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Querying simplexes in quasi-triangulation
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DOI：10.1016/j.cad.2011.09.010.png)
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Given a quasi-triangulation, the dual structure of the Voronoi diagram of a molecule, querying its simplexes of a particular bounding state for the spherical probe of a given radius occurs frequently. The beta-complex and the beta-shape are such examples with various applications for reasoning the spatial structure of molecules. While such simplexes can be found by linearly scanning all simplexes in the quasi-triangulation, it is desirable to do the query faster. This paper introduces the Q2P-transformation that transforms the simplex query problem in the three-dimensional space to the orthogonal range search problem of points in another three-dimensional space. Based on this observation, we show that the kd-tree data structure suits very well for developing efficient query algorithms for the quasi-triangulation of molecules using the set of sixteen primitive and the set of ten high-level query operators. In particular, the proposed method shows an extremely efficient performance for incremental queries. A subset of the presented observations facilitates efficient simplex queries for any simplicial complexes including the Delaunay and regular triangulations. (C) 2011 Elsevier Ltd. All rights reserved.
Keyword:
Voronoi diagram of spheres
Beta-shape
Beta-complex
Quasi-triangulation
Simplex query
Particle proximity
Orthogonal range search
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