Graph drawing with spring embedders employs a V x V computation phase over the graph's vertex set to compute repulsive forces. Here, the efficacy of forces diminishes with distance: a vertex can effectively only influence other vertices in a certain radius around its position. Therefore, the algorithm lends itself to an implementation using search data structures to reduce the runtime complexity. NVIDIA RT cores implement hierarchical tree traversal in hardware. We show how to map the problem of finding graph layouts with force-directed methods to a ray tracing problem that can subsequently be implemented with dedicated ray tracing hardware. With that, we observe speedups of 4x to 13x over a CUDA software implementation.
We consider the problem of placing arrow heads in directed graph drawings without them overlapping other drawn objects. This gives drawings where edge directions can be deduced unambiguously. We show hardness of the problem, present exact and heuristic algorithms, and report on a practical study.
We study planar drawings of directed graphs in the L-drawing standard. We provide necessary conditions for the existence of these drawings and show that testing for the existence of a planar L-drawing is an NP-complete problem. Motivated by this result, we focus on upward-planar L-drawings. We show that directed st-graphs admitting an upward- (resp. upward-rightward-) planar L-drawing are exactly those admitting a bitonic (resp. monotonically increasing) st-ordering. We give a linear-time algorithm that computes a bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or rep…
We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of matrix representations. In an L-drawing, vertices have exclusive $x$- and $y$-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-completeness and provide a heuristics based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. W…
In this paper we present a new approach to visualize directed graphs and their hierarchies that completely departs from the classical four-phase framework of Sugiyama and computes readable hierarchical visualizations that contain the complete reachability information of a graph. Additionally, our approach has the advantage that only the necessary edges are drawn in the drawing, thus reducing the visual complexity of the resulting drawing. Furthermore, most problems involved in our framework require only polynomial time. Our framework offers a suite of solutions depending upon the requirements,…
Force-directed algorithms are among the most flexible methods for calculating layouts of simple undirected graphs. Also known as spring embedders, such algorithms calculate the layout of a graph using only information contained within the structure of the graph itself, rather than relying on domain-specific knowledge. Graphs drawn with these algorithms tend to be aesthetically pleasing, exhibit symmetries, and tend to produce crossing-free layouts for planar graphs. In this survey we consider several classical algorithms, starting from Tutte's 1963 barycentric method, and including recent scal…
We consider the problem of layering general directed graphs under height and possibly also width constraints. Given a directed graph G = (V,A) and a maximal height, we propose a layering approach that minimizes a weighted sum of the number of reversed arcs, the arc lengths, and the width of the drawing. We call this the Compact Generalized Layering Problem (CGLP). Here, the width of a drawing is defined as the maximum sum of the number of vertices placed on a layer and the number of dummy vertices caused by arcs traversing the layer. The CGLP is NP-hard. We present two MIP models for this prob…
We introduce the problem of Weak Dominance Drawing for general directed acyclic graphs and we show the connection with the linear extension diameter of a partial order P. We present complexity results and bounds.
In a planar confluent orthogonal drawing (PCOD) of a directed graph (digraph) vertices are drawn as points in the plane and edges as orthogonal polylines starting with a vertical segment and ending with a horizontal segment. Edges may overlap in their first or last segment, but must not intersect otherwise. PCODs can be seen as a directed variant of Kandinsky drawings or as planar L-drawings of subdivisions of digraphs. The maximum number of subdivision vertices in an edge is then the split complexity. A PCOD is upward if each edge is drawn with monotonically increasing y-coordinates and quasi…
In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge $e$ is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of $e$ and of a horizontal segment ending at the head of $e$. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for $st$-graphs, i.e., planar DAGs with a single source $s$ and a single sink $t$ containing an edge directed from $s$ to $t$. It is known that a plane $st$-graph, i.e., an embedded $st$-graph in which the edge $(s,t)$ is incident to the outer face, admits an upward-…
We study the problem of selecting a maximum-weight subgraph of a given graph such that the subgraph can be drawn within a prescribed drawing area subject to given non-uniform vertex sizes. We develop and analyze heuristics both for the general (undirected) case and for the use case of (directed) calculation graphs which are used to analyze the typical mistakes that high school students make when transforming mathematical expressions in the process of calculating, for example, sums of fractions.
Shape-based metrics measure how faithfully a drawing D represents the structure of a graph G, using the proximity graph S of D. While some limited graph classes admit proximity drawings (i.e., optimally shape-faithful drawings, where S = G), algorithms for shape-faithful drawings of general graphs have not been investigated. In this paper, we present the first study for shape-faithful drawings of general graphs. First, we conduct extensive comparison experiments for popular graph layouts using the shape-based metrics, and examine the properties of highly shape-faithful drawings. Then, we prese…
In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing $Γ_H$ of a subgraph $H$ of a directed graph $G$ and asks whether $Γ_H$ can be extended to an upward planar drawing of $G$. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results. First, we prove that the Upward Planarity Extension problem is NP-complete, even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides wit…
A major factor affecting the readability of a graph drawing is its resolution. In the graph drawing literature, the resolution of a drawing is either measured based on the angles formed by consecutive edges incident to a common node (angular resolution) or by the angles formed at edge crossings (crossing resolution). In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution. To the best of our knowledge, this is the first time where the problem of maximizing the total resolution of a drawing is studied. The main…
We describe a graph visualization tool for visualizing Java bytecode. Our tool, which we call J-Viz, visualizes connected directed graphs according to a canonical node ordering, which we call the sibling-first recursive (SFR) numbering. The particular graphs we consider are derived from applying Shiver's k-CFA framework to Java bytecode, and our visualizer includes helpful links between the nodes of an input graph and the Java bytecode that produced it, as well as a decompiled version of that Java bytecode. We show through several case studies that the canonical drawing paradigm used in J-Viz …
While orthogonal drawings have a long history, smooth orthogonal drawings have been introduced only recently. So far, only planar drawings or drawings with an arbitrary number of crossings per edge have been studied. Recently, a lot of research effort in graph drawing has been directed towards the study of beyond-planar graphs such as 1-planar graphs, which admit a drawing where each edge is crossed at most once. In this paper, we consider graphs with a fixed embedding. For 1-planar graphs, we present algorithms that yield orthogonal drawings with optimal curve complexity and smooth orthogonal…
The visualization of any graph plays important role in various aspects, such as graph drawing software. Complex systems (like large databases or networks) that have a graph structure should be properly visualized in order to avoid obfuscation. One way to provide an aesthetic improvement to a graph visualization is to apply a force-directed drawing algorithm to it. This method, that emerged in the 60's views graphs as spring systems that exert forces (repulsive or attractive) to the nodes. A Lombardi drawing of a graph is a drawing where the edges are drawn as circular arcs (straight edges are …
The crossing resolution of a non-planar drawing of a graph is the value of the minimum angle formed by any pair of crossing edges. Recent experiments have shown that the larger the crossing resolution is, the easier it is to read and interpret a drawing of a graph. However, maximizing the crossing resolution turns out to be an NP-hard problem in general and only heuristic algorithms are known that are mainly based on appropriately adjusting force-directed algorithms. In this paper, we propose a new heuristic algorithm for the crossing resolution maximization problem and we experimentally compa…
In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We show that every plane bimodal graph without 2-cycles admits a planar L-drawing. This includes the class of upward-plane graphs. Finally, outerplanar digraphs admit a planar L-drawing - although the…
A weak dominance drawing $Γ$ of a DAG $G=(V,E)$, is a $d$-dimensional drawing such that there is a directed path from a vertex $u$ to a vertex $v$ in $G$ if $D(u) <D(v)$ for every dimension $D$ of $Γ$. We have a \emph{falsely implied path (fip)} when $D(u) < D(v)$ for every dimension $D$ of~$Γ$, but there is no path from $u$ to $v$. Minimizing the number of fips is an important theoretical and practical problem, which is NP-hard. We show that it is an FPT~problem for parameter $k$, where $k$ is the maximum degree of a vertex of the \emph{modular~decomposition~tree} of~$G$. Namely, for any cons…
In this paper we study the problem of designing a distributed graph visualization algorithm for large graphs. The algorithm must be simple to implement and the computing infrastructure must not require major hardware or software investments. We design, implement, and experiment a force-directed algorithm in Giraph, a popular open source framework for distributed computing, based on a vertex-centric design paradigm. The algorithm is tested both on real and artificial graphs with up to million edges, by using a rather inexpensive PaaS (Platform as a Service) infrastructure of Amazon. The experim…
We present algorithms and experiments for the visualization of directed graphs that focus on displaying their reachability information. Our algorithms are based on the concepts of the path and channel decomposition as proposed in the framework presented in GD 2018 (pp. 579-592) and focus on showing the existence of paths clearly. In this paper we customize these concepts and present experimental results that clearly show the interplay between bends, crossings and clarity. Additionally, our algorithms have direct applications to the important problem of showing and storing transitivity informat…
We present a new heuristic algorithm for computing a minimum Feedback Arc Set in directed graphs. The new technique produces solutions that are better than the ones produced by the best previously known heuristics, often reducing the FAS size by more than 50%. It is based on computing the PageRank score of the nodes of the directed line graph of the input directed graph. Although the time required by our heuristic is heavily influenced by the size of the produced line graph, our experimental results show that it runs very fast even for very large graphs used in graph drawing.
We present a novel algorithm to match GPS trajectories onto maps offline (in batch mode) using techniques borrowed from the field of force-directed graph drawing. We consider a simulated physical system where each GPS trajectory is attracted or repelled by the underlying road network via electrical-like forces. We let the system evolve under the action of these physical forces such that individual trajectories are attracted towards candidate roads to obtain a map matching path. Our approach has several advantages compared to traditional, routing-based, algorithms for map matching, including th…
In this paper, we introduce and study the multilevel-planarity testing problem, which is a generalization of upward planarity and level planarity. Let $G = (V, E)$ be a directed graph and let $\ell: V \to \mathcal P(\mathbb Z)$ be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of $G$ is a planar drawing of $G$ such that the $y$-coordinate of each vertex $v \in V$ is $y(v) \in \ell(v)$, and each edge is drawn as a strictly $y$-monotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source a…
MIT（麻省理工学院）；Harvard University（哈佛大学）；University of Waterloo（滑铁卢大学）
Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai obj…
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Fraunhofer Institute of Optronics, System Technologies and Image Exploitation（弗劳恩霍夫光电子、系统技术与图像利用研究所）
This paper addresses the following basic question: given two layouts of the same graph, which one is more aesthetically pleasing? We propose a neural network-based discriminator model trained on a labeled dataset that decides which of two layouts has a higher aesthetic quality. The feature vectors used as inputs to the model are based on known graph drawing quality metrics, classical statistics, information-theoretical quantities, and two-point statistics inspired by methods of condensed matter physics. The large corpus of layout pairs used for training and testing is constructed using force-d…
Given a consumer data-set, the axioms of revealed preference proffer a binary test for rational behaviour. A natural (non-binary) measure of the degree of rationality exhibited by the consumer is the minimum number of data points whose removal induces a rationalisable data-set.We study the computational complexity of the resultant consumer rationality problem in this paper. This problem is, in the worst case, equivalent (in terms of approximation) to the directed feedback vertex set problem. Our main result is to obtain an exact threshold on the number of commodities that separates easy cases …
