), 2364–2373 (2011)Google Scholar
  14. 14.
    Felsner, S., Valtr, P.: Coding and counting arrangements of pseudolines. Discret. Comput. Geom. 46(3), 405–416 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fink, M., Hershberger, J., Suri, S., Verbeek, K.: Bundled crossings in embedded graphs. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) LATIN 2016. LNCS, vol. 9644, pp. 454–468. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49529-2_34 CrossRefGoogle Scholar
  16. 16.
    Fink, M., Pupyrev, S., Wolff, A.: Ordering metro lines by block crossings. JGAA 19(1), 111–153 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gansner, E. Hu, Y., North, S., Scheidegger, C.: Multilevel agglomerative edge bundling for visualizing large graphs. In: PacificVis 2011, pp. 187–194. IEEE (2011)Google Scholar
  18. 18.
    Gansner, E.R., Koren, Y.: Improved circular layouts. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 386–398. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-70904-6_37 CrossRefGoogle Scholar
  19. 19.
    Giordano, F., Liotta, G., Mchedlidze, T., Symvonis, A., Whitesides, S.: Computing upward topological book embeddings of upward planar digraphs. J. Discret. Algorithms 30, 45–69 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Holten, D.: Hierarchical edge bundles: visualization of adjacency relations in hierarchical data. TVCG 12(5), 741–748 (2006)Google Scholar
  21. 21.
    Holten, D., van Wijk, J.J.: Force-directed edge bundling for graph visualization. Comput. Graph. Forum 28(3), 983–990 (2009)CrossRefGoogle Scholar
  22. 22.
    Lambert, A., Bourqui, R., Auber, D.: Winding roads: routing edges into bundles. Comput. Graph. Forum 29(3), 853–862 (2010)CrossRefGoogle Scholar
  23. 23.
    Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an orientable triangulated surface. In: SoCG 2001, pp. 80–89. ACM (2001)Google Scholar
  24. 24.
    Mohar, B.: The genus crossing number. ARS Math. Contempo. 2(2), 157–162 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Pach, J., Tóth, G.: Degenerate crossing numbers. Discret. Comput. Geom. 41(3), 376–384 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pupyrev, S., Nachmanson, L., Bereg, S., Holroyd, A.E.: Edge routing with ordered bundles. Comput. Geom. 52, 18–33 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schaefer, M.: The graph crossing number and its variants: a survey. Electron. J. Comb. Dyn. Surv. 21 (2013)Google Scholar
  28. 28.
    Schaefer, M., Štefankovič, D.: The degenerate crossing number and higher-genus embeddings. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 63–74. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-27261-0_6 CrossRefGoogle Scholar
  29. 29.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: Book embeddings and crossing numbers. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds.) WG 1994. LNCS, vol. 903, pp. 256–268. Springer, Heidelberg (1995). doi: 10.1007/3-540-59071-4_53 CrossRefGoogle Scholar
  30. 30.
    Thomassen, C.: The graph genus problem is NP-complete. J. Algorithms 10(4), 568–576 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yamanaka, K., Nakano, S., Matsui, Y., Uehara, R., Nakada, K.: Efficient enumeration of all ladder lotteries and its application. Theor. Comput. Sci. 411(16–18), 1714–1722 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • Martin Fink
    • 2
  • Sergey Pupyrev
    • 3
    • 4
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  3. 3.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  4. 4.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia

Personalised recommendations