Geometric Heuristics for Rectilinear Crossing Minimization

Abstract

In this article, we consider the rectilinear crossing minimization problem , i.e., we seek a straight-line drawing Γ of a graph G =( V , E ) with a small number of edge crossings. Crossing minimization is an active field of research [1, 10]. 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 Γ, 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 number 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 O pen G raph D rawing F ramework  [11] on a varied set of benchmark instances. Additionally, we show that the difference of the number of crossings of topological drawings computed with the edge insertion approach [10, 13] and the number of crossings in straight-line drawings computed by our heuristic is relatively small. All experiments are evaluated with a statistical significance level of α = 0.05.