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Experimental Evaluation of Book Drawing Algorithms

  • Jonathan Klawitter
  • Tamara Mchedlidze
  • Martin Nöllenburg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

Abstract

A k-page book drawing of a graph \(G=(V,E)\) consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the k pages, which are half-planes bounded by the spine. In a book drawing, two edges cross if and only if they are assigned to the same page and their vertices alternate along the spine. Crossing minimization in a k-page book drawing is NP-hard, yet book drawings have multiple applications in visualization and beyond. Therefore several heuristic book drawing algorithms exist, but there is no broader comparative study on their relative performance. In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing algorithms.

References

  1. 1.
    Alam, M.J., Brandenburg, F.J., Kobourov, S.G.: On the book thickness of 1-planar graphs. CoRR, abs/1510.05891, October 2015. http://arxiv.org/abs/1510.05891
  2. 2.
    Bannister, M.J., Eppstein, D.: Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 210–221. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45803-7_18 Google Scholar
  3. 3.
    Bansal, R., Srivastava, K., Varshney, K., Sharma, N., et al.: An evolutionary algorithm for the 2-page crossing number problem. In: Evolutionary Computation (CEC 2008), pp. 1095–1102. IEEE, June 2008.  https://doi.org/10.1109/CEC.2008.4630933
  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).  https://doi.org/10.1007/978-3-540-30559-0_28 CrossRefGoogle Scholar
  5. 5.
    Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theor. Ser. B 27(3), 320–331 (1979). http://dx.doi.org/10.1016/0095-8956(79)90021-2 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to vlsi design. SIAM J. Algebraic Discrete Methods 8(1), 33–58 (1987). http://dx.doi.org/10.1137/0608002 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cibulka, J.: Simulated annealing book embedder (2015). https://github.com/josefcibulka/book-embedder
  8. 8.
    Cimikowski, R.: Algorithms for the fixed linear crossing number problem. Discrete Appl. Math. 122(1), 93–115 (2002). http://dx.doi.org/10.1016/S0166-218X(01)00314-6 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cimikowski, R.: An analysis of some linear graph layout heuristics. J. Heuristics 12(3), 143–153 (2006). http://dx.doi.org/10.1007/s10732-006-4294-9 CrossRefzbMATHGoogle Scholar
  10. 10.
    Cimikowski, R., Mumey, B.: Approximating the fixed linear crossing number. Discrete Appl. Math. 155(17), 2202–2210 (2007). http://dx.doi.org/10.1016/j.dam.2007.05.009 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Clote, P., Dobrev, S., Dotu, I., Kranakis, E., Krizanc, D., Urrutia, J.: On the page number of RNA secondary structures with pseudoknots. J. Math. Biol. 65(6–7), 1337–1357 (2012). http://dx.doi.org/10.1007/s00285-011-0493-6 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    de Klerk, E., Pasechnik, D.V., Salazar, G.: Improved lower bounds on book crossing numbers of complete graphs. SIAM J. Discrete Math. 27(2), 619–633 (2013). http://dx.doi.org/10.1137/120886777 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dujmović, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discrete Comput. Geom. 37(4), 641–670 (2007). http://dx.doi.org/10.1007/s00454-007-1318-7 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dynnikov, I.A.: Three-page approach to knot theory. Encoding and local moves. Funct. Anal. Appl. 33(4), 260–269 (1999). http://dx.doi.org/10.1007/BF02467109 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ganley, J.L., Heath, L.S.: The pagenumber of \(k\)-trees is \({O}(k)\). Discrete Appl. Math. 109(3), 215–221 (2001). http://dx.doi.org/10.1016/S0166-218X(00)00178-5 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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).  https://doi.org/10.1007/978-3-540-70904-6_37 CrossRefGoogle Scholar
  17. 17.
    Giacomo, E.D., Didimo, W., Liotta, G., Wismath, S.K.: Book embeddability of series-parallel digraphs. Algorithmica 45(4), 531–547 (2006). http://dx.doi.org/10.1007/s00453-005-1185-7 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    He, H., Sýkora, O.: New circular drawing algorithms. In: Workshop on Information Technologies - Applications and Theory (ITAT 2004) (2004). https://dspace.lboro.ac.uk/2134/2386
  19. 19.
    He, H., Sýkora, O., Mäkinen, E.: Genetic algorithms for the 2-page book drawing problem of graphs. J. Heuristics 13(1), 77–93 (2007). http://dx.doi.org/10.1007/s10732-006-9000-4 CrossRefGoogle Scholar
  20. 20.
    He, H., Sýkora, O., Mäkinen, E.: An improved neural network model for the two-page crossing number problem. IEEE Trans. Neural Netw. 17(6), 1642–1646 (2006). http://dx.doi.org/10.1109/TNN.2006.881486 CrossRefGoogle Scholar
  21. 21.
    He, H., Sýkora, O., Salagean, A., Vrt’o, I.: Heuristic crossing minimisation algorithms for the two-page drawing problem. Technical report, Loughborough University (2006). https://dspace.lboro.ac.uk/2134/2377
  22. 22.
    He, H., Sýkora, O., Vrt’o, I.: Crossing minimisation heuristics for 2-page drawings. Electron. Notes Discrete Math. 22, 527–534 (2005). http://dx.doi.org/10.1016/j.endm.2005.06.088 CrossRefzbMATHGoogle Scholar
  23. 23.
    He, H., Sălăgean, A., Mäkinen, E., Vrt’o, I.: Various heuristic algorithms to minimise the two-page crossing numbers of graphs. Open Comput. Sci. 5(1), 22–40 (2015). https://doi.org/10.1515/comp-2015-0004 CrossRefGoogle Scholar
  24. 24.
    Kapoor, N., Russell, M., Stojmenovic, I., Zomaya, A.Y.: A genetic algorithm for finding the pagenumber of interconnection networks. J. Parallel Distrib. Comput. 62(2), 267–283 (2002). http://dx.doi.org/10.1006/jpdc.2001.1789 CrossRefzbMATHGoogle Scholar
  25. 25.
    Kindermann, P., Löffler, M., Nachmanson, L., Rutter, I.: Graph drawing contest report. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 531–537. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-27261-0_43 CrossRefGoogle Scholar
  26. 26.
    Klawitter, J.: Algorithms for crossing minimisation in book drawings. Master’s thesis, Karlsruhe Institute of Technology (2016)Google Scholar
  27. 27.
    Klawitter, J., Mchedlidze, T., Nöllenburg, M.: Experimental evaluation of book drawing algorithms. CoRR, abs/1708.09221, August 2017. http://arxiv.org/abs/1708.09221
  28. 28.
    Konoe, M., Hagihara, K., Tokura, N.: Page-number of hypercubes and cube-connected cycles. Syst. Comput. Jpn. 20(4), 34–47 (1989). http://dx.doi.org/10.1002/scj.4690200404 MathSciNetCrossRefGoogle Scholar
  29. 29.
    López-Rodríguez, D., Mérida-Casermeiro, E., Ortíz-de-Lazcano-Lobato, J.M., Galán-Marín, G.: K-Pages graph drawing with multivalued neural networks. In: de Sá, J.M., Alexandre, L.A., Duch, W., Mandic, D. (eds.) ICANN 2007. LNCS, vol. 4669, pp. 816–825. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74695-9_84 CrossRefGoogle Scholar
  30. 30.
    Masuda, S., Nakajima, K., Kashiwabara, T., Fujisawa, T.: Crossing minimization in linear embeddings of graphs. IEEE Trans. Comput. 39(1), 124–127 (1990). http://dx.doi.org/10.1109/12.46286 MathSciNetCrossRefGoogle Scholar
  31. 31.
    McShine, L., Tetali, P.: On the mixing time of the triangulation walk and other catalan structures. Randomization Methods Algorithm Des. 43, 147–160 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Poranen, T., Mäkinen, E., He, H.: A simulated annealing algorithm for the 2-page crossing number problem. In: Proceedings of International Network Optimization Conference (INOC) (2007)Google Scholar
  33. 33.
    Satsangi, D., Srivastava K., Gursaran: A hybrid evolutionary algorithm for the page number minimization problem. In: Nagamalai, D., Renault, E., Dhanuskodi, M. (eds.) Trends in Computer Science, Engineering and Information Technology. Communications in Computer and Information Science, vol. 204. pp. 463–475. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-24043-0_47
  34. 34.
    Satsangi, D., Srivastava, K., Srivastava, G.: K-page crossing number minimization problem: an evaluation of heuristics and its solution using gesakp. Memetic Comput. 5(4), 255–274 (2013). http://dx.doi.org/10.1007/s12293-013-0115-5 CrossRefGoogle Scholar
  35. 35.
    Shahrokhi, F., Székely, L.A., Sýkora, O., Vrt’o, I.: The book crossing number of a graph. J. Graph Theor. 21(4), 413–424 (1996).  https://doi.org/10.1002/(SICI)1097-0118(199604)21:4%3c413::AID-JGT7%3e3.0.CO;2-S MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Six, J.M., Tollis, I.G.: A framework and algorithms for circular drawings of graphs. J. Discrete Algorithms 4(1), 25–50 (2006). http://dx.doi.org/10.1016/j.jda.2005.01.009 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tanaka, Y., Shibata, Y.: On the pagenumber of the cube-connected cycles. Math. Comput. Sci. 3(1), 109–117 (2010). http://dx.doi.org/10.1007/s11786-009-0012-y MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, J.: Hopfield neural network based on estimation of distribution for two-page crossing number problem. IEEE Trans. Circ. Syst. II Express Briefs 55(8), 797–801 (2008). http://dx.doi.org/10.1109/TCSII.2008.922373 Google Scholar
  39. 39.
    Wattenberg, M.: Arc diagrams: visualizing structure in strings. In: Information Visualization (INFOVIS 2002), pp. 110–116. IEEE (2002)Google Scholar
  40. 40.
    Yannakakis, M.: Linear and book embeddings of graphs. In: Makedon, F., Mehlhorn, K., Papatheodorou, T., Spirakis, P. (eds.) AWOC 1986. LNCS, vol. 227, pp. 226–235. Springer, Heidelberg (1986).  https://doi.org/10.1007/3-540-16766-8_20 CrossRefGoogle Scholar
  41. 41.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989). http://dx.doi.org/10.1016/0022-0000(89)90032-9 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.University of AucklandAucklandNew Zealand
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.TU WienViennaAustria

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