Martin Nöllenburg

Drawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming

Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing nonoverlapping station labels. In this paper, we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NP-hard) using mixed-integer programming (MIP). We identify seven design rules used in most real-world metro maps. We split these rules into hard and soft constraints and translate them into an MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods.

Optimizing active ranges for consistent dynamic map labeling

Map labeling encounters unique issues in the context of dynamic maps with continuous zooming and panning-an application with increasing practical importance. In consistent dynamic map labeling, distracting behavior such as popping and jumping is avoided. In the model for consistent dynamic labeling that we use, a label becomes a 3d-solid, with scale as the third dimension. Each solid can be truncated to a single scale interval, called its active range, corresponding to the scales at which the label will be selected. The active range optimization (ARO) problem is to select active ranges so that no two truncated solids overlap and the sum of the heights of the active ranges is maximized. The simple ARO problem is a variant in which the active ranges are restricted so that a label is never deselected when zooming in. We investigate both the general and simple variants, for 1d- as well as 2d-maps. The 1d-problem can be seen as a scheduling problem with geometric constraints, and is also closely related to geometric maximum independent set problems. Different label shapes define different ARO variants. We show that 2d-ARO and general 1d-ARO are NP-complete, even for quite simple shapes. We solve simple 1d-ARO optimally with dynamic programming, and present a toolbox of algorithms that yield constant-factor approximations for a number of 1d- and 2d-variants.

Morphing polylines: A step towards continuous generalization

We study the problem of morphing between two polylines that represent linear geographical features like roads or rivers generalized at two different scales. This problem occurs frequently during continuous zooming in interactive maps. Situations in which generalization operators like typification and simplification replace, for example, a series of consecutive bends by fewer bends are not always handled well by traditional morphing algorithms. We attempt to cope with such cases by modeling the problem as an optimal correspondence problem between characteristic parts of each polyline. A dynamic programming algorithm is presented that solves the matching problem in O(nm) time, where n and m are the respective numbers of characteristic parts of the two polylines. In a case study we demonstrate that the algorithm yields good results when being applied to data from mountain roads, a river and a region boundary at various scales.

Boundary Labeling with Octilinear Leaders

A major factor affecting the readability of an illustration that contains textual labels is the degree to which the labels obscure graphical features of the illustration as a result of spatial overlaps. Boundary labeling addresses this problem by attaching the labels to the boundary of a rectangle that contains all features. Then, each feature should be connected to its associated label through a polygonal line, called leader, such that no two leaders intersect. In this paper we study the boundary labeling problem along a new line of research, according to which different pairs of type leaders (i.e. doand pd, odand pd) are combined to produce boundary labelings. Thus, we are able to overcome the problem that there might be no feasible solution when labels are placed on different sides and only one type of leaders is allowed. Our main contribution is a new algorithm for solving the total leader length minimization problem (i.e., the problem of finding a crossing free boundary labeling, such that the total leader length is minimized) assuming labels of uniform size. We also present an NP-completeness result for the case where the labels are of arbitrary size.

Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability

A binary tanglegram is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number.We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.

Drawing metro maps using bézier curves

The automatic layout of metro maps has been investigated quite intensely over the last few years. Previous work has focused on the octilinear drawing style where edges are drawn horizontally, vertically, or diagonally at 45°. Inspired by manually created curvy metro maps, we advocate the use of the curvilinear drawing style; we draw edges as Bezier curves. Since we forbid metro lines to bend (even in stations), the user of such a map can trace the metro lines easily. In order to create such drawings, we use the force-directed framework. Our method is the first that directly represents and operates on edges as curves.

Drawing trees with perfect angular resolution and polynomial area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.

Lombardi drawings of graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artistMark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.

Algorithms for Multi-Criteria Boundary Labeling

We present new algorithms for labeling a set P of n points in the plane with labels that are aligned to one side of the bounding box of P . The points are connected to their labels by curves (leaders) that consist of two segments: a horizontal segment, and a second segment at a xed angle with the rst. Our algorithms nd a collection of crossing-free leaders that minimizes the total number of bends, the total length, or any other ‘badness’ function of the leaders. A generalization to labels on two opposite sides of the bounding box of P is considered and an experimental evaluation of the performance is included.

On the usability of lombardi graph drawings

A recent line of work in graph drawing studies Lombardi drawings, i.e., drawings with circular-arc edges and perfect angular resolution at vertices. Little is known about the effects of curved edges versus straight edges in typical graph reading tasks. In this paper we present the first user evaluation that empirically measures the readability of three different layout algorithms (traditional spring embedder and two recent near-Lombardi force-based algorithms) for three different tasks (shortest path, common neighbor, vertex degree). The results indicate that, while users prefer the Lombardi drawings, the performance data do not present such a positive picture.