An improved algorithm for the metro-line crossing minimization problem

• Authors: Martin Nöllenburg
• Venue: GD'09 Proceedings of the 17th international conference on Graph Drawing
• Year: 2009

In the metro-line crossing minimization problem, we are given a plane graph G=(V,E) and a set $\mathcal{L}$ of simple paths (or lines) that coverG, that is, every edge e∈E belongs to at least one path in $\mathcal{L}$. The problem is to draw all paths in $\mathcal{L}$ 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 $O(|\mathcal{L}|^2\cdot |V|)$ 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 $O(|\mathcal{L}|^3\cdot |E|^{2.5})$-time algorithm. For the second variant, in which all lines are paths between degree-1 vertices of G, Argyriou et al. [2] gave an $O((|E|+|\mathcal{L}|^2)\cdot |E|)$-time algorithm.