Quasi-triangulation and interworld data structure in three dimensions

Kim, Deok-Soo; Kim, Donguk; Cho, Youngsong; Sugihara, Kōkichi

doi:10.1016/j.cad.2006.04.008

Cited by 71 publications

(49 citation statements)

References 10 publications

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“…A quasi-triangulation may have a hierarchy of worlds where there can be one or more small world(s) underneath the root world. [27,58,59] Being the subset of a quasi-triangulation, the zero beta-complex has a root world which may or may not contain one or more small world(s). By definition, the vdW-volume corresponding to the atoms in small worlds in the quasi-triangulation is a subset of the vdWvolume corresponding to the atoms of the root world.…”

Section: Interior Tetrahedron (Tetra4): β-Cellmentioning

confidence: 99%

“…The topology of quasi-triangulation is stored in the (extended) Interworld data structure that requires only O(n) memory where n is the number of atoms in A. [27,58] Unlike the Delaunay triangulation and the regular triangulation, the quasi-triangulation is not necessarily a simplicial complex due to anomalies and small worlds. For details, see Refs.…”

Section: Voronoi Diagram Quasi-triangulation and The Beta-complexmentioning

confidence: 99%

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Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

et al. 2012

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Given a set of spherical balls, called atoms, in three-dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well-solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta-decomposition, based on the recent theory of the beta-complex. Given the beta-complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta-complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta-complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in ℝ(3) that m = O(n) on average for biomolecules and m = O(n(2)) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ(2) and extended to ℝ(3). The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.

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Section: Interior Tetrahedron (Tetra4): β-Cellmentioning

confidence: 99%

Section: Voronoi Diagram Quasi-triangulation and The Beta-complexmentioning

confidence: 99%

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

et al. 2012

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“…On the other hand, the topology of the quasi-triangulation can be stored in a much simpler data structure called the Inter-world data structure which consists of three arrays (i.e. a vertex array, a tetrahedron array, and a gate array) (21) . For the definition and properties of the quasi-triangulation, see (21), (22).…”

Section: Quasi-triangulationmentioning

confidence: 99%

“…The computation of the Voronoi diagram can take O(n 3 ) time in the worst case for general three-dimensional spheres and takes O(n) time on average for molecular atoms where n is the number of input atoms (19) . The conversion between the Voronoi diagram and the quasi-triangulation takes O(m) time in the worst case where m is the number of geometric entities in either structure (21) . Fig.…”

Section: Topology Structure Computation For Voronoi Diagram/quasi-trimentioning

confidence: 99%

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BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

Cho

Kim

Ryu

et al. 2012

JAMDSM

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Molecular shape is one of the most critical factors that determines molecular function. Therefore, it is frequently desirable to understand geometric characteristics of a molecule more precisely and efficiently. In this paper, we introduce the BetaMol, a molecular modeling, analysis, and visualization software based on the recent theory of the beta-complex and the quasi-triangulation that are derived from the Voronoi diagram of three-dimensional spherical atoms. The powerful features of the BetaMol are solely based on a unified, single framework of the mathematically rigorous and computationally efficient beta-complex theory. The BetaMol is implemented in the standard C++ language with OpenGL graphics library and freely available at Voronoi Diagram Research Center web site (http://voronoi.hanyang.ac.kr).

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Voronota: A fast and reliable tool for computing the vertices of the Voronoi diagram of atomic balls

Olechnovič

Venclovas

2014

J Comput Chem

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The Voronoi diagram of balls, corresponding to atoms of van der Waals radii, is particularly well-suited for the analysis of three-dimensional structures of biological macromolecules. However, due to the shortage of practical algorithms and the corresponding software, simpler approaches are often used instead. Here, we present a simple and robust algorithm for computing the vertices of the Voronoi diagram of balls. The vertices of Voronoi cells correspond to the centers of the empty tangent spheres defined by quadruples of balls. The algorithm is implemented as an open-source software tool, Voronota. Large-scale tests show that Voronota is a fast and reliable tool for processing both experimentally determined and computationally modeled macromolecular structures. Voronota can be easily deployed and may be used for the development of various other structure analysis tools that utilize the Voronoi diagram of balls. Voronota is available at: http://www.ibt.lt/bioinformatics/voronota.

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Quasi-triangulation and interworld data structure in three dimensions

Cited by 71 publications

References 10 publications

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

Voronota: A fast and reliable tool for computing the vertices of the Voronoi diagram of atomic balls

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