Topological sweep of the complete graph

We present a novel, simple and easily implementable algorithm to report all intersections in an embedding of a complete graph. For graphs with NN vertices and complexity KK measured as the number of segments of the embedding, the running time of the algorithm is Θ(K+NM)Θ(K+NM), where MM is the maximum number of edges cut by any vertical line. Our algorithm handles degeneracies, such as vertical edges or multiply intersecting edges, without requiring numerical perturbations to achieve general position.The algorithm is based on the sweep line technique, one of the most fundamental techniques in computational geometry, where an imaginary line passes through a given set of geometric objects, usually from left to right. The algorithm sweeps the graph using a topological line, borrowing the concept of horizon trees from the topological sweep method [H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Syst. Sci. 38 (1989) 165–194; J. Comput. Syst. Sci. 42 (1991) 249–251 (corrigendum)].The novelty in our approach is to control the topological line through the use of the moving wall that separates at any time the graph into two regions: the region of known structure, in front of the moving wall, and the region that may contain intersections generated by edges–that have not yet been registered in the sweep process–behind the wall.Our method has applications to graph drawing and for depth-based statistical analysis, for computing the simplicial depth median for a set of NN data points [G. Aloupis, S. Langerman, M. Soss, G. Toussaint, Algorithms for bivariate medians and a Fermat-Torricelli problem for lines, Comp. Geom. Theory Appl. 26 (1) (2003) 69–79].We present the algorithm, its analysis, experimental results and extension of the method to general graphs.