Section: Scientific Foundations

Fundamental geometric data structures and algorithms

Geometrica is pursuing long standing research on fundamental geometric data structures and algorithms. Geometrica has a large expertise in Voronoi diagrams and Delaunay triangulations, randomized algorithms, combinatorial geometry and related fields. Recently, we devoted efforts to developing the field of computational geometry beyond linear objects. We are especially interested in developing a theory of curved Voronoi diagrams. Such diagrams allow to model growing processes and have important applications in biology, ecology, chemistry and other fields. They also play a role in some optimization problems and in anisotropic mesh generation. Euclidean Voronoi diagrams of non punctual objects are also non affine diagrams. They are of particular interest to robotics, CAD and molecular biology. Even for the simplest diagrams, e.g. Euclidean Voronoi diagrams of lines, triangles or spheres in 3-space, obtaining tight combinatorial bounds and efficient algorithms are difficult research questions. In addition, effective implementations require to face specific algebraic and arithmetic questions. Working out carefully the robustness issues is a central objective of Geometrica (see below).

In the past years, the main objective of computational geometry has been the design of time efficient algorithms, either from the theoretical point of view of the asymptotic complexity or the more practical aspect of running efficient benchmarks. Surprisingly, less interest has been devoted to improve the space behavior of such algorithms although the problem may become of importance when the main memory is not enough and the system has to swap to find extra memory space; this may happen either for massive data or for small memory devices such as PDA. geometrica intends to attack this problem from several aspects:

— the compression of geometrical objects for storage or network transmission purposes

— the design of algorithms accessing locally the memory to reduce (but not remove) the swapping in memory

— the design of new data-structures to represent geometrical objects using less memory.

For all these aspects we are interested in both the theoretical asymptotic sizes involved and the practical aspect for reasonable input size. Such a distinction between asymptotic behavior and practical interest may appear strange since we are interested in massive data, but even for a very big mesh of several millions points which can be called ``massive data'', we are still far from the asymptotic behavior of some theoretical structures.