Euclidean Voronoi Diagrams of 3D Spheres: Their Construction and Related Problems from Biochemistry

Voronoi diagrams have several important applications in science and engineering. While the properties and algorithms for the ordinary Voronoi diagrams of point sets have been well-known, their counterparts for a set of spheres have not been sufficiently studied. In this paper, we present properties and two algorithms for Voronoi diagrams of 3D spheres based on the Euclidean distance from the surface of spheres. Starting from a valid initial Voronoi vertex, the edge-tracing algorithm follows Voronoi edges until the construction is completed. The region-expansion algorithm constructs the desired diagram by successively expanding the Voronoi region of each sphere, one after another, via a series of topology operations, starting from the ordinary Voronoi diagram for the centres of spheres. In the worst-case, the edge-tracing algorithm takes O(mn) time, and the region-expansion algorithm takes O(n3 log n) time, where m and n are the numbers of edges and spheres, respectively. It should, however, be noted that the worst-case time complexity for both algorithms reduce to O(n2) for proteins since the number of immediate neighbor atoms for an atom is constant. Adapting appropriate filtering techniques to reduce search space, the expected time complexities can even reduce to linear. Then, we show how such a Voronoi diagram can be used for solving various important geometric problems in biological systems by illustrating two examples: the computation of surfaces defined on a protein, and the extraction and characterization of interaction interfaces between multiple proteins.