Towards Dynamic Randomized Algorithms in Computational Geometry

Computational geometry aims to design and analyze algorithms for solving geometric problem. It is a recent field of theorical computer science, that rapidly developed since it appeared in M.I Shamos' thesis in 1978. Randomization allows to avoid the use of complicated data structures, and has been proved to be very efficient, from both points of view of theoretical complexity and pratical results. We take particular interest in designing dynamic algorithms : in practice, the data of a problem are often acquired progressively. It is obviously not reasonable to compute the whole result again each time a new data is inserted, (semi-) dynamic schemes are thus necessary. We introduce a very general data structure, the influence graph, that allows us to construct various geometric structures : Voronoi diagrams, arrangements of line segments... We study both theoretical complexity and pratical efficiency of the algorithms.