A Semi-Dynamic Construction of Higher Order Vorono Diagrams and its Randomized Analysis Une construction dynamique des diagrammes de Vorono

The k-Delaunay tree extends the Delaunay tree introduced in [1,2]. It is a hierarchical data structure that allows the semi-dynamic construction of the higher order Vorono diagrams of a nite set of n points in any dimension. In this paper, we prove that a randomized construction of the k-Delaunay tree, and thus, of all the order k Vorono diagrams, can be done in O(n logn + k 3 n) expected time and O(k 2 n) expected storage in the plane, which is asymptotically optimal for xed k. Our algorithm extends to d dimensional space with expected time complexity O k d d+1 2 e +1 n b d+1 2 c and space complexity O k d d+1 2 e n b d+1 2 c . The algorithm is simple and experimental results are given.