Anomaly Occurrences in Quasi-triangulations and Beta-complexes

Voronoi diagrams, quasi-triangulations, and beta-complexes are powerful for solving spatial problems among spherical particles with different radii. However, a quasi-triangulation, and thus a beta-complex as well, can be a non-simplicial complex due to an anomaly condition. While a beta-complex is straightforward to use when it is a simplicial complex, it may not seem obvious if it is not. In this paper, we report the experimental statistics of showing the frequency of anomaly case occurrences in both quasi-triangulations and beta-complexes of molecular structures and randomly generated models in three-dimension. The experiment was based on 100 molecular structures from the protein data bank (PDB) and four random sets where each set consists of 100 models of three-dimensional spheres. Anomalies extremely rarely occur in molecular structures and rarely occur even in random sphere sets.