ON THE GRANULARITY AND CLUSTERING OF DIRECTED ACYCLIC TASK GRAPHS

In this paper we consider the impact of the granularity on scheduling task graphs. Scheduling consists of two parts, the processors assignment of tasks and the ordering of tasks for execution in each processor. The processor assignment part is also known as clustering in the literature when there is no limitation in the number of processors and the architecture is completely connected. We introduce two types of clusterings, the nonlinear and linear clusterings. A clustering is nonlinear if two parallel tasks are mapped in the same cluster otherwise is linear. Linear clustering fully exploits the natural parallelism of a given DAG while nonlinear clustering sequentializes independent tasks to reduce parallelism. We also introduce a new quantification of the granularity of a DAG and define a coarse grain DAG as the one whose granularity is greater than one. We prove the following interesting result: Every nonlinear clustering of a coarse grain DAG can be transformed into a linear clustering which has less or equal parallel time than the nonlinear. We use this result to prove the optimality of some important linear clusterings used in parallel numerical computing. We also present experiments with an actual architecture that verify our theoretical results. These results provide a justification for the popularity of linear clustering in the literature.