arXiv subjects

Thomas C. van Dijk

Publications and source records attributed to Thomas C. van Dijk.

9 recordsLinked to original sources

arXiv2026

Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints

Geophylogenies are a common type of diagram for visualizing the evolutionary history of species in a geographic context. As a drawing problem, these diagrams are commonly modeled as a rooted binary ''phylogenetic'' tree $T$ where every leaf is associated with a point feature (''site'') in a rectangular map range. The tree is drawn downward planar, its leaves are placed at equidistant positions on the upper boundary of the map, and each leaf is connected to its corresponding site by a straight-line leader. Prior work focuses on minimizing the number of leader crossings, or avoiding leaders altogether. In this paper, we explore paged geophylogenies, where the leaders can be partitioned into multiple pages where only crossings within a page are counted. For the general case, where each page can contain an arbitrary subset of the leaders, we provide an integer linear programming (ILP) formulation for minimizing the number of pages, and a polynomial-time algorithm for a special case that is equivalent to a one-sided tanglegram. We argue that, from a visualization perspective, instead each page must contain only leaders for a single subtree, and provide an $\mathcal{O}(n^7)$ time algorithm for minimizing pages in this setting - which also involves improving the fastest known algorithm for testing the existence of a crossing-free labeling. Counter to what our worst case bound suggests, our implementation can handle instances with hundreds of sites within a second, as we show in our experimental evaluation, which further investigates the trade-offs between leader crossings and the number of pages.

cs.CG↗

arXiv2025

Visualizing Geophylogenies -- Internal and External Labeling with Phylogenetic Tree Constraints

A geophylogeny is a phylogenetic tree (or dendrogram) where each leaf (e.g. biological taxon) has an associated geographic location (site). To clearly visualize a geophylogeny, the tree is typically represented as a crossing-free drawing next to a map. The correspondence between the taxa and the sites is either shown with matching labels on the map (internal labeling) or with leaders that connect each site to the corresponding leaf of the tree (external labeling). In both cases, a good order of the leaves is paramount for understanding the association between sites and taxa. We define several quality measures for internal labeling and give an efficient algorithm for optimizing them. In contrast, minimizing the number of leader crossings in an external labeling is NP-hard. On the positive side, we show that crossing-free instances can be solved in polynomial time and give a fixed-parameter tractable (FPT) algorithm. Furthermore, optimal solutions can be found in a matter of seconds on realistic instances using integer linear programming. Finally, we provide several efficient heuristic algorithms and experimentally show them to be near optimal on real-world and synthetic instances.

cs.DM↗

arXiv2019

Bundled Crossings Revisited

An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into \emph{bundles}. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a \emph{bundled crossing}. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most $k$ bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in $k$) when we require a simple circular layout. These results make use of the connection between bundled crossings and graph genus.

cs.CG↗

arXiv2018

Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far. Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. A constrained variant of Stabbing turns out to be even APX-hard. While for general set cover the best possible approximation ratio is $Θ(\log n)$, it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity. Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.

cs.CG↗

arXiv2017

Computing Storyline Visualizations with Few Block Crossings

Storyline visualizations show the structure of a story, by depicting the interactions of the characters over time. Each character is represented by an x-monotone curve from left to right, and a meeting is represented by having the curves of the participating characters run close together for some time. There have been various approaches to drawing storyline visualizations in an automated way. In order to keep the visual complexity low, rather than minimizing pairwise crossings of curves, we count block crossings, that is, pairs of intersecting bundles of lines. Partly inspired by the ILP-based approach of Gronemann et al. [GD 2016] for minimizing the number of pairwise crossings, we model the problem as a satisfiability problem (since the straightforward ILP formulation becomes more complicated and harder to solve). Having restricted ourselves to a decision problem, we can apply powerful SAT solvers to find optimal drawings in reasonable time. We compare this SAT-based approach with two exact algorithms for block crossing minimization, using both the benchmark instances of Gronemann et al. and random instances. We show that the SAT approach is suitable for real-world instances and identify cases where the other algorithms are preferable.

cs.CG↗

arXiv2016

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straight-line drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneous drawing with six bends per edge each, in quadratic area. For more restricted classes of planar graphs (i.e., matchings, paths, cycles, outerplanar graphs and subhamiltonian graphs), we manage to significantly reduce the required number of bends per edge, while keeping the area quadratic.

cs.CG↗

arXiv2016

Block Crossings in Storyline Visualizations

Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally.

cs.CG↗

arXiv2016

Snapping Graph Drawings to the Grid Optimally

In geographic information systems and in the production of digital maps for small devices with restricted computational resources one often wants to round coordinates to a rougher grid. This removes unnecessary detail and reduces space consumption as well as computation time. This process is called snapping to the grid and has been investigated thoroughly from a computational-geometry perspective. In this paper we investigate the same problem for given drawings of planar graphs under the restriction that their combinatorial embedding must be kept and edges are drawn straight-line. We show that the problem is NP-hard for several objectives and provide an integer linear programming formulation. Given a plane graph G and a positive integer w, our ILP can also be used to draw G straight-line on a grid of width w and minimum height (if possible).

cs.CG↗

arXiv2014

Improved Approximation Algorithms for Box Contact Representations

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called \textsc{Contact Representation of Word Networks} (\textsc{Crown}) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. \textsc{Crown} is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, \textsc{Max-Crown}, in which realizing each desired adjacency yields a certain profit. We present the first $O(1)$-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

cs.DS↗