Graph Drawing via Gradient Descent, (GD)2(GD)^2

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple readability criteria. $(GD)^2$ can optimize any criterion that can be described by a smooth function. If the criterion cannot be captured by a smooth function, a non-smooth function for the criterion is combined with another smooth function, or auto-differentiation tools are used for the optimization. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., vertex resolution, angular resolution, aspect ratio). We provide quantitative and qualitative evidence of the effectiveness of $(GD)^2$ with experimental data and a functional prototype: \url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020

A Geometric Heuristic for Rectilinear Crossing Minimization

\u3cp\u3eIn this paper we consider the rectilinear crossing minimization problem, i.e., we seek a straight-line drawing ? of a graph G = (V, E) with a small number of edge crossings. Crossing minimization is an active field of research [1,9]. While there is a lot of work on heuristics for topological drawings, these techniques are typically not transferable to the rectilinear (i.e., straight-line) setting. We introduce and evaluate three heuristics for rectilinear crossing minimization. The approaches are based on the primitive operation of moving a single vertex to its crossing-minimal position in the current drawing ?, for which we give an O (kn + m)\u3csup\u3e2\u3c/sup\u3e log (kn + m)-time algorithm, where k is the degree of the vertex and n and m are the numbers of vertices and edges of the graph, respectively. In an experimental evaluation, we demonstrate that our algorithms compute straight-line drawings with fewer crossings than energy-based algorithms implemented in the Open Graph Drawing Framework [10] on a varied set of benchmark instances. All experiments are evaluated with a statistical significance level of ? = 0.05.\u3c/p\u3

A geometric heuristic for rectilinear crossing minimization

\u3cp\u3eIn this paper we consider the rectilinear crossing minimization problem, i.e., we seek a straight-line drawing ? of a graph G = (V, E) with a small number of edge crossings. Crossing minimization is an active field of research [1,9]. While there is a lot of work on heuristics for topological drawings, these techniques are typically not transferable to the rectilinear (i.e., straight-line) setting. We introduce and evaluate three heuristics for rectilinear crossing minimization. The approaches are based on the primitive operation of moving a single vertex to its crossing-minimal position in the current drawing ?, for which we give an O (kn + m)\u3csup\u3e2\u3c/sup\u3e log (kn + m)-time algorithm, where k is the degree of the vertex and n and m are the numbers of vertices and edges of the graph, respectively. In an experimental evaluation, we demonstrate that our algorithms compute straight-line drawings with fewer crossings than energy-based algorithms implemented in the Open Graph Drawing Framework [10] on a varied set of benchmark instances. All experiments are evaluated with a statistical significance level of ? = 0.05.\u3c/p\u3

A geometric heuristic for rectilinear crossing minimization

In this paper we consider the rectilinear crossing minimization problem, i.e., we seek a straight-line drawing ? of a graph G = (V, E) with a small number of edge crossings. Crossing minimization is an active field of research [1,9]. While there is a lot of work on heuristics for topological drawings, these techniques are typically not transferable to the rectilinear (i.e., straight-line) setting. We introduce and evaluate three heuristics for rectilinear crossing minimization. The approaches are based on the primitive operation of moving a single vertex to its crossing-minimal position in the current drawing ?, for which we give an O (kn + m)2 log (kn + m)-time algorithm, where k is the degree of the vertex and n and m are the numbers of vertices and edges of the graph, respectively. In an experimental evaluation, we demonstrate that our algorithms compute straight-line drawings with fewer crossings than energy-based algorithms implemented in the Open Graph Drawing Framework [10] on a varied set of benchmark instances. All experiments are evaluated with a statistical significance level of ? = 0.05