User interfaces for inspecting spatio-temporal events often allow their users to filter the events by specifying a time window with a time slider. We consider the case that filtered events are visualized on a map using textual or iconic labels. However, to ensure a clear visualization, not all filtered events are annotated with a label. We present algorithms for setting up a data structure that encodes for every possible time window the set of displayed labels. Our algorithms ensure that the displayed labels never overlap and guarantee the stability of the labeling during certain basic interactions with the time slider. Assuming that the labels have different priorities (weights), we aim to maximize the weight of the displayed labels integrated over all possible time windows. As basic interactions, we consider moving the entire time window, symmetrically scaling it, and dragging one of its endpoints. We consider two stability requirements: (1) during a basic interaction, a label should appear and disappear at most once; (2) if a label is displayed for a time window $Q$Q, then it is also displayed for all the time windows contained in $Q$Q and that contain its timestamp. We prove that finding an optimal solution is NP-hard and propose efficient constant-factor approximation algorithms for unit-square and unit-disk labels, as well as a fast greedy heuristic for arbitrarily shaped labels. In experiments on real-world data, we compare the non-exact algorithms with an exact approach through integer linear programming.

Graph visualizations help with complex data analysis but often require expert knowledge to apply and configure advanced algorithms. yFiles, a diagramming SDK, bridges this gap by enabling developers to create interactive visualizations easily. This work demonstrates how yFiles helps transform raw data into accessible graph visualizations.

Context Providing accessible urban green spaces is crucial for planning and ensuring healthy, resilient, and sustainable cities. The importance of visually accessible urban green spaces increases due to inner urban development processes. Objectives This article proposes a new index, the Green Window View Index (GWVI) for analyzing and assessing visible vegetation, that promotes an integrated planning of urban green spaces and buildings at different scales and levels. It is defined as the proportion of visible vegetation area in a field of view when looking out of a specific window with a defined distance to the window. Methods The method for estimating GWVI consists of three steps: (a) the modeling of the three-dimensional environment, (b) the simulation of the two-dimensional window views using modern rendering engines for three-dimensional graphics, (c) the computation of the GWVI . The method is proposed and tested through a case study of the urban area of Bonn, Germany, using a Digital Terrain Model (DTM), CityGML-based semantic 3D City Model at level of detail (LoD) 2, airborne Light Detection and Ranging (LiDAR) data, and 2D land use data from the official German property cadaster information system (ALKIS). Results With an average processing time of 0.05 s per window view, an average GWVI of 26.00% could be calculated for the entire study area and visualized in both 2D and 3D. Conclusion The proposed engine generates multi-scale visibility values for various vegetation shapes. These values are intended for use in participatory citizenship and decision-making processes for analysis by architects, real-estate appraisers, investors, and urban as well as landscape planners.

Let P be a polygon and C a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another polygon that encloses P and whose oriented boundary is composed of elements from C. We require P and the output shortcut hull to be weakly simple polygons, which we define as a generalization of simple polygons. Shortcut hulls find their application in cartography, where a common task is to compute simplified representations of area features. We aim at a shortcut hull that has a small area and a small perimeter. Our optimization objective is to minimize a convex combination of these two criteria. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via computation of shortest paths. For the more challenging case in which the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for the schematization of polygons.

Today's Large Language Models (LLM) provide the possibility of translating user requests given in natural language into executable code. Based on this, we present an approach for interactively modifying graph visualizations. We explain how to prompt such LLMs and how to tackle technical restrictions as limitations of the LLMs on input sizes.

The optimal path between two vertices in a graph depends on the optimization objective, which is often defined as a weighted sum of multiple criteria. When integrating two criteria, their relative importance is expressed with a balance factor α. We present a new approach for inferring α from trajectories. The core of our approach is a compression algorithm that requires a graph G representing a transportation network, two edge costs modeling routing criteria, and a path P in G representing the trajectory. It yields a minimum subsequence S of the sequence of vertices of P and a balance factor α, such that the path P can be fully reconstructed from S, G, its edge costs, and α. By minimizing the size of S over α, we learn the balance factor that corresponds best to the user's routing preferences. In an evaluation with crowd-sourced cycling trajectories, we weigh the usage of official signposted cycle routes against other routes. More than 50% of the trajectories can be segmented into five optimal sub-paths or less. Almost 40% of the trajectories indicate that the cyclist is willing to take a detour of 50% over the geodesic shortest path to use an official cycle path.

In this chapter we discuss various visual aspects that affect the aesthetic quality and usability of external labeling. In general, it is far from obvious which type of external labeling is the best choice and this highly depends on the application. We discuss several important criteria that must be taken into account when deciding on a type of external labeling. We distinguish between the style of the labeling and the actual placement of the labels based on it.

Ortho-Radial drawings are a generalization of orthogonal drawings to grids that are formed by concentric circles and straight-line spokes emanating from the circles' center. Such drawings have applications in schematic graph layouts, e.g., for metro maps and destination maps. A plane graph is a planar graph with a fixed planar embedding. We give a combinatorial characterization of the plane graphs that admit a planar ortho-radial drawing without bends. Previously, such a characterization was only known for paths, cycles, and theta graphs, and in the special case of rectangular drawings for cubic graphs, where the contour of each face is required to be a rectangle. The characterization is expressed in terms of an ortho-radial representation that, similar to Tamassia's orthogonal representations for orthogonal drawings describes such a drawing combinatorially in terms of angles around vertices and bends on the edges. In this sense our characterization can be seen as a first step towards generalizing the Topology-Shape-Metrics framework of Tamassia to ortho-radial drawings.

In this paper, we analyze the approximation quality of a greedy heuristic for automatic map labeling. As input, we have a set of events, each associated with a label at a fixed position, a timestamp, and a weight. Let a time-window labeling be a selection of these labels such that all corresponding timestamps lie in a queried time window and no two labels overlap. A solution to the time-window labeling problem consists of a data structure that encodes a time-window labeling for each possible time window; when a user specifies a time window of interest using a slider interface, we query the data structure for the corresponding labeling. We define the quality of a time-window labeling solution as the sum of the weights of the labels in each time-window labeling, integrated over all time windows. We aim at maximizing the quality under the condition that a label may never disappear when the user shrinks the time window. In this paper, we analyze how well a greedy heuristic approximates the maximum quality that can be realized under this condition. On the one hand, we present an instance with square labels of equal size and equal weight for which the greedy heuristic fails to find a solution of at least 1/4 of the quality of an optimal solution. On the other hand, we prove that the greedy heuristic does guarantee a solution with at least 1/8 of the quality of an optimal solution. In the case of disk-shaped labels of equal size and equal weight, the greedy heuristic gives a solution with at least 1/10 of the quality of an optimal solution. If the labels are squares or disks of equal size and the maximum weight divided by the minimum weight is at most b, then the greedy heuristic has approximation ratio Theta(log b).

This article addresses the augmentation of existing building models by a priori not observable structures such as electric installations. The aim is to unambiguously determine an electric network in an incremental manner with a minimum number of local measurements, e.g. using wire detectors, by suggesting the next measurement. Different reasoning strategies, e.g. utilizing graph-theoretical algorithms, have been presented and tested based on a hypothesis which is generated using Mixed Integer Linear Programming (MILP) while incorporating standards regarding the installation of electric wiring and findings from previous measurements. The presented method has been successfully applied on simulated and real-world buildings, it saves up to 80% of the necessary measurements compared to an exhaustive verification of the whole existing electric network, and paves the way for efficiently extending existing models, e.g. GIS models, with information on hidden utilities. This opens up new opportunities to model further infrastructures, e.g. water pipes, in future research.

Schematic maps are in daily use to show the connectivity of subway systems and to facilitate travellers to plan their journeys effectively. This study surveys up-to-date algorithmic approaches in order to give an overview of the state of the art in schematic network mapping. The study investigates the hypothesis that the choice of algorithmic approach is often guided by the requirements of the mapping application. For example, an algorithm that computes globally optimal solutions for schematic maps is capable of producing results for printing, while it is not suitable for computing instant layouts due to its long running time. Our analysis and discussion, therefore, focus on the computational complexity of the problem formulation and the running times of the schematic map algorithms, including algorithmic network layout techniques and station labeling techniques. The correlation between problem complexity and running time is then visually depicted using scatter plot diagrams. Moreover, since metro maps are common metaphors for data visualization, we also investigate online tools and application domains using metro map representations for analytics purposes, and finally summarize the potential future opportunities for schematic maps.

This book focuses on techniques for automating the procedure of creating external labelings, also known as callout labelings. In this labeling type, the features within an illustration are connected by thin leader lines (called leaders) with their labels, which are placed in the empty space surrounding the image.

An ortho-radial grid is described by concentric circles and straight-line spokes emanating from the circles' center. An ortho-radial drawing is the analog of an orthogonal drawing on an ortho-radial grid. Such a drawing has an unbounded outer face and a central face that contains the origin. Building on the notion of an ortho-radial representation (Barth et al., SoCG, 2017), we describe an integer-linear program (ILP) for computing bend-free ortho-radial representations with a given embedding and fixed outer and central face. Using the ILP as a building block, we introduce a pruning technique to compute bend-optimal ortho-radial drawings with a given embedding and a fixed outer face, but freely choosable central face. Our experiments show that, in comparison with orthogonal drawings using the same embedding and the same outer face, the use of ortho-radial drawings reduces the number of bends by 43.8% on average. Further, our approach allows us to compute ortho-radial drawings of embedded graphs such as the metro system of Beijing or London within seconds.

Visualizing spatial data on small-screen devices such as smartphones and smartwatches poses new challenges in computational cartography. The current interfaces for map exploration require their users to zoom in and out frequently. Indeed, zooming and panning are tools suitable for choosing the map extent corresponding to an area of interest. They are not as suitable, however, for resolving the graphical clutter caused by a high feature density since zooming in to a large map scale leads to a loss of context. Therefore, in this paper, we present new external labeling methods that allow a user to navigate through dense sets of points of interest while keeping the current map extent fixed. We provide a unified model, in which labels are placed at the boundary of the map and visually associated with the corresponding features via connecting lines, which are called leaders. Since the screen space is limited, labeling all features at the same time is impractical. Therefore, at any time, we label a subset of the features. We offer interaction techniques to change the current selection of features systematically and, thus, give the user access to all features. We distinguish three methods, which allow the user either to slide the labels along the bottom side of the map or to browse the labels based on pages or stacks. We present a generic algorithmic framework that provides us with the possibility of expressing the different variants of interaction techniques as optimization problems in a unified way. We propose both exact algorithms and fast and simple heuristics that solve the optimization problems taking into account different criteria such as the ranking of the labels, the total leader length as well as the distance between leaders. In experiments on real-world data we evaluate these algorithms and discuss the three variants with respect to their strengths and weaknesses proving the flexibility of the presented algorithmic framework.

Polyline leaders form a natural generalization of straight-line leaders by supporting bends. To facilitate readability, polyline leaders usually have a schematic and tidy appearance caused by the restriction to segment orientations that are aligned with the Cartesian coordinate axes or possibly with their two bisecting diagonals. As one would expect, the most natural requirement in labelings with polyline leaders is the avoidance of crossings between leaders. However, bend minimization is another criterion which becomes important, as has also been observed in the early survey on boundary labeling by Kaufmann in 2009 [60].

In this chapter we discuss algorithmic techniques that are frequently applied in external labeling. We split them into non-exact (Section 4.1) and exact algorithms (Section 4.2). While exact algorithms solve instances of external labeling problems and provide an optimality guarantee for the solution, non-exact approaches refrain from such guarantees in favor of faster running time. In Section 4.3 we discuss complexity results mainly focusing on problems in external labeling that are NP-hard. We conclude this chapter in Section 4.4 with guidelines regarding the applicability of each technique.

Straight-line leaders form the simplest way to establish the visual association between features and their external labels. Further, they can be easily traced by the reader [6]. In this chapter, we first give an overview of different aspects of labelings with straight-line leaders (Section 5.1) and then discuss important contributions (Section 5.2). We note that a survey focusing on external labeling in medical visualizations was published by Preim and Oeltze-Jafra [82] in 2014.

Map applications on mobile devices such as smartphones and smartwatches have become ubiquitous. When visualizing spatial data on such small-screen devices, one major challenge is annotating the data with labels (e.g., small icons). The restricted space requires new visualization techniques as established strategies, such as maximizing the number of placed labels, easily lead to the omission of information. We propose an approach that distributes all labels contained in a temporarily fixed map section on multiple pages. Applying interaction techniques for navigating through the pages, a user can access all information both without any overlapping labels and without the need for zooming. We propose a method with two phases; a pre-processing phase and a query phase . We use an optimization approach to pre-compute labelings on the level of a whole city and provide the on-demand querying of individual labelings at a more local level. Our approach provides a consistent label-page assignment, meaning that labels do not appear and disappear when the user pans the map. Further, our model provides quick access to important information and a spatially balanced distribution of labels on pages. In experiments on real-world data we analyze different parameter settings and show that our model yields high-quality labelings. (c) 2021 Elsevier Ltd. All rights reserved.