On scheduling collaborative computations on the Internet, I: mesh-DAGs and their close relatives

Advancing technology has rendered the Internet a viable medium for collaborative computing, via mechanisms such as Web-based computing and Grid computing. We present a "pebble game" that abstracts the process of scheduling a computation-DAG (directed acyclic graph) for computing over the Internet, including a novel formal criterion for comparing the qualities of competing schedules. Within this formal setting, we identify a strategy for scheduling the task-nodes of a computation-DAG whose dependencies have the structure of a mesh of any finite dimensionality (a mesh-DAG), that is optimal to within a small constant factor (to within a low-order additive term for 2- and 3-dimensional mesh-DAG). We show that this strategy remains nearly optimal for a generalization of 2-dimensional mesh-DAG whose structures are determined by abelian monoids (a monoid-based version of Cayley graphs).