5 Conclusions and Open Problems 4.4 an Algorithm with Subquadratic Overhead

A singly-exponential stratiication scheme for real semi-algebraic varieties and its applications. 28 If we stop because of (i), this means that throughout the way z + (y) and z ? (y) are visible to one another, and therefore the face is y-monotone. If we stop because z + (y) and z ? (y) stop seeing one another, then it must be the case that the intrusion is due to the left endpoint of a curve of S, which induces a vertical segment there. But then necessarily the union of segments z + (y)z ? (y) for y ranging from y ? through the point where we stopped constitute the face f and therefore we must have also reached y = y + , and f is evidently y-monotone in this case. 2 In summary, we have the following result Theorem 4.3 Given a collection T of n xy-monotone algebraic surface patches of constant maximum degree, bounded by algebraic curves of constant maximum degree as well, and in general position in three-dimensional space, the time needed to compute the full vertical decomposition of the arrangement A(T) is O(nn q (n) log n+ V log n), where V is the combinatorial complexity of the vertical decomposition, and q is a constant that depends on the degree of the surface patches and their boundaries. We have proved bounds on the maximum combinatorial complexity of the vertical decomposition of sets of triangles in 3-space that are sensitive to the complexity of the arrangement of triangles. We have also given a simple deterministic output-sensitive algorithm for computing the vertical decomposition, and we have extended the algorithm to handle surface patches. The paper raises several open problems: The generalization of the combinatorial result to the case of surface patches; this seems to be a major open problem in the study of general arrangements. Tightening the small gap that remains between the upper and lower bounds for vertical decompositions. We believe that the n "-factor in the upper bound is only an artifact of our proof technique and that the lower bound is closer to the truth than the upper bound. Towards the end of Subsection 4.3, we have discussed the issue of navigating around the decomposed arrangement. An interesting problem that arises in this respect is to devise an eecient decomposition (i.e., a decomposition with a small number of cells) with a constant number of neighbors …