The geometric problems encountered in this chapter are again stated in the abstract framework of objects, regions, and conflicts introduced in chapter 4. A dynamic algorithm maintains the set of regions defined and without conflict over the current set of objects, when the objects can be removed from the current set as well as added. In contrast, on-line algorithms that support insertions but not deletions are sometimes called semi-dynamic.

Throughout this chapter, we denote by S the current set of objects and use the notation introduced in the previous two chapters to denote the different subsets of regions defined over S. In particular, F0(S) stands for the set of regions defined and without conflict over S. To design a dynamic algorithm that maintains the set F0(S) is a much more delicate problem than its static counterpart. In the previous chapter, we have shown how randomized incremental methods provide simple solutions to static problems. In addition, the influence graph techniques naturally lead to the design of semi-dynamic algorithms. In this chapter, we propose to show how the combined use of both conflict and influence graphs can yield fully dynamic algorithms.

The general idea behind our approach is to maintain a data structure that meets the following two requirements:

• It allows conflicts to be detected between any object and the regions defined and without conflict over the current subset.

• After deleting an object, the structure is identical to what it would have been, had the deleted object never been inserted.