Static randomized incremental algorithms

Abstract

We presented previous works on incremental randomized algorithms. A common point to all these algorithms is that the results hold whatever the point distribution might be, contrary to more classical analyses such as in [Dwy91]. The only condition is that the points must be inserted in random order.

These algorithms are all static because they use the conflict graph. This structure has been successfully used by a lot of authors, to compute various geometric structures. Let us mention [MMO91] who compute abstract Voronoi diagrams, [Mul88, Mul89a] who constructs arrangements in the plane, [Mul89b, Mul91a] who studies levels in arrangements of hyperplanes, and of course [CS89] who develop numerous applications. One of the aims of following chapters is to design a fully dynamic structure.

Let us notice that, in the theorems proved up to now, the cost of inserting an object is dominated by the cost of updating the conflict graph, that is not directly a function of the size of the change in the desired structure. We can roughly say that the cost of inserting the first objects is high, while the change in the output is of constant size.

Numerous other results have been obtained in parallel or after ours. Some of these results will be summarized in the chapters presenting our work, and the other ones together form Chapter 7.