A Survey on Computing Schematic Network Maps: The Challenge to Interactivity Hsiang-Yun Wu Benjamin Niedermann Shigeo Takahashi Martin Nollenburg¨ TU Wien, Austria University of Bonn, Germany University of Aizu, Japan TU Wien, Austria [email protected] [email protected] [email protected] [email protected] Abstract—Schematic maps are in daily use to show the connec- years after Nollenburg’s¨ report [26] and discuss the correlation tivity of subway systems and to facilitate travellers to plan their in terms of computational complexity and interactivity, as journeys effectively. This study surveys up-to-date algorithmic well as the potential techniques that can be used in different approaches in order to give an overview of the state of the art in schematic network mapping. The study investigates the research domains. hypothesis that the choice of algorithmic approach is often guided by the requirements of the mapping application. For example, an A. Problem Definition algorithm that computes globally optimal solutions for schematic The initial idea for a metro map lies on simplifying the maps is capable of producing results for printing, while it is not layout geometry to facilitate users’ comprehensive understand- suitable for computing instant layouts due to its long running time. Our analysis and discussion, therefore, focus on the compu- ing of connectivity between metro stations. This allows us tational complexity of the problem formulation and the running to formulate the drawing problem as network visualization times of the schematic map algorithms, including algorithmic problems, which are often studied to untangle visual clutter network layout techniques and station labeling techniques. The of the layout to improve its readability [51]. correlation between problem complexity and running time is Metro Map Problem: Let us formulate the metro map then visually depicted using scatter plot diagrams. Moreover, since metro maps are common metaphors for data visualization, problem by introducing an undirected graph G = (V; E) 2 we also investigate online tools and application domains using embedded in a plane R . Each vertex v 2 V represents a metro map representations for analytics purposes, and finally metro station and an edge e = (vi; vj) 2 E indicates the summarize the potential future opportunities for schematic maps. physical connection between two metro stations. In addition, a line l 2 L is a metro line containing a set of metro stations and Index Terms—Metro Maps, Graph Drawing, Metaphors edges that are defined from the corresponding metro system. Note that l is a set cover, which implies each edge e belongs to I. INTRODUCTION at least one l 2 L. We call MG = (G; L) here a Metro Graph A metro map is a schematic visual representation of an un- as previously defined by Nollenburg¨ [26]. The input of the derlying transit network that depicts the connectivity between proposed problem is the connectivity of a Metro Graph MG metro stations and lines of a public transportation system [30]. together with its geographically accurate embedding, and the With well-designed metro maps, travellers can effectively solution aims to find a schematic layout of MG that maximally identify their locations and find their way or perform routing satisfies several user-defined aesthetic drawing criteria. planning on a complex transportation system in a big city, such as London, Paris, or Tokyo. These maps are especially helpful B. Our Taxonomy Design since travellers often look for a quick solution for the shortest In this study, we investigate the trade-off between com- or cheapest path from station A to B, how to transfer from putation times and layout quality as well as application A to B, and how many stations left to B [33]. To support requirements. This is motivated by our observation that a these tasks, Henry Beck introduced the so-called Tube Map user who is trying to create a map would accept longer of London Underground, and has proposed several drawing computation times for an exact solution if the generated map criteria to achieve this [14], and many extended versions have is expected to be printed, while the user will not be patient if been evaluated [34] [35]. the application is expected to be an interactive application. To Nonetheless, the need for automatic drawing algorithms investigate our assumption systematically, we analyze several is still increasing due to the high cost and limited adapt- factors in this survey (see Tables I and III) that influence the ability of hand-drawn maps. This has been considered as computational complexity and optimality requirements of a a difficult problem because several subproblems, including problem together with the corresponding running time of the layout schematization, line crossing minimisation, and map algorithmic approach (Table III). Our two primary topics cover label placement have been studied and proved as complex algorithmic network layout techniques and station labeling problems [26]. Two state-of-the-art reports from 2007 and techniques. The correlation between problem complexity and 2014 have surveyed similar approaches before [50] [26]. time complexity is then given using a scatter plot diagram Hence we focus on relatively new approaches from the last 5 visually. Copyright of this manuscript is retained by the authors 1 This is done by categorizing publications along two primary [45] Fast coordinates, problem complexity and running time. These [49] [48] [43] include (1) the range from local to global optimality in terms of [8] the incorporated aesthetic criteria and objective functions, and [18][42] (2) the range of suitability from static to interactive visualiza- [46] tions based on the computation speed. The values on the first Local Global coordinate are computed using the scoring Tables I and III. [9] [28] For simplicity, we assume that the degree of interactivity, [39] [52] running time in other words, of a schematic map algorithm is [29] [12] paper ≥ 2014 [27] potentially linearly-correlated to the degree of criteria selected paper < 2014 Slow by the proposed approach. In other words, we expect the (a) (b) developed techniques would find a good balance between the efficiency of the algorithms and the quality of the schematic Fig. 1. Network map visualization, including an example of (1) Vienna maps generated by those approaches. For better instructing metro map [27], and (b) layout techniques with respect to their globality readers along this assumption, we will focus our discussion and potential for interactiveness. on these two aspects in the following sections. TABLE I THE POINT SYSTEM FOR EVALUATING THE CRITERION EFFECTIVENESS. C. Tasks and Design Rules ID S Description (x-coordinate in Figs. 1- 3) It is known that drawing criteria are often designed based (N1) Combinatorial property on users’ effectiveness to accomplish tasks on a map. These (N1.1) 4 Overall combinatorial optimization. tasks are similar to tasks on graphs, such as Topology-Based (N1.2) 3 Combinatorial criteria for sets of vertices or edges. (N1.3) 2 Combinatorial criteria for pairs of vertices or edges. Tasks, Attribute-Based Tasks, and Browsing Tasks since finding (N1.4) 1 Combinatorial criteria for single vertices or edges. shortest or cheapest paths, identifying a station of a line, (N2) Geometry property and navigating along a specific route are mostly performed (N2.1) 4 Uniform geometric optimization. by the travellers. We revisit Nollenburg’s¨ list [26] of design (N2.2) 3 Geometric criteria for sets of vertices or edges. (N2.3) 2 Geometric criteria for pairs of vertices or edges. principles and investigate which of these serve as dominant (N2.4) 1 Geometric criteria for single vertices or edges. constraints, for generating a globally optimal map, in the (N3) Approach optimality coming sections. In this survey, we primarily focus on two (N3.1) 4 Global optimality and exact global optimization. (N3.2) 3 Global optimality, but local optimization. directions in this field, including network layout techniques (N3.3) 2 Local optimality, but global optimality for sub-problems. (N) and labeling techniques (L) as summarized in Table I (N3.4) 1 Local optimality, and local optimization. and Table III, respectively. Note that the selected criteria are ordered in the sense that the criterion with higher scores influences global structures of the layout, while the one with computed solutions, i.e., whether an global optimum is com- lower scores affects the layout in a local fashion. This scoring puted or just a local optimum. Table I lists the three criteria, scheme will then be used as an indicator for guiding readers to which are ranked by scores ranging from 1 (high locality) to 4 navigate the diagrams created in the coming sections. Based (high globality). Additionally, we assess each technique by the on the aforementioned scoring scheme, we derive our novel required computational resources and resulting degree of suit- taxonomy of metro map techniques in this survey paper. ability for interactive applications (summarized in Table II). The remainder of this paper is structured as follows. In Each of the papers discussed in the following paragraphs Section II, we summarize relevant schematic network layout thus receives two scores that we use as coordinates in the algorithms, and in Section III, research approaches integrating two-dimensional scatter plot of Figure 1. map features, mainly on text and image labels. In Section IV, We start with four classic papers that serve as represen- we then introduce several tools that are accessible online tatives of the main algorithmic techniques that were already and demonstrate the usability of the collected techniques in discussed in the 2014 survey of Nollenburg¨ [26]. Since our different research domains. Finally in Section VI, we conclude focus is on new results from 2014 onward, we refer to the 2014 this paper and list several future directions to this topic.