Drawing Metro Maps on Concentric Circles

This thesis examines algorithms for the drawing of metro maps. The most important elements of a metro map are stations and lines connecting the stations. In the context of this thesis, we restrict the elements used for representing lines to one of two classes: On the one hand, circular segments lying on circles with a common center S are allowed. On the other hand, line segments lying on lines passing through S are allowed. To make the metro maps as legible as possible, we try to create a drawing in which the lines bend as little as possible. To plan a journey in the metro network, the users of metro maps must be able to quickly and correctly follow the lines visually. In this undertaking, bends are a disturbance. To this end, we adapt the Topology-ShapeMetrics framework by Roberto Tamassia, the purpose of which originally is bend minimization in orthogonal drawings of graphs. A second objective is producing drawings that take up as little space as possible. For doing so, we present a compaction approach based on Simulated Annealing. The compaction metaheuristic proposed by us is also suitable for performing the Shape step of the unaltered Topology-ShapeMetrics framework, too. Thus, we present a general improvement of this framework in itself. We practically evaluate the adapted framework for drawing metro maps with concentric circles and demonstrate its usability even for complex metro networks such as Berlin or London, at the same time also determining reasonable values for the parameters of our framework. For two subproblems that must be solved when drawing metro maps, we show NPhardness: First, a given metro network must be converted into a graph of maximum degree 4 since vertices with a larger degree cannot be represented in the desired drawing style. It would be desirable to do so in such a way that the best drawing of the resulting graph has as few edge-bends as possible; we show this problem to be NP-hard. Second, the final step of our framework compactifies a preliminary drawing. We show that achieving an optimal compaction is also NP-hard. Furthermore, we examine another naturally arising idea: While the framework used by us minimizes the number of edge-bends for an orthogonal drawing style, which we then transform into the desired drawing style, it would be desirable to perform bend-minimization with a concentric-circle drawing style directly. We model the problem and illustrate some arising difficulties which make direct transferral of the used techniques from the orthogonal case impossible. Deutsche Zusammenfassung In dieser Arbeit geht es um das algorithmische Zeichnen von Nahverkehrsnetzplänen. Die wichtigsten zu zeichnenden Elemente eines solchen Netzplanes sind Stationen und Linien, die diese Stationen verbinden. In dieser Arbeit beschränken wir uns darauf, die Linien nur mit den folgenden zwei Elementen darzustellen: Zum einen sind Kreissegmente erlaubt, die auf Kreisen mit einem gemeinsamen Zentrum S liegen. Zum anderen sind Strecken erlaubt, die auf Linien durch S liegen. Um die Netzpläne möglichst lesbar zu gestalten, versuchen wir eine Zeichnung zu erzeugen, in der die Linien möglichst selten abknicken. Beim Planen von Fahrten müssen die Benutzer der Pläne in der Lage sein, diesen Linien schnell und sicher