Achieving Good Angular Resolution in 3D Arc Diagrams

We study a three-dimensional analogue to the well-known graph visualization approach known as arc diagrams. We provide several algorithms that achieve good angular resolution for 3D arc diagrams, even for cases when the arcs must project to a given 2D straight-line drawing of the input graph. Our methods make use of various graph coloring algorithms, including an algorithm for a new coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on Graph Drawing (GD 2013

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

A Tabu Search Based Approach for Graph Layout

This paper describes an automated tabu search based method for drawing general graph layouts with straight lines. To our knowledge, this is the first time tabu methods have been applied to graph drawing. We formulated the task as a multi-criteria optimization problem with a number of metrics which are used in a weighted fitness function to measure the aesthetic quality of the graph layout. The main goal of this work is to speed up the graph layout process without sacrificing layout quality. To achieve this, we use a tabu search based method that goes through a predefined number of iterations to minimize the value of the fitness function. Tabu search always chooses the best solution in the neighbourhood. This may lead to cycling, so a tabu list is used to store moves that are not permitted, meaning that the algorithm does not choose previous solutions for a set period of time. We evaluate the method according to the time spent to draw a graph and the quality of the drawn graphs. We give experimental results applied on random graphs and we provide statistical evidence that our method outperforms a fast search-based drawing method (hill climbing) in execution time while it produces comparably good graph layouts.We also demonstrate the method on real world graph datasets to show that we can reproduce similar results in a real world setting

An Interactive Tool to Explore and Improve the Ply Number of Drawings

Given a straight-line drawing $\Gamma$ of a graph $G=(V,E)$, for every vertex $v$ the ply disk $D_v$ is defined as a disk centered at $v$ where the radius of the disk is half the length of the longest edge incident to $v$. The ply number of a given drawing is defined as the maximum number of overlapping disks at some point in $\mathbb{R}^2$. Here we present a tool to explore and evaluate the ply number for graphs with instant visual feedback for the user. We evaluate our methods in comparison to an existing ply computation by De Luca et al. [WALCOM'17]. We are able to reduce the computation time from seconds to milliseconds for given drawings and thereby contribute to further research on the ply topic by providing an efficient tool to examine graphs extensively by user interaction as well as some automatic features to reduce the ply number.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Incremental Grid-like Layout Using Soft and Hard Constraints

We explore various techniques to incorporate grid-like layout conventions into a force-directed, constraint-based graph layout framework. In doing so we are able to provide high-quality layout---with predominantly axis-aligned edges---that is more flexible than previous grid-like layout methods and which can capture layout conventions in notations such as SBGN (Systems Biology Graphical Notation). Furthermore, the layout is easily able to respect user-defined constraints and adapt to interaction in online systems and diagram editors such as Dunnart.Comment: Accepted to Graph Drawing 201

GPU-Based Parallel Particle Swarm Optimization Methods for Graph Drawing

Particle Swarm Optimization (PSO) is a population-based stochastic search technique for solving optimization problems, which has been proven to be effective in a wide range of applications. However, the computational efficiency on large-scale problems is still unsatisfactory. A graph drawing is a pictorial representation of the vertices and edges of a graph. Two PSO heuristic procedures, one serial and the other parallel, are developed for undirected graph drawing. Each particle corresponds to a different layout of the graph. The particle fitness is defined based on the concept of the energy in the force-directed method. The serial PSO procedure is executed on a CPU and the parallel PSO procedure is executed on a GPU. Two PSO procedures have different data structures and strategies. The performance of the proposed methods is evaluated through several different graphs. The experimental results show that the two PSO procedures are both as effective as the force-directed method, and the parallel procedure is more advantageous than the serial procedure for larger graphs

Compressed sensing using sparse binary measurements: a rateless coding perspective

Compressed Sensing (CS) methods using sparse binary measurement matrices and iterative message-passing re- covery procedures have been recently investigated due to their low computational complexity and excellent performance. Drawing much of inspiration from sparse-graph codes such as Low-Density Parity-Check (LDPC) codes, these studies use analytical tools from modern coding theory to analyze CS solutions. In this paper, we consider and systematically analyze the CS setup inspired by a class of efficient, popular and flexible sparse-graph codes called rateless codes. The proposed rateless CS setup is asymptotically analyzed using tools such as Density Evolution and EXIT charts and fine-tuned using degree distribution optimization techniques