The high-level structure of a graph is a crucial ingredient for the analysis and visualization of relational data. However, discovering the salient graph patterns that form this structure is notoriously difficult for two reasons. (1) Finding important patterns, such as cliques and bicliques, is computationally hard. (2) Real-world graphs contain noise, and therefore do not always exhibit patterns in their pure form. Defining meaningful noisy patterns and detecting them efficiently is a currently unsolved challenge. In this paper, we propose to use well-ordered matrices as a tool to both define and effectively detect noisy patterns. Specifically, we represent a graph as its adjacency matrix and optimally order it using Moran's $I$. Standard graph patterns (cliques, bicliques, and stars) now translate to rectangular submatrices. Using Moran's $I$, we define a permitted level of noise for such patterns. A combination of exact algorithms and heuristics allows us to efficiently decompose the matrix into noisy patterns. We also introduce a novel motif simplification that visualizes noisy patterns while explicitly encoding the level of noise. We showcase our techniques on several real-world data sets.

Chorematic diagrams are highly reduced schematic maps of geospatial data and processes. They can visually summarize complex situations using only a few simple shapes (choremes) placed upon a simplified base map. Due to the extreme reduction of data in chorematic diagrams, they tend to be produced manually; few automated solutions exist. In this paper we consider the algorithmic problem of summarizing classed region maps, such as choropleth or land use maps, using a chorematic diagram with a single disk choreme. It is infeasible to solve this problem exactly for large maps. Hence, we propose several point sampling strategies and use algorithms for classed point sets to efficiently find the best disk that represents one of the classes. We implemented our algorithm and experimentally compared sampling strategies and densities. The results show that with the right sampling strategy, high-quality results can be obtained already with moderately sized point sets and within seconds of computation time. Les chor & egrave;mes sont des cartes sch & eacute;matiques tr & egrave;s simplifi & eacute;es de donn & eacute;es et processus g & eacute;ospatiaux. Ils peuvent r & eacute;sumer visuellement des situations tr & egrave;s complexes en utilisant uniquement quelques formes simples (les chor & egrave;mes) repr & eacute;sent & eacute;es sur une carte de base simplifi & eacute;e. Compte tenue de la r & eacute;duction extr & ecirc;me des donn & eacute;es dans les cartes chor & egrave;mes, ces cartes sont g & eacute;n & eacute;ralement r & eacute;alis & eacute;es & agrave; la main, peu de solutions automatiques existent. Dans cet article, nous consid & eacute;rons le probl & egrave;me algorithmique de la synth & egrave;se de cartes de r & eacute;gions class & eacute;es, telles que des cartes choropl & egrave;thes ou des cartes d'utilisation des sols, en utilisant une carte chor & egrave;me avec un seul chor & egrave;me en forme de disque. Il n'est pas possible de r & eacute;soudre de fa & ccedil;on exacte ce probl & egrave;me pour de grandes cartes. Ainsi nous proposons plusieurs strat & eacute;gies d'& eacute;chantillonnage de points et nous utilisons des algorithmes pour des ensembles de points class & eacute;s pour trouver efficacement le meilleur disque repr & eacute;sentant l'une des classes. Nous avons impl & eacute;ment & eacute; notre algorithme et nous avons compar & eacute; exp & eacute;rimentalement les strat & eacute;gies et densit & eacute;s d'& eacute;chantillonnage. Les r & eacute;sultats montrent qu'avec la bonne strat & eacute;gie d'& eacute;chantillonnage, on peut obtenir des r & eacute;sultats de grande qualit & eacute; avec un ensemble de points de taille mod & eacute;r & eacute; et en quelques secondes de temps de calcul.

BioFabrics were introduced by Longabaugh in 2012 as a way to draw large graphs in a clear and uncluttered manner. The visual quality of BioFabrics crucially depends on the order of vertices and edges, which can be chosen independently. Effective orders can expose salient patterns, which in turn can be summarized by motifs, allowing users to take in complex networks at-a-glance. However, so far there is no efficient layout algorithm which automatically recognizes patterns and delivers both a vertex and an edge ordering that allows these patterns to be expressed as motifs. In this paper we show how to use well-ordered matrices as a tool to efficiently find good vertex and edge orders for BioFabrics. Specifically, we order the adjacency matrix of the input graph using Moran's $I$ and detect (noisy) patterns with our recent algorithm. In this note we show how to "unfold" the ordered matrix and its patterns into a high-quality BioFabric. Our pipelines easily handles graphs with up to 250 vertices.

A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with “dependent” links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in O(nm) time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an O(n^4m)-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar.

Thematic maps visually communicate statistical information about spatial units such as countries or states. They must balance the individual readability of those map elements that carry the statistical information and the overall cartographic context. Nowadays, most maps are not static images, but must flexibly respond to a range of device types and display sizes. Current approaches to responsive thematic mapping are limited: they are labor-intensive for practitioners and often rely on combining disjointed visual encodings to cover different device types. In this paper we introduce the first algorithmic framework to efficiently compute responsive thematic maps that smoothly adapt to different display sizes. A key component of our framework is the layout guide: a combinatorial structure which encodes the two essential aspects of a thematic map. The first aspect are the visual requirements of each statistical map element (at least their desired width and height), the second aspect is the cartographic context in the form of relative positions of map elements. Our main algorithmic contribution is the map arranger which takes a visual container as input and returns a suitable layout guide. The map arranger does so in a stable and consistent manner: if the container changes only a little, then so does the layout guide, and the same input container always results in the same layout guide. To use our framework, one needs three ingredients: (1) a reference layout, which corresponds to the “ideal” thematic map, (2) a total vertical and horizontal order for all map elements (the desired layouts for containers with extreme aspect ratios), and (3) a thematic mapping algorithm that can construct a thematic map from a layout guide. We demonstrate our framework on two types of thematic maps, namely rectangular and Demers cartograms.

Information visualization is inherently a field that brings together various research domains. Roughly speaking, we may identify two perspectives: the design perspective, revolving around how to ensure that a human can work effectively with the visual representations of data and the tools that offer them, and the algorithmic perspective, focusing on how to automatically create such visual representations. Munzner's model for visualization design places design choices before algorithmic considerations. It offers predominantly a design perspective; as a consequence, applications of this model may consider the algorithmic perspective as an afterthought, bypassing a step that translates the design into the formalism necessary for algorithmic study. As a result, the design may be entangled with the algorithms used to compute a visualization. Focusing on layout algorithms, we explore the ramifications of this entanglement: quality often goes undefined and unmeasured, and ad-hoc heuristics tend to be applied, reducing trustworthiness and potentially leading to incorrect conclusions. We look at how we may complement Munzner's model—the design perspective—with a clear model of the algorithmic perspective, to obtain a formal, measured understanding of the interplay between visualizations and the algorithms used to create them. Paradoxically, the solution lies in a clearer separation of concerns between design and algorithm. We argue that this model leads to better comparison between approaches, a more fine-grained understanding of their strengths and weaknesses, and allows for uncovering new opportunities, as to eventually lead to a better understanding of visualization.

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider unit square symbols that need to be placed at specified $y$-coordinates. We optimize the drawing order of the symbols as well as their $x$-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most $2$, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in $O(n\log n)$ time. If the width is at most $2$, there is a vertical line that stabs all squares, and in this case, we give a 2-approximation algorithm that runs in $O(n\log n)$ time. As a minimum visible perimeter of $2$ is always trivially achievable, we measure this approximation with respect to the visible perimeter exceeding $2$. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

We propose a method for visualizing uncertain set systems, which differs from previous set visualization approaches that are based on certainty (an element either belongs to a set or not). Our method is inspired by storyline visualizations and parallel coordinate plots: (a) each element is represented by a vertical glyph, subdivided into bins that represent different levels of uncertainty; (b) each set is represented by an x-monotone curve that traverses element glyphs through the bins representing the level of uncertainty of their membership. Our implementation also includes optimizations to reduce visual complexity captured by the number of turns for the set curves and the number of crossings. Although several of the natural underlying optimization problems are NP-hard in theory (e.g., optimal element order, optimal set order), in practice, we can compute near-optimal solutions with respect to curve crossings with the help of a new exact algorithm for optimally ordering set curves within each element's bins. With these optimizations, the proposed method makes it easy to see set containment (the smaller set's curve is strictly below the larger set's curve). A brief design-space exploration using uncertain set-membership data, as well as multi-dimensional discrete data, shows the flexibility of the proposed approach.

We study the visual complexity of animated transitions between point sets. Although there exist many metrics for point set similarity, these metrics are not adequate for this purpose, as they typically treat each point separately. Instead, we propose to look at translations of entire subsets/groups of points to measure the visual complexity of a transition between two point sets. Specifically, given two labeled point sets A and B in R^d, the goal is to compute the cheapest transformation that maps all points in A to their corresponding point in B, where the translation of a group of points counts as a single operation in terms of complexity. In this paper we identify several problem dimensions involving group translations that may be relevant to various applications, and study the algorithmic complexity of the resulting problems. Specifically, we consider different restrictions on the groups that can be translated, and different optimization functions. For most of the resulting problem variants we are able to provide polynomial time algorithms, or establish that they are NP-hard. For the remaining open problems we either provide an approximation algorithm or establish the NP-hardness of a restricted version of the problem. Furthermore, our problem classification can easily be extended with additional problem dimensions giving rise to new problem variants that can be studied in future work.

Isolines visually characterize scalar fields by connecting all points of the same value by a closed curve at repeated intervals. They work only as a set which gives the viewer an indication of the shape of the underlying field. Hence, when simplifying isolines it is important that the correspondence - the harmony - between adjacent isolines is preserved whenever it is present. The majority of state-of-the-art simplification methods treat isolines independently; at best they avoid collisions between adjacent simplified isolines. A notable exception is the work by Van Goethem et al. (2021) who were the first to introduce the concept of harmony between adjacent isolines explicitly as an algorithmic design principle. They presented a proof-of-concept algorithm that harmoniously simplifies a sequence of polylines. However, the sets of isolines of scalar fields, most notably terrain, consist of closed curves which are nested in arbitrarily complex ways and not of an ordered sequence of polylines. In this paper we significantly extend the work by Van Goethem et al. (2021) to capture harmony in general sets of isolines. Our new simplification algorithm can handle sets of isolines describing arbitrary scalar fields and is more efficient, allowing us to harmoniously simplify terrain with hundreds of thousands of vertices. We experimentally compare our method to the results of Van Goethem et al. (2021) on bundles of isolines and to general simplification methods on isolines extracted from DEMs of Antartica. Our results indicate that our method efficiently preserves the harmony in the simplified maps, which are thereby less noisy, cartographically more meaningful, and easier to read.

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k \geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of $S$ such that $\mathcal{CH}(I) \cap S = I$, where $\mathcal{CH}(I)$ denotes the convex hull of $I$. We identify an island with its convex hull; therefore, a partition into islands has the additional requirement that the convex hulls of the islands are pairwise-disjoint. We present three greedy algorithms for constructing island partitions and analyze their approximation ratios.

Points of interest on a map such as restaurants, hotels, or subway stations, give rise to categorical point data: data that have a fixed location and one or more categorical attributes. Consequently, recent years have seen various set visualization approaches that visually connect points of the same category to support users in understanding the spatial distribution of categories. Existing methods use complex and often highly irregular shapes to connect points of the same category, leading to high cognitive load for the user. In this paper we introduce SimpleSets, which uses simple shapes to enclose categorical point patterns, thereby providing a clean overview of the data distribution. SimpleSets is designed to visualize sets of points with a single categorical attribute; as a result, the point patterns enclosed by SimpleSets form a partition of the data. We give formal definitions of point patterns that correspond to simple shapes and describe an algorithm that partitions categorical points into few such patterns. Our second contribution is a rendering algorithm that transforms a given partition into a clean set of shapes resulting in an aesthetically pleasing set visualization. Our algorithm pays particular attention to resolving intersections between nearby shapes in a consistent manner. We compare SimpleSets to the state-of-the-art set visualizations using standard datasets from the literature.

Grid maps are a well-known technique to visualize data associated with spatial regions. A grid map assigns each region to a tile in a grid (often orthogonal or hexagonal) and then represents the associated data values within this tile. Good grid maps represent the underlying geographic space well: regions that are geographically close are close in the grid map and vice versa. Though Tobler’s law suggests that spatial proximity relates to data similarity, local variations may obscure clusters and patterns in the data. For example, there are often clear differences between urban centers and adjacent rural areas with respect to socio-economic indicators. To get a better view of the data distribution, we propose grid-map layouts that take data values into account and place regions with similar data into close proximity. In the limit, such a data layout is essentially a chart and loses all spatial meaning. We present an algorithm to create hybrid layouts, allowing for trade-offs between data values and geographic space when assigning regions to tiles. Our algorithm also handles hierarchical grid maps and allows us to focus either on data or on geographic space on different levels of the hierarchy. Leveraging our algorithm we explore the design space of (hierarchical) grid maps with a hybrid layout and their semantics.

Set systems can be visualized in various ways. An important distinction between techniques is whether the elements have a spatial location that is to be used for the visualization; for example, the elements are cities on a map. Strictly adhering to such location may severely limit the visualization and force overlay, intersections and other forms of clutter. On the other hand, completely ignoring the spatial dimension omits information and may hide spatial patterns in the data. We study layouts for set systems (or hypergraphs) in which spatial locations are displaced onto concentric circles or a grid, to obtain schematic set visualizations. We investigate the tractability of the underlying algorithmic problems adopting different optimization criteria (e.g. crossings or bends) for the layout structure, also known as the support of the hypergraph. Furthermore, we describe a simulated‐annealing approach to heuristically optimize a combination of such criteria. Using this method in computational experiments, we explore the trade‐offs and dependencies between criteria for computing high‐quality schematic set visualizations.

Cartograms are popular for visualizing numerical data for administrative regions in thematic maps. When there are multiple data values per region (over time or from different datasets) shown as animated or juxtaposed cartograms, preserving the viewer's mental map in terms of stability between multiple cartograms is another important criterion alongside traditional cartogram criteria such as maintaining adjacencies. We present a method to compute stable stable Demers cartograms, where each region is shown as a square scaled proportionally to the given numerical data and similar data yield similar cartograms. We enforce orthogonal separation constraints using linear programming, and measure quality in terms of keeping adjacent regions close (cartogram quality) and using similar positions for a region between the different data values (stability). Our method guarantees the ability to connect most lost adjacencies with minimal-length planar orthogonal polylines. Experiments show that our method yields good quality and stability on multiple quality criteria.

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.

Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n log n log(2) k)-time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk)-time algorithm for any model where consistency is "concatenable": a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models that are not concatenable. We show how to compute the maximum subsequence for such models in O(nk(2) log k) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets.

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 R^d 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 R^m (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 R^(d-1) 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.

Grid maps are spatial arrangements of simple tiles (often squares or hexagons), each of which represents a spatial element. They are an established, effective way to show complex data per spatial element, using visual encodings within each tile ranging from simple coloring to nested small-multiples visualizations. An effective grid map is coherent with the underlying geographic space: the tiles maintain the contiguity, neighborhoods and identifiability of the corresponding spatial elements, while the grid map as a whole maintains the global shape of the input. Of particular importance are salient local features of the global shape which need to be represented by tiles assigned to the appropriate spatial elements. State-of-the-art techniques can adequately deal only with simple cases, such as close-to-uniform spatial distributions or global shapes that have few characteristic features. We introduce a simple fully-automated 3-step pipeline for computing coherent grid maps. Each step is a well-studied problem: shape decomposition based on salient features, tile-based Mosaic Cartograms, and point-set matching. Our pipeline is a seamless composition of existing techniques for these problems and results in high-quality grid maps. We provide an implementation, demonstrate the efficacy of our approach on various complex datasets, and compare it to the state-of-the-art.

GPS trajectories of vehicles moving on a road network are a valuable source of traffic information. However, the sheer volume of available data makes it challenging to identify and visualize salient patterns. Meaningful visual summaries of trajectory collections require that both the trajectories and the underlying network are aggregated and simplified in a coherent manner. In this paper we propose a coordinated fully-automated pipeline for computing a schematic overview of mobility patterns from a collection of trajectories on a street network. Our pipeline utilizes well-known building blocks from GIS, automated cartography, and trajectory analysis: map matching, road selection, schematization, movement patterns, and metro-map style rendering. We showcase the results of our pipeline on two real-world trajectory collections around The Hague and Beijing.