We present new graph-theoretical conditions for inscribable polyhedra and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is inscribable. These results have several consequences:

All 4-connected polyhedra are inscribable.

All simplical polyhedra in which all vertex degrees are between 4 and 6, inclusive, are inscribable.

All triangulations without chords or nonfacial triangles are realizable as Delaunay triangulations.

We also strengthen some earlier results about matchings in inscribable polyhedra. Specifically, we show that any nonbipartite inscribable polyhedron has a perfect matching containing any specified edge, and that any bipartite inscribable polyhedron has a perfect matching containing any two specified disjoint edges. We give examples showing that these results are best possible.

We characterize the conditions under which completing a Delaunay tessellation produces a configuration which is a nondegenerate Delaunay triangulation of an arbitrarily small perturbation of the original sites. One consequence of this result is a simple method for resolving degeneracies in Delaunay triangulations that does not require symbolic perturbation of the data.

Computational grids have the potential for solving large-scale scientific problems using heterogeneous and geographically distributed resources. However, a number of major technical hurdles must be overcome before this potential can be realized. One problem that is critical to effective utilization of computational grids is the efficient scheduling of jobs. This work addresses this problem by describing and evaluating a grid scheduling architecture and three job migration algorithms. The architecture is scalable and does not assume control of local site resources. The job migration policies use the availability and performance of computer systems, the network bandwidth available between systems, and the volume of input and output data associated with each job. An extensive performance comparison is presented using real workloads from leading computational centers. The results, based on several keymetrics, demonstrate that the performance of our distributed migration algorithms is significantly greater than that of a local scheduling framework and comparable to a non-scalable global scheduling approach.