Analysis of Layered Hierarchy for Necklaces

Molecular configurations can be modeled as sets of spheres in 3D. By imposing geometric constraints on the spheres based on molecular biology one can apply efficient techniques [3], for example, to analyze the complexity of molecular surface [1]. Guibas et al. [2] introduced a model called necklace that finds applications in computer graphics, computer vision, robotics, geographic information systems and molecular biology. They studied two data structures, wrapped hierarchy and layered hierarchy, for representing necklaces and for performing collision tests. Guibas et al. [2] proved that the wrapped hierarchy admits a separating family of size O(n2−2/d). This is the first subquadratic bound proved for collision detection using predefined hierarchies. Although the layered hierarchy can be used for collision detection, “no subquadratic bound on the size of a separating family based on the layered hierarchy is currently known” [2]. The main result of this paper is that the same upper bound holds for the layered hierarchy. One of the advantages of layered hierarchy over wrapped hierarchy is the “local” definition of the cages. We propose a modification of the layered hierarchy so that some deformations of a necklace can be maintained efficiently. In particular, we show that rigid-body conformational changes can be handled in O(log n) time only.