Two Polynomial Time Algorithms for the Metro-line Crossing Minimization Problem

Abstract

Abstract. The metro-line crossing minimization (MLCM) problem was recently introduced in [5] as a response to the problem of drawing metro maps or public transportation networks, in general. According to this problem, we are given a planar, embedded graph G = (V, E) and a set L of simple paths on G, called lines. The main task is to place the lines on the embedding of G, so that the number of crossings among pairs of lines is minimized. Our main contribution is two polynomial time algorithms. The first solves the general case of the MLCM problem, where the lines that traverse a particular vertex of G are allowed to use any side of it to either "enter" or "exit", assuming that the endpoints of the lines are located at vertices of degree one. The second one -which is more efficient in terms of time complexity-solves the restricted case, where only the left and the right side of each vertex can be used. To the best of our knowledge, this is the first time where the general case of the MLCM problem is solved. Previous work in the graph drawing literature was devoted to the restricted case of the MLCM problem under the additional assumption that the endpoints of the lines are either the topmost or the bottommost in their corresponding vertices, i.e., they are either on top or below the lines that pass through the vertex. Even for this case, we improve a known result of Asquith et al. [3] from O(|E| 5/2 |L| 3 ) to O(|V |(|E| + |L|)).

An Improved Algorithm for the Metro-line Crossing Minimization Problem

“…Our algorithm improves the algorithm of Asquith et al [1] for the same problem, which has a running time of O(|L| 3 · |E| 2.5 ). Our algorithm can also be used to solve a closely related problem considered by Argyriou et al [2], where all lines must be paths connecting two degree-1 vertices in G. Hence, it also improves the algorithm of Argyriou et al, which has a running time of O((|E| + |L| 2 ) · |E|). These are the only two variants of MLCM that are known to be efficiently solvable, and our algorithm is to the best of our knowledge currently the fastest method to solve both of them for general plane underlying graphs.…”

“…This is exactly the situation in which lines terminate at leaves of the underlying graph. Argyriou et al [2] presented an algorithm to solve MLCM-T1 in general plane graphs in O((|E| + |L| 2 ) · |E|) time. For MLCM-T1 in the previously mentioned 2-side model, they improved the running time to O((|E| + |L|) · |V |).…”

“…Subsequently, MLCM was considered in several variants and for different classes of under- lying graphs [1,2,9], which are discussed in detail in Section 3. One important variant, posed as an open problem by Benkert et al [8] and addressed by Bekos et al [9] and by Asquith et al [1], restricts the positions of each line's start and end point (called termini ) to be left-or rightmost in the ordered sequence of lines along the underlying edges leading to its termini.…”

Efficient Generation of Geographically Accurate Transit Maps

“…In that paper, optimal layouts for path and tree networks were investigated but arbitrary graphs were left as an open problem. In [1,2,13], several variants of MLCM were defined and efficient algorithms were presented for some of these variants, often with a restriction to planar graphs. In [3], an ILP formulation for MLCM under the periphery condition (see Sect.…”

Orthogonal Connector Routing

“…They do not consider the problem of how to avoid introducing unnecessary crossings when separating connectors with a shared path. Our algorithm for ordering connectors in shared paths so as to avoid introducing unnecessary crossings is related to algorithms for metro-line crossing [12,13]. The main difference is that we have the additional requirement that the ordering should not introduce unnecessary bends in the layout and so crossings are only allowed to occur when a connector enters or leaves the shared path, but not in the shared path itself.…”