Merge trees are a popular topological descriptor for scalar field data. A common measure to compare two merge trees is the interleaving distance, which relies on a mapping between the two merge trees, also referred to as an interleaving. Despite its desirable properties, the interleaving distance has not been used much in practice, largely due to the fact that computing the exact interleaving distance is NP-hard. In this paper, we show that the exact interleaving distance can be computed efficiently for merge trees encountered in practice: we present the first implementation of the exact fixed-parameter tractable (FPT) algorithm by Touli and Wang [Touli and Wang, 2022]. This algorithm uses a dynamic program to test if a specific interleaving distance δ is feasible. They bound the running time using a parameter τ that captures the number of mapping options between the two merge trees for the output distance δ. Our experiments show that, even though τ can become quite large for real-world merge trees, the running time of our implementation does not depend very heavily on τ. Furthermore, we modify the FPT algorithm into a sweepline algorithm that runs much faster in practice. Finally, we introduce a natural restriction for the interleaving distance capturing the geometric similarity between the underlying scalar fields. This restricted interleaving distance can be computed more efficiently and can, in some settings, also result in more meaningful interleavings. We extend our implementations to support these restrictions and demonstrate their effect on the running time of the algorithms.

Merge trees are a powerful tool from topological data analysis that is frequently used to analyze scalar fields. The similarity between two merge trees can be captured by an interleaving: a pair of maps between the trees that jointly preserve ancestor relations in the trees. Interleavings can have a complex structure; visualizing them requires a sense of (drawing) order which is not inherent in this purely topological concept. However, in practice it is often desirable to introduce additional geometric constraints, which leads to variants such as labeled or monotone interleavings. Monotone interleavings respect a given order on the leaves of the merge trees and hence have the potential to be visualized in a clear and comprehensive manner.In this paper, we introduce ParkView: a schematic, scalable encoding for monotone interleavings. ParkView captures both maps of the interleaving using an optimal decomposition of both trees into paths and corresponding branches. We prove several structural properties of monotone interleavings, which support a sparse visual encoding using active paths and hedges that can be linked using a maximum of 6 colors for merge trees of arbitrary size. We show how to compute an optimal path-branch decomposition in linear time and illustrate ParkView on a number of real-world datasets.

As more and more moving objects are tracked, the amount and variety in trajectory data is ever increasing. Visual summaries provide an at-a-glance overview of such trajectories and are therefore a useful tool for exploring large trajectory collections. Typically, such a visual summary visualizes the spatial positions of moving entities using a one-dimensional representation and combines such representations by placing them in temporal order along a time line. However, existing summaries are generally not tailored to specific patterns that arise in the trajectory data. The formation of groups is a quintessential pattern that emerges in the collective motion of many types of tracked objects, such as humans, birds, and other animals. Our main contribution is GroupRugs, a visual summary technique for collective movement data that highlights the structure of the emerging groups and their evolution over time, while still summarizing the spatial relations in the data. Specifically, we introduce two methods to produce GroupRugs: a naive baseline approach, and a pipeline that optimizes several aspects of GroupRugs. The quality of a visual summary is usually assessed via two main criteria: spatial quality, which measures how well the one-dimensional representations capture the structure of the data points at each time step, and stability, which captures the coherence of consecutive one-dimensional representations over time. For GroupRugs, we additionally care for how well the structure of the emerging groups is expressed in the visualization. In extensive computational experiments, we quantitatively evaluate GroupRugs against state-of-the-art techniques for summarizing trajectories, using well-established metrics for the three important quality criteria. Our evaluation shows that GroupRugs greatly outperform existing techniques in expressing the structure of emerging groups, while sacrificing only little spatial quality or stability.

Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving distance for merge trees is solely based on the hierarchical structure, and disregards any other geometrical or topological properties that might be present in the underlying data. Furthermore, the interleaving distance is NP-hard to compute.

Abstract Polycube segmentations for 3D models effectively support a wide variety of applications such as seamless texture mapping, spline fitting, structured multi‐block grid generation, and hexahedral mesh construction. However, the automated construction of valid polycube segmentations suffers from robustness issues: state‐of‐the‐art methods are not guaranteed to find a valid solution. In this paper we present DualCube: an iterative algorithm which is guaranteed to return a valid polycube segmentation for 3D models of any genus. Our algorithm is based on a dual representation of polycubes. Starting from an initial simple polycube of the correct genus, together with the corresponding dual loop structure and polycube segmentation, we iteratively refine the polycube, loop structure, and segmentation, while maintaining the correctness of the solution. DualCube is robust by construction: at any point during the iterative process the current segmentation is valid. Its iterative nature furthermore facilitates a seamless trade‐off between quality and complexity of the solution. DualCube can be implemented using comparatively simple algorithmic building blocks; our experimental evaluation establishes that the quality of our polycube segmentations is on par with, or exceeding, the state‐of‐the‐art.

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all. We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance. We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

Bias elimination and recent probing studies attempt to remove specific information from embedding spaces.Here it is important to remove as much of the target information as possible, while preserving any other information present.INLP is a popular recent method which removes specific information through iterative nullspace projections.Multiple iterations, however, increase the risk that information other than the target is negatively affected.We introduce two methods that find a single targeted projection: Mean Projection (MP, more efficient) and Tukey Median Projection (TMP, with theoretical guarantees).Our comparison between MP and INLP shows that (1) one MP projection removes linear separability based on the target and (2) MP has less impact on the overall space.Further analysis shows that applying random projections after MP leads to the same overall effects on the embedding space as the multiple projections of INLP.Applying one targeted (MP) projection hence is methodologically cleaner than applying multiple (INLP) projections that introduce random effects.

An important data source for traffic analysis is GPS trajectory data. However, due to measurement inaccuracies, such data does not necessarily align well with data describing the road network. Hence, GPS data typically needs to be aligned with the road network before further analysis can take place; this process is called map matching. The challenges in map matching are exacerbated when trajectories are sparse (have a low measurement frequency); we then need to fill the gaps between the measurements with a realistic estimate of the actual route between two points. As vehicle movement is subject to physical constraints (such as acceleration and speed bounds), an estimated route should be physically consistent. We present a method that creates physically consistent estimated routes to perform map matching for sparse trajectories.

Topological Data Analysis (TDA) focuses on the inherent shape of (spatial) data. As such, it may provide useful methods to explore spatial representations of linguistic data (embeddings) which have become central in NLP. In this paper we aim to introduce TDA to researchers in language technology. We use TDA to represent document structure as so-called story trees. Story trees are hierarchical representations created from semantic vector representations of sentences via persistent homology. They can be used to identify and clearly visualize prominent components of a story line. We showcase their potential by using story trees to create extractive summaries for news stories.

Abstract We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses $$O\bigl (n \log n \log \log n\bigr )$$ O ( n log n log log n ) space, supports queries in worst case $$O\bigl (\log ^2 n\bigr )$$ O ( log 2 n ) time, and updates in $$O\bigl (\log ^5 n\bigr )$$ O ( log 5 n ) amortized time. This leads to an $$O\bigl (n\,\alpha (n)\log ^5 n\bigr )$$ O ( n α ( n ) log 5 n ) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $$O\bigl (n^2\bigr )$$ O ( n 2 ) time algorithms.

The availability of devices that track moving objects has led to an explosive growth in trajectory data. When exploring the resulting large trajectory collections, visual summaries are a useful tool to identify time intervals of interest. A typical approach is to represent the spatial positions of the tracked objects at each time step via a one-dimensional ordering; visualizations of such orderings can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality - how well does the ordering capture the structure of the data at each time step, and stability - how coherent are the orderings over consecutive time steps or temporal ranges?In this paper we introduce a new Stable Principal Component (SPC) method to compute such orderings, which is explicitly parameterized for stability, allowing a trade-off between the spatial quality and stability. We conduct extensive computational experiments that quantitatively compare the orderings produced by ours and other stable dimensionality-reduction methods to various state-of-the-art approaches using a set of well-established quality metrics that capture spatial quality and stability. We conclude that stable dimensionality reduction outperforms existing methods on stability, without sacrificing spatial quality or efficiency; in particular, our new SPC method does so at a fraction of the computational costs.

We study a fundamental question from graph drawing: given a pair $(G,C)$ of a graph $G$ and a cycle $C$ in $G$ together with a simple polygon $P$, is there a straight-line drawing of $G$ inside $P$ which maps $C$ to $P$? We say that such a drawing of $(G,C)$ respects $P$. We fully characterize those instances $(G,C)$ which are polygon-universal, that is, they have a drawing that respects $P$ for any simple (not necessarily convex) polygon $P$. Specifically, we identify two necessary conditions for an instance to be polygon-universal. Both conditions are based purely on graph and cycle distances and are easy to check. We show that these two conditions are also sufficient. Furthermore, if an instance $(G,C)$ is planar, that is, if there exists a planar drawing of $G$ with $C$ on the outer face, we show that the same conditions guarantee for every simple polygon $P$ the existence of a planar drawing of $(G,C)$ that respects $P$. If $(G,C)$ is polygon-universal, then our proofs directly imply a linear-time algorithm to construct a drawing that respects a given polygon $P$.

Understanding human mobility patterns is an important aspect of traffic analysis and urban planning. Trajectory data provide detailed views on specific routes, but typically do not capture all traffic. On the other hand, loop detectors built into the road network capture all traffic flow at specific locations, but provide no information on the individual routes. Given a set of loop-detector measurements as well as a (small) set of representative trajectories, our goal is to investigate how one can effectively combine these two partial data sources to create a more complete picture of the underlying mobility patterns. Specifically, we want to reconstruct a realistic set of routes from the loop-detector data, using the given trajectories as representatives of typical behavior.

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space Rd and each point is labeled with k binary-valued properties. A priori we assume that it is "easy"to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space Rm (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to Rd1 suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear "inseparability"of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities.

Rectangular treemaps visualize hierarchical numerical data by recursively partitioning an input rectangle into smaller rectangles whose areas match the data. Numerical data often has uncertainty associated with it. To visualize uncertainty in a rectangular treemap, we identify two conflicting key requirements: (i) to assess the data value of a node in the hierarchy, the area of its rectangle should directly match its data value, and (ii) to facilitate comparison between data and uncertainty, uncertainty should be encoded using the same visual variable as the data, that is, area. We present Uncertainty Treemaps, which meet both requirements simultaneously by introducing the concept of hierarchical uncertainty masks. First, we define a new cost function that measures the quality of Uncertainty Treemaps. Then, we show how to adapt existing treemapping algorithms to support uncertainty masks. Finally, we demonstrate the usefulness and quality of our technique through an expert review and a computational experiment on real-world datasets.

Abstract Rectangular treemaps are often the method of choice to visualize large hierarchical datasets. Nowadays such datasets are available over time, hence there is a need for (a) treemaps that can handle time‐dependent data, and (b) corresponding quality criteria that cover both a treemap's visual quality and its stability over time. In recent years a wide variety of (stable) treemapping algorithms has been proposed, with various advantages and limitations. We aim to provide insights to researchers and practitioners to allow them to make an informed choice when selecting a treemapping algorithm for specific applications and data. To this end, we perform an extensive quantitative evaluation of rectangular treemaps for time‐dependent data. As part of this evaluation we propose a novel classification scheme for time‐dependent datasets. Specifically, we observe that the performance of treemapping algorithms depends on the characteristics of the datasets used. We identify four potential representative features that characterize time‐dependent hierarchical datasets and classify all datasets used in our experiments accordingly. We experimentally test the validity of this classification on more than 2000 datasets, and analyze the relative performance of 14 state‐of‐the‐art rectangular treemapping algorithms across varying features. Finally, we visually summarize our results with respect to both visual quality and stability to aid users in making an informed choice among treemapping algorithms. All datasets, metrics, and algorithms are openly available to facilitate reuse and further comparative studies.

We study an agglomerative clustering problem motivated by visualizing disjoint glyphs (represented by geometric shapes) centered at specific locations on a geographic map. As we zoom out, the glyphs grow and start to overlap. We replace overlapping glyphs by one larger merged glyph to maintain disjointness. Our goal is to compute the resulting hierarchical clustering efficiently in practice. A straightforward algorithm for such spatial agglomerative clustering runs in O(n2 log n) time, where n is the number of glyphs. This is not efficient enough for many real-world datasets which contain up to tens or hundreds of thousands of glyphs. Recently the theoretical upper bound was improved to O(nα(n) log7 n) time [10], where α(n) is the extremely slow growing inverse Ackermann function. Although this new algorithm is asymptotically much faster than the naive algorithm, from a practical point of view, it does not perform better for n ≤ 106. In this paper we present a new agglomerative clustering algorithm which works efficiently in practice. Our algorithm relies on the use of quadtrees to speed up spatial computations. Interestingly, even in non-pathological datasets we can encounter large glyphs that intersect many quadtree cells and that are involved in many clustering events. We therefore devise a special strategy to handle such large glyphs. We test our algorithm on several synthetic and real-world datasets and show that it performs well in practice.

We propose a novel type of low distortion radial embedding which focuses on one specific entity and its closest neighbors. Our embedding preserves near-exact distances to the focus entity and aims to minimize distortion between the other entities. We present an interactive exploration tool SolarView which places the focus entity at the center of a "solar system" and embeds its neighbors guided by concentric circles. SolarView provides an implementation of our novel embedding and several state-of-the-art dimensionality reduction and embedding techniques, which we adapted to our setting in various ways. We experimentally evaluated our embedding and compared it to these state-of-the-art techniques. The results show that our embedding competes with these techniques and achieves low distortion in practice. Our method performs particularly well when the visualization, and hence the embedding, adheres to the solar system design principle of our application. Nonetheless-as with all dimensionality reduction techniques-the distortion may be high. We leverage interaction techniques to give clear visual cues that allow users to accurately judge distortion. We illustrate the use of SolarView by exploring the high-dimensional metric space of bibliographic entity similarities.

Linear layouts are a simple and natural way to draw a graph: all vertices are placed on a single line and edges are drawn as arcs between the vertices. Despite its simplicity, a linear layout can be a very meaningful visualization if there is a particular order defined on the vertices. Common examples of such ordered - and often also directed - graphs are event sequences and processes. A main drawback of linear layouts are the usually (very) large aspect ratios of the resulting drawings, which prevent users from obtaining a good overview of the whole graph. In this paper we present a novel and versatile algorithm to optimally fold a linear layout of a graph such that it can be drawn nicely in a specified aspect ratio, while still clearly communicating the linearity of the layout. Our algorithm allows vertices to be drawn as blocks or rectangles of specified sizes to incorporate different drawing styles, label sizes, and even recursive structures. For reasonably-sized drawings the folded layout can be computed interactively. We demonstrate the applicability of our algorithm on graphs that represent process trees, a particular type of process model. Our algorithm arguably produces much more readable layouts than existing methods.

We describe an algorithm that morphs between two planar orthogonal drawings Gamma_I and Gamma_O of a connected graph G, while preserving planarity and orthogonality. Necessarily Gamma_I and Gamma_O share the same combinatorial embedding. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. [Biedl et al., 2013]. Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of Gamma_O. We can find corresponding wires in Gamma_I that share topological properties with the wires in Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in Gamma_I, which guides our morph from Gamma_I to Gamma_O.

Linear features on terrains model the boundaries of ground cover regions, delineate glaciers, or form the boundary of rivers and lakes. When computing the similarity between such linear features, it is important to also take their context into account: the terrain. We hence explore the possibilities of volume-based distance measures for linear features on a terrain. Our measures construct suitable base surfaces between the linear features, which can slice through the input terrain and also hover above. The similarity between two linear features is then captured by the volume of "earth" above the base surface and below the terrain, and possibly also by the volume of "air" below the base surface and above the terrain. We suggest six ways of choosing a suitable base surface. These choices give rise to different measured volumes and can be useful in different application scenarios.

Treemaps are a popular tool to visualize hierarchical data: items are represented by nested rectangles and the area of each rectangle corresponds to the data being visualized for this item. The visual quality of a treemap is commonly measured via the aspect ratio of the rectangles. If the data changes, then a second important quality criterion is the stability of the treemap: how much does the treemap change as the data changes. We present a novel stable treemapping algorithm that has very high visual quality. Whereas existing treemapping algorithms generally recompute the treemap every time the input changes, our algorithm changes the layout of the treemap using only local modifications. This approach not only gives us direct control over stability, but it also allows us to use a larger set of possible layouts, thus provably resulting in treemaps of higher visual quality compared to existing algorithms. We further prove that we can reach all possible treemap layouts using only our local modifications. Furthermore, we introduce a new measure for stability that better captures the relative positions of rectangles. We finally show via experiments on real-world data that our algorithm outperforms existing treemapping algorithms also in practice on either visual quality and/or stability. Our algorithm scores high on stability regardless of whether we use an existing stability measure or our new measure.

Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from the one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are delta-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum delta-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.

Motivated by the question of simultaneous embedding of several flow maps, we consider the problem of drawing multiple geometric Steiner arborescences with no crossings in the rectilinear and in the angle-restricted setting. When terminal-to-root paths are allowed to turn freely, we show that two rectilinear Steiner arborescences have a non-crossing drawing if neither tree necessarily completely disconnects the other tree and if the roots of both trees are "free". If the roots are not free, then we can reduce the decision problem to 2SAT. If terminal-to-root paths are allowed to turn only at Steiner points, then it is NP-hard to decide whether multiple rectilinear Steiner arborescences have a non-crossing drawing. The setting of angle-restricted Steiner arborescences is more subtle than the rectilinear case. Our NP-hardness result extends, but testing whether there exists a non-crossing drawing if the roots of both trees are free requires additional conditions to be fulfilled.

We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.

The Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.

The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n / logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.

We consider the problem of computing k shortest paths in a two-dimensional environment with polygonal obstacles, where the jth path, for 1 ≤ j ≤ k, is the shortest path in the free space that is also homotopically distinct from each of the first j – 1 paths. In fact, we consider a more general problem: given a source point s, construct a partition of the free space, called the kth shortest path map (k-SPM), in which the homotopy of the kth shortest path in a region has the same structure. Our main combinatorial result establishes a tight bound of Θ(k2h + kn) on the worst-case complexity of this map. We also describe an O((k3h + k2n) log (kn)) time algorithm for constructing the map. In fact, the algorithm constructs the jth map for every j ≤ k. Finally, we present a simple visibility-based algorithm for computing the k shortest paths between two fixed points. This algorithm runs in O(m log n + k) time and uses O(m + k) space, where m is the size of the visibility graph. This latter algorithm can be extended to compute k shortest simple (non-self-intersecting) paths, taking O(k2 m(m + kn) log (kn)) time. We invite the reader to play with our applet demonstrating k-SPMs [10].

We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 − ϵ) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.

Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an L-contact representation, that is, planar Laman graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an L-contact representation of G. The overall running time is (n2), where n is the number of vertices of G, and the L-contact representation is realized on the n × n grid.

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. These layouts are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists. Furthermore we show how to impose restrictions on the orientations of edges and junctions of the rectangular layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.

Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. Most flow maps are still drawn by hand and only few automated methods exist. Some of the known algorithms do not support edge-bundling and those that do, cannot guarantee crossing-free flows. We present a new algorithmic method that uses edge-bundling and computes crossing-free flows of high visual quality. Our method is based on so-called spiral trees, a novel type of Steiner tree which uses logarithmic spirals. Spiral trees naturally induce a clustering on the targets and smoothly bundle lines. Our flows can also avoid obstacles, such as map features, region outlines, or even the targets. We demonstrate our approach with extensive experiments.

Statistical data associated with geographic regions is nowadays globally available in large amounts and hence automated methods to visually display these data are in high demand. There are several well-established thematic map types for quantitative data on the ratio-scale associated with regions: choropleth maps, cartograms, and proportional symbol maps. However, all these maps suffer from limitations, especially if large data values are associated with small regions. To overcome these limitations, we propose a novel type of quantitative thematic map, the necklace map. In a necklace map, the regions of the underlying two-dimensional map are projected onto intervals on a one-dimensional curve (the necklace) that surrounds the map regions. Symbols are scaled such that their area corresponds to the data of their region and placed without overlap inside the corresponding interval on the necklace. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. They visualize data sets well which are not proportional to region sizes. The linear ordering of the symbols along the necklace facilitates an easy comparison of symbol sizes. One map can contain several nested or disjoint necklaces to visualize clustered data. The advantages of necklace maps come at a price: the association between a symbol and its region is weaker than with other types of maps. Interactivity can help to strengthen this association if necessary. We present an automated approach to generate necklace maps which allows the user to interactively control the final symbol placement. We validate our approach with experiments using various data sets and maps.

No abstract available.

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. They are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.