Abstract
The \(K\)-page crossing number minimization problem (KPMP) is to determine the minimum number of edge crossings over all \(K\)-page book drawings of a graph \(G\) with the vertices placed in a sequence along the spine and the edges on the \(K\)-pages of the book. In this paper we have (a) statistically evaluated five heuristics for ordering vertices on the spine for minimum number of edge crossings with all the edges placed on a single page, (b) statistically evaluated five heuristics for distributing edges on \(K\)-pages with minimum number of crossings for a fixed ordering of vertices on the spine and (c) implemented and experimentally evaluated an instance of guided evolutionary simulated annealing (GESA) called GESAKP here for solving the KPMP. In accordance with the results of (a) and (b) above, in GESAKP, placement of vertices on the spine is decided using a random depth first search of the graph and an edge embedding heuristic is used to distribute the edges on \(K\)-pages of a book. Extensive experiments have been carried out on a suite of benchmark, standard and random graphs to compare the performance of GESAKP with variants of the simple genetic algorithm and other existing approaches. In order to improve the results for some graphs, simple extensions to GESAKP were made. Experiments show that in almost all cases, zero or low cost could be achieved for \(K\le 5\). Also, for \(K \le \) ‘known upper bound’ i.e. upper bound for minimum number of pages necessary to draw or embed the edges of a graph without crossings, zero crossings were obtained. In general, GESAKP outperformed the other techniques. From our experimental results we also present the conjectures for the \(K\)-page crossing number of some complete tripartite graphs and pagenumber of toroidal meshes and a class of complete bipartite and tripartite graphs.
Similar content being viewed by others
References
Lopez-Rodriguez D, Merida-Casermeiro E, Ortiz-de-Lazcano-Lobato JM, Galan-Marin G (2007) K-pages graph drawing with multivalued neural networks. In: ICANN’07 in Proceedings of the 17th international conference on artificial neural networks, pp 816–825
Bansal R, Srivastava K, Shweta Varshney K, Sharma N (2008) An evolutionary algorithm for the 2-page crossing number problem. In: Proceedings of the IEEE congress on evolutionary computation (CEC), pp 1095–1102
Masuda S, Nakajima K, Kashiwabara T, Fujisawa T (1990) Crossing minimization in linear embeddings of graphs. IEEE Trans Comput 39:124–127
Poranen T, Erkki M, He H, Miller EH (2007) A simulated annealing algorithm for the 2-page crossing number problem. In: Proceedings of International Network Optimization Conference (INOC)
He H, Sykora O, Makinen E (2007) Genetic algorithms for the 2-page book drawing problem of graphs. J Heuristics 13:77–93
He H, Sýkora O, Salagean A, Makinen E (2007) Parallelisation of genetic algorithms for the 2-page crossing number problem. J Parallel Distrib Comput 67:229–241
He H, Sykora O, Makinen E (2006) An improved neural network model for the two-page crossing number problem. IEEE Trans Neural Netw 17(6):1642–1646
Chung FRK, Leighton FT, Rosenberg AL (1987) Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J Algebraic Discrete Methods 8(1):33–58
He H, Salagean A, Makinen E (2010) One- and two-page crossing numbers for some types of graphs. Int J Comput Math 87(8):1667–1679
Shahrokhi F, Sykora O, Szekely LA, Vrto I (1997) Crossing numbers: bounds and applications. In: Barany I, Boroczky K (eds) Intuitive geometry. Bolyai society mathematical studies, vol 6. Akademia Kiado, Budapest, pp 179–206
Cimikowski R (2001) Algorithms for the fixed linear crossing number problem. Discrete Appl Math 122(1–3):93–115
Games RA (1986) Optimal book embeddings of the FFT, Benes, and barrel shifter networks. Algorithmica 1:233–250
Yip PPC, Pao Yoh-Han (1995) Combinatorial optimization with use of guided evolutionary simulated annealing. IEEE Trans Neural Netw 6(2):290–295
Sharma R, Srivastava K (2007) A level structure based hybrid evolutionary algorithm for the edge bisection problem. In: International conference on computational intelligence and multimedia applications (ICCIMA 2007), vol 2, pp 146–150
Sykora O, Vrto I (1993) On crossing numbers of hypercubes and cube connected cycles. BIT Numer Math 33(2):232–237
Guan Y, Williams KL (2003) Profile minimization on triangulated triangles. Discrete Math 260(1–3):69–76
Hasunuma T, Shibata Y (1997) Embedding de Brujin, Kautz and Shuffle-exchange networks in books. Discrete Appl Math 78(1): 103–116
GDToolkit. http://www.dia.uniroma3.it\(\sim \)gdt/
Levin RI, Rubin DS (2001) Statistics for management. Pearson, San Francisco
SYSTAT 9, Statistics I (1999) by SPSS Inc., USA
Cimikowski R, Mumey B (2007) Approximating the fixed linear crossing number. Discrete Appl Math 155(17)
Kapoor N, Russell M, Stojmenovic I (2002) A genetic algorithm for finding the pagenumber of interconnection networks. J Parallel Distrib Comput 62:267–283
Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680
Glover F, Laguna M (1997) Tabu search. Kluwer, Boston
Gendreau M, Potvin J-Y (2003) Handbook of metaheuristics. Springer, Berlin
Satsangi D, Srivastava K, Gursaran (2011) A hybrid evolutionary algorithm for the page number minimization problem. In: Trends in computer science, engineering and information technology (CCSEIT), vol 204. Springer, Berlin, pp 463–475
Enomoto H, Nakamigawa T, Ota K (1997) On the pagenumber of complete bipartite graphs. J Comb theory 71(1):111–120
Acknowledgments
This work is supported by University Grants Commission (UGC), New Delhi vide letter no 36-66/2008 (SR). The authors would like to thank two anonymous referees for their valuable comments and suggestions which have helped improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendix
Appendix
See Table 29.
Rights and permissions
About this article
Cite this article
Satsangi, D., Srivastava, K. & Srivastava, G. \(K\)-page crossing number minimization problem: An evaluation of heuristics and its solution using GESAKP. Memetic Comp. 5, 255–274 (2013). https://doi.org/10.1007/s12293-013-0115-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12293-013-0115-5