A β-Shape from the Voronoi Diagram of Atoms for Protein Structure Analysis

Abstract

In this paper, we present a β-shape and a β-complex for a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known α-shape and α-complex (and their weighted counterparts as well). To compute a β-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasi-triangulation which is the topological dual of the Voronoi diagram. Then, we compute a β-complex from the quasi-triangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a β-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m is the number of simplices in the quasi-triangulation. Then, a β-shape for a particular β consisting of k simplices can be located in O(log m + k) time in the worst case from the simplicies in the β-complex sorted according to the interval values.