Abstract
Given a bipartite graph G = (L 0,L 1,E) and a fixed ordering of the nodes in L 0, the problem of finding an ordering of the nodes in L 1 that minimizes the number of crossings has received much attention in literature. The problem is NP-complete in general and several practically efficient heuristics and polynomial-time algorithms with a constant approximation ratio have been suggested. We generalize the problem and consider the version where the edges have nonnegative weights. Although this problem is more general and finds specific applications in automatic graph layout problems similar to those of the unweighted case, it has not received as much attention. We provide a new technique that efficiently approximates a solution to this more general problem within a constant approximation ratio of 3. In addition we provide appropriate generalizations of some common heuristics usually employed for the unweighted case and compare their performances.
Partially supported by TUBITAK-The Scientific and Technological Research Council of Turkey, grant 106E071.
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References
Wolf implementations, http://www2.isikun.edu.tr/personel/cesimerten/wolf.rar
Biedl, T., Brandenburg, F.J., Deng, X.: Crossings and permutations. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 1–12. Springer, Heidelberg (2006)
Brandes, U., Dwyer, T., Schreiber, F.: Visualizing related metabolic pathways in two and a half dimensions (long paper). In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 111–122. Springer, Heidelberg (2004)
Brandes, U., Wagner, D.: visone - analysis and visualization of social networks. In: Graph Drawing Software, pp. 321–340. Springer, Heidelberg (2003)
Demestrescu, C., Finocchi, I.: Breaking cycles for minimizing crossings. J. Exp. Algorithmics 6(2) (2001)
Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11, 379–403 (1994)
Finocchi, I.: Layered drawings of graphs with crossing constraints. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 357–367. Springer, Heidelberg (2001)
Forster, M.: Applying crossing reduction strategies to layered compound graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 276–284. Springer, Heidelberg (2002)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)
Jünger, M., Mutzel, P.: 2-layer straightline crossing minimization: Performance of exact and heuristic algorithms. Journal of Graph Algorithms and Applications 1(1), 1–25 (1997)
Li, X.Y., Stallmann, M.F.: New bounds on the barycenter heuristic for bipartite graph drawing. Inf. Process. Lett. 82(6), 293–298 (2002)
Mehlhorn, K., Näher, S.: The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999)
Munoz, X., Unger, W., Vrto, I.: One sided crossing minimization is np-hard for sparse graphs. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 115–123. Springer, Heidelberg (2002)
Nagamochi, H.: An improved bound on the one-sided minimum crossing number in two-layered drawings. Discrete Comput. Geom. 33(4), 569–591 (2005)
Smith, B.S., Lim, S.K.: Qca channel routing with wire crossing minimization. In: GLSVSLI ’05: Proceedings of the 15th ACM Great Lakes symposium on VLSI, pp. 217–220. ACM Press, New York (2005)
Stallmann, M., Brglez, F., Ghosh, D.: Heuristics, experimental subjects, and treatment evaluation in bigraph crossing minimization. J. Exp. Algorithmics 6(8) (2001)
Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Transactions on Syst. Man. and Cybernetics 11(2), 109–125 (1981)
Yamaguchi, A., Sugimoto, A.: An approximation algorithm for the two-layered graph drawing problem. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 81–91. Springer, Heidelberg (1999)
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Çakıroḡlu, O.A., Erten, C., Karataş, Ö., Sözdinler, M. (2007). Crossing Minimization in Weighted Bipartite Graphs. In: Demetrescu, C. (eds) Experimental Algorithms. WEA 2007. Lecture Notes in Computer Science, vol 4525. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72845-0_10
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DOI: https://doi.org/10.1007/978-3-540-72845-0_10
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