Représentations compactes de structures de données géométriques

Abstract : We consider the problem of designing compact and succint representations for geometric data structures. As opposed to raw compression issues,the focus is here on designing data structures that preserve the possibility of answering local incidence queries in constant time, while using little as memory resources as possible. One of the main contribution of this work is to present a general algorithmic framework for designing compact representations of structures as planar graphs and 3D meshes. As application we propose some solutions for compactly representing the connectivity (combinatorial information) of some classes of local planar graphs. For planar triangulations with m faces, we propose a compact representation of the connectivity information that improves to 2:175 bits per triangle the asymptotic amount of space required and that supports navigation between adjacent triangles in constant time : this is optimal for the class of triangulations with a boundary of arbitrary sizes. For triangulations of a surface with bounded genus g, our representation is still valid and optimal, requiring O(g logm) extra bits. Such a representation can be adapted to allow local updates of the triangulation. More precisely, a triangulation with m triangles can be maintained under vertex insertions in O(1) amortized time and under vertex deletions/edge °ips in O(log2m) amortized time. We also propose the ¯rst optimal representations for 3-connected planar graphs and triangulations, which are the standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1:62 bits per triangle (or equivalently 3:24 bits per vertex) for triangulations. These structures allow constant time access to vertex specific data, like coordinates, but our work does not address the compression of this geometric information.
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Luca Castelli Aleardi. Représentations compactes de structures de données géométriques. Informatique [cs]. Ecole Polytechnique X, 2006. Français. ⟨tel-00336188⟩

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