We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

The two-dimensional Movers' Problem may be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional robot system B, determine whether one can move B from a given placement to another without touching any obstacle, and plan such a motion when one exists. Efficient algorithms are presented for the two special cases in which B is either a disc or a straightline segment, running respectively in time 0(n log n) and 0(n2 log n). To solve the problem for a disc one uses the planar Voronoi diagram determined by the obstacles; in the case of a line-segment one generalizes the notion of Voronoi diagram to the 3-dimensional configuration space of the moving segment.

We consider the notion of a (data) format where each format defines a family of data structures. These formats arose from the theory of databases. Previous works have investigated the notion of generic transformations of data structures between formats. We give a novel grouptheoretic view of genericity which unifies the original approaches of Hull-Yap and Aho-Ullman. Among the results are: A necessary and sufficient condition for the existence of generic embeddings; the fact that digraphs cannot be generically embedded in hypergraphs; the striking fact that there is no hypergraph on more than two vertices with the alternating group as its automorphism group, and combinatorial techniques for counting structures with a prescribed automorphism group.