arXiv subjects

Jules Wulms

Publications and source records attributed to Jules Wulms.

At least 19 recordsLinked to original sources

arXiv2026

Clarity and Computational Efficiency of Orbital Boundary Labeling

Circular interfaces such as those found on smartwatches, automotive dashboards, cockpit instruments, or in radial visualizations pose unique challenges for placing readable labels. Traditional rectangular labeling methods waste screen space and create visual clutter on these constrained displays. In orbital boundary labeling, the labels (e.g., the features' names) are placed in an annulus-shaped orbit outside of the figure, and each label is connected to its feature using a short, crossing-free leader line. We contribute algorithms to compute two leader styles, orbital-radial and straight-line, for uniform and non-uniform label sizes, optimizing for crossing-free shortest leaders. We evaluate the model and the algorithms with computational experiments and a controlled user experiment. The user experiment reveals that both leader types exhibit similar accuracy, but straight-line leaders yield faster response times.

cs.HC↗

arXiv2026

Unfolding Ordered Matrices into BioFabric Motifs

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.

cs.HC↗

arXiv2026

Noisy Graph Patterns via Ordered Matrices

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.

cs.HC↗

arXiv2024

Visualizing Extensions of Argumentation Frameworks as Layered Graphs

The visualization of argumentation frameworks (AFs) is crucial for enabling a wide applicability of argumentative tools. However, their visualization is often considered only as an accompanying part of tools for computing semantics and standard graphical representations are used. We introduce a new visualization technique that draws an AF, together with an extension (as part of the input), as a 3-layer graph layout. Our technique supports the user to more easily explore the visualized AF, better understand extensions, and verify algorithms for computing semantics. To optimize the visual clarity and aesthetics of this layout, we propose to minimize edge crossings in our 3-layer drawing. We do so by an exact ILP-based approach, but also propose a fast heuristic pipeline. Via a quantitative evaluation, we show that the heuristic is feasible even for large instances, while producing at most twice as many crossings as an optimal drawing in most cases.

cs.HC↗

arXiv2024

Boundary Labeling in a Circular Orbit

Boundary labeling is a well-known method for displaying short textual labels for a set of point features in a figure alongside the boundary of that figure. Labels and their corresponding points are connected via crossing-free leaders. We propose orbital boundary labeling as a new variant of the problem, in which (i) the figure is enclosed by a circular contour and (ii) the labels are placed as disjoint circular arcs in an annulus-shaped orbit around the contour. The algorithmic objective is to compute an orbital boundary labeling with the minimum total leader length. We identify several parameters that define the corresponding problem space: two leader types (straight or orbital-radial), label size and order, presence of candidate label positions, and constraints on where a leader attaches to its label. Our results provide polynomial-time algorithms for many variants and NP-hardness for others, using a variety of geometric and combinatorial insights.

cs.CG↗

arXiv2024

The Complexity of Geodesic Spanners using Steiner Points

A geometric $t$-spanner $\mathcal{G}$ on a set $S$ of $n$ point sites in a metric space $P$ is a subgraph of the complete graph on $S$ such that for every pair of sites $p,q$ the distance in $\mathcal{G}$ is a most $t$ times the distance $d(p,q)$ in $P$. We call a connection between two sites a \emph{link}. In some settings, such as when $P$ is a simple polygon with $m$ vertices and a link is a shortest path in $P$, links can consist of $\Theta (m)$ segments and thus have non-constant complexity. The spanner complexity is a measure of how compact a spanner is, which is equal to the sum of the complexities of all links in the spanner. In this paper, we study what happens if we are allowed to introduce $k$ Steiner points to reduce the spanner complexity. We study such Steiner spanners in simple polygons, polygonal domains, and edge-weighted trees. We show that Steiner points have only limited utility. For a spanner that uses $k$ Steiner points, we provide an $\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the worst-case complexity of any $(t-\varepsilon)$-spanner, for any constant $\varepsilon \in (0,1)$ and integer constant $t \geq 2$. Additionally, we show NP-hardness for the problem of deciding whether a set of sites in a polygonal domain admits a $3$-spanner with a given maximum complexity using $k$ Steiner points. On the positive side, for trees we show how to build a $2t$-spanner that uses $k$ Steiner points of complexity $O(mn^{1/t}/k^{1/t} + n \log (n/k))$, for any integer $t \geq 1$. We generalize this to forests, and use it to obtain a $2\sqrt{2}t$-spanner in a simple polygon with complexity $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} + n\log^2 n)$. When a link can be any path between two sites, we show how to improve the spanning ratio to $(2k+\varepsilon)$, for any constant $\varepsilon \in (0,2k)$, and how to build a $6t$-spanner in a polygonal domain with the same complexity.

cs.CG↗

arXiv2024

Capturing the Shape of a Point Set with a Line Segment

Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line segment. Specifically, given a radius $r$, we say a line segment is representative of a point set $P$ if it is within distance $r$ of each point $p \in P$. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius $r$ using the shortest line segment. We describe an algorithm to find the shortest representative segment in $O(n \log h + h \log^3 h)$ time. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in $P$ move.

cs.CG↗

arXiv2024

Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4

A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart pseudo-triangulations ($k$-DPTs), that is, pseudo-triangulations in which exactly $k$ faces are darts and all other faces are triangles. We study the flip graph for such pseudo-triangulations, in which a flip exchanges the diagonals of a pseudo-quadrilatral. Our results are as follows. We prove that the flip graph of $1$-DPTs is generally not connected, and show how to compute its connected components. Furthermore, for $k$-DPTs on a point configuration called the double chain we analyze the structure of the flip graph on a more fine-grained level.

cs.CG↗

arXiv2024

Competitive Searching over Terrains

We study a variant of the searching problem where the environment consists of a known terrain and the goal is to obtain visibility of an unknown target point on the surface of the terrain. The searcher starts on the surface of the terrain and is allowed to fly above the terrain. The goal is to devise a searching strategy that minimizes the competitive ratio, that is, the worst-case ratio between the distance traveled by the searching strategy and the minimum travel distance needed to detect the target. For $1.5$D terrains we show that any searching strategy has a competitive ratio of at least $\sqrt{82}$ and we present a nearly-optimal searching strategy that achieves a competitive ratio of $3\sqrt{19/2} \approx \sqrt{82} + 0.19$. This strategy extends directly to the case where the searcher has no knowledge of the terrain beforehand. For $2.5$D terrains we show that the optimal competitive ratio depends on the maximum slope $\lambda$ of the terrain, and is hence unbounded in general. Specifically, we provide a lower bound on the competitive ratio of $\Omega(\sqrt{\lambda})$. Finally, we complement the lower bound with a searching strategy based on the maximum slope of the known terrain, which achieves a competitive ratio of $O(\sqrt{\lambda})$.

cs.CG↗

arXiv2023

Minimizing Corners in Colored Rectilinear Grids

Given a rectilinear grid $G$, in which cells are either assigned a single color, out of $k$ possible colors, or remain white, can we color white grid cells of $G$ to minimize the total number of corners of the resulting colored rectilinear polygons in $G$? We show how this problem relates to hypergraph visualization, prove that it is NP-hard even for $k=2$, and present an exact dynamic programming algorithm. Together with a set of simple kernelization rules, this leads to an FPT-algorithm in the number of colored cells of the input. We additionally provide an XP-algorithm in the solution size, and a polynomial $\mathcal{O}(OPT)$-approximation algorithm.

cs.CG↗

arXiv2023

Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

A fundamental question is whether one can maintain a maximum independent set in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of $n$ disks of unit radius in the plane, we show that a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log n)$, and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension $d$, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain $O(1+\varepsilon)$-approximate maximum independent set in truly sublinear update time, under standard complexity assumptions.

cs.CG↗

arXiv2023

Crossing Minimization in Time Interval Storylines

Storyline visualizations are a popular way of visualizing characters and their interactions over time: Characters are drawn as x-monotone curves and interactions are visualized through close proximity of the corresponding character curves in a vertical strip. Existing methods to generate storylines assume a total ordering of the interactions, although real-world data often do not contain such a total order. Instead, multiple interactions are often grouped into coarser time intervals such as years. We exploit this grouping property by introducing a new model called storylines with time intervals and present two methods to minimize the number of crossings and horizontal space usage. We then evaluate these algorithms on a small benchmark set to show their effectiveness.

cs.CG↗

arXiv2022

The Influence of Dimensions on the Complexity of Computing Decision Trees

A decision tree recursively splits a feature space $\mathbb{R}^{d}$ and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work treats heuristic algorithms to compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number $d$ of dimensions of the feature space. We show that it can be solved in $O(n^{2d + 1}d)$ time, but under reasonable complexity-theoretic assumptions it is not possible to achieve $f(d) \cdot n^{o(d / \log d)}$ running time, where $n$ is the number of training examples. The problem is solvable in $(dR)^{O(dR)} \cdot n^{1+o(1)}$ time, if there are exactly two classes and $R$ is an upper bound on the number of tree leaves labeled with the first~class.

cs.CC↗

arXiv2022

Planarizing Graphs and their Drawings by Vertex Splitting

The splitting number of a graph $G=(V,E)$ is the minimum number of vertex splits required to turn $G$ into a planar graph, where a vertex split removes a vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the edges formerly incident to $v$ among its two split copies $v_1, v_2$. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in $\mathbb R^2$, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform fixed-parameter tractable (FPT) algorithm for the splitting number problem (without drawings). Then we show the NP-completeness of the splitting number problem for graph drawings, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.

cs.CG↗

arXiv2022

Transitions in Dynamic Point Labeling

The labeling of point features on a map is a well-studied topic. In a static setting, the goal is to find a non-overlapping label placement for (a subset of) point features. In a dynamic setting, the set of point features and their corresponding labels change, and the labeling has to adapt to such changes. To aid the user in tracking these changes, we can use morphs, here called transitions, to indicate how a labeling changes. Such transitions have not gained much attention yet, and we investigate different types of transitions for labelings of points, most notably consecutive transitions and simultaneous transitions. We give (tight) upper bounds on the number of overlaps that can occur during these transitions. When each label has a non-negative weight associated to it, and each overlap imposes a penalty proportional to the weight of the overlapping labels, we show that it is NP-complete to decide whether the penalty during a simultaneous transition has weight at most $k$. Finally, we consider geotagged data on a map, by labeling points with rectangular or square labels. We developed a prototype implementation to evaluate different transition styles in practice, measuring both number of overlaps and transition duration.

cs.CG↗

arXiv2021

Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares

A well-established theoretical model for modular robots in two dimensions are edge-connected configurations of square modules, which can reconfigure through so-called sliding moves. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of $n$ squares into any other using at most $O(n^2)$ sliding moves, while keeping the configuration connected at all times. For certain pairs of configurations, reconfiguration may require $Ω(n^2)$ sliding moves. However, significantly fewer moves may be sufficient. We prove that it is NP-hard to minimize the number of sliding moves for a given pair of edge-connected configurations. On the positive side we present Gather&Compact, an input-sensitive in-place algorithm that requires only $O(\bar{P} n)$ sliding moves to transform one configuration into the other, where $\bar{P}$ is the maximum perimeter of the two bounding boxes. The squares move within the bounding boxes only, with the exception of at most one square at a time which may move through the positions adjacent to the bounding boxes. The $O(\bar{P} n)$ bound never exceeds $O(n^2)$, and is optimal (up to constant factors) among all bounds parameterized by just $n$ and $\bar{P}$. Our algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid $xy$-monotone component. We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacristán [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.

cs.CG↗

arXiv2021

Worbel: Aggregating Point Labels into Word Clouds

Point feature labeling is a classical problem in cartography and GIS that has been extensively studied for geospatial point data. At the same time, word clouds are a popular visualization tool to show the most important words in text data which has also been extended to visualize geospatial data (Buchin et al. PacificVis 2016). In this paper, we study a hybrid visualization, which combines aspects of word clouds and point labeling. In the considered setting, the input data consists of a set of points grouped into categories and our aim is to place multiple disjoint and axis-aligned rectangles, each representing a category, such that they cover points of (mostly) the same category under some natural quality constraints. In our visualization, we then place category names inside the computed rectangles to produce a labeling of the covered points which summarizes the predominant categories globally (in a word-cloud-like fashion) while locally avoiding excessive misrepresentation of points (i.e., retaining the precision of point labeling). We show that computing a minimum set of such rectangles is NP-hard. Hence, we turn our attention to developing heuristics and exact SAT models to compute our visualizations. We evaluate our algorithms quantitatively, measuring running time and quality of the produced solutions, on several artificial and real-world data sets. Our experiments show that the heuristics produce solutions of comparable quality to the SAT models while running much faster.

cs.CG↗

arXiv2021

Layered Area-Proportional Rectangle Contact Representations

We investigate two optimization problems on area-proportional rectangle contact representations for layered, embedded planar graphs. The vertices are represented as interior-disjoint unit-height rectangles of prescribed widths, grouped in one row per layer, and each edge is ideally realized as a rectangle contact of positive length. Such rectangle contact representations find applications in semantic word or tag cloud visualizations, where a collection of words is displayed such that pairs of semantically related words are close to each other. In this paper, we want to maximize the number of realized rectangle contacts or minimize the overall area of the rectangle contact representation, while avoiding any false adjacencies. We present a network flow model for area minimization, a linear-time algorithm for contact maximization of two-layer graphs, and an ILP model for maximizing contacts of $k$-layer graphs.

cs.CG↗