An Improved Algorithm for the Metro-line Crossing Minimization Problem

Nöllenburg, Martin (2010) An Improved Algorithm for the Metro-line Crossing Minimization Problem. In: Graph Drawing 17th International Symposium, GD 2009, September 22-25, 2009, Chicago, IL, USA , pp. 381-392 (Official URL:

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In the metro-line crossing minimization problem, we are given a plane graph G = (V, E) and a set L of simple paths (or lines) that cover G, that is, every edge e ∈ E belongs to at least one path in L. The problem is to draw all paths in L along the edges of G such that the number of crossings between paths is minimized. This crossing minimization problem arises, for example, when drawing metro maps, in which multiple transport lines share parts of their routes. We present a new line-layout algorithm with O(|L|2 · |V |) running time that improves the best previous algorithms for two variants of the metro-line crossing minimization problem in unrestricted plane graphs. For the first variant, in which the so-called periphery condition holds and terminus side assignments are given in the input, Asquith et al. [1] gave an O(|L|3 · |E|2.5 )-time algorithm. For the second variant, in which all lines are paths between degree-1 vertices of G, Argyriou et al. [2] gave an O((|E| + |L|2 ) · |E|)-time algorithm.

Item Type:Conference Paper
Additional Information:10.1007/978-3-642-11805-0_36
Classifications:G Algorithms and Complexity > G.420 Crossings
P Styles > P.600 Poly-line
ID Code:1096

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Asquith, M., Gudmundsson, J., Merrick, D.: An ILP for the metro-line crossing problem. In: Harland, J., Manyem, P. (eds.) Proc. 14th Computing: The Australasian Theory Symp. (CATS 2008). CRPIT, vol. 77, pp. 49–56. Australian Comput. Soc. (2008)

Argyriou, E., Bekos, M.A., Kaufmann, M., Symvonis, A.: Two polynomial time algorithms for the metro-line crossing minimization problem. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 336–347. Springer, Heidelberg (2009)

Ovenden, M.: Metro Maps of the World. Capital Transport Publishing (2003)

Stott, J.M., Rodgers, P.: Metro map layout using multicriteria optimization. In: Proc. 8th Internat. Conf. Information Visualisation (IV 2004), pp. 355–362. IEEE, Los Alamitos (2004)

Hong, S.H., Merrick, D., do Nascimento, H.A.D.: Automatic visualization of metro maps. J. Visual Languages and Computing 17, 203–224 (2006)

Nöllenburg, M., Wolff, A.: A mixed-integer program for drawing high-quality metro maps. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 321–333. Springer, Heidelberg (2006)

Hahn, W.C., Weinberg, R.A.: A subway map of cancer pathways (2002); Poster in Nature Reviews Cancer


Benkert, M., Nöllenburg, M., Uno, T., Wolff, A.: Minimizing intra-edge crossings in wiring diagrams and public transportation maps. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 270–281. Springer, Heidelberg (2007)

Bekos, M.A., Kaufmann, M., Potika, K., Symvonis, A.: Line crossing minimization on metro maps. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 231–242. Springer, Heidelberg (2008)