Trash removal algorithm for fast construction of the elliptic Gabriel graph using Delaunay triangulation

Given a set of points P, finding near neighbors among the points is an important problem in many applications in CAD/CAM, computer graphics, computational geometry, etc. In this paper, we propose an efficient algorithm for constructing the elliptic Gabriel graph (EGG), which is a generalization of the well-known Gabriel graph and parameterized by a non-negative value α. Our algorithm is based on the observation that a candidate point which may define an edge of an EGG with a given point p∈P is always in the scaled Voronoi region of p with a scale factor 2/α² when the parameter α<1. From this observation, we can reduce the search space for locating neighbors of p in the EGG. For the case of α≥1, due to the fact that EGG is a subgraph of the Delaunay graph of P, EGG can be efficiently computed by watching the validity of each edge in the Delaunay graph. The proposed algorithm is shown to have its time complexity as O(n³) in the worst case and O(n) in the average case when α is moderately close to unity. The idea presented in this paper may similarly apply to other problems for the proximity search for point sets.