This paper considers the problem of scheduling dynamic parallel computations to achieve linear speedup without using significantly more space per processor than that required for a single processor execution. Earlier research in the Cilk project proposed the "strict" computational model, in which every dependency goes from a thread x only to one of x's ancestor threads, and guaranteed both linear speedup and linear expansion of space. However, Cilk threads are stateless, and the task graph that Cilk language expresses is series-parallel graph, which is a proper subset of arbitrary task graph. Moreover, Cilk does not support applications with pipelining. We propose the "aligned" multithreaded computational model, which extends the "strict" computational model in Cilk. In the aligned multithreaded computational model, dependencies can go from arbitrary thread x not only to x's ancestor threads, but also to x's younger brother threads, that are spawned by x's parent thread but after x. We use the same measures of time and space as those used in Cilk: T1 is the time required for executing the computation on 1 processor, T∞ is the time required by an infinite number of processors, and S1 is the space required to execute the computation on 1 processor. We show that for any aligned computation, there exists an execution schedule that achieves both efficient time and efficient space. Specifically, we show that for an execution of any aligned multithreaded computation on P processors, the time required is bounded by O(T1/P + T∞), and the space required can be loosely bounded by O(λ·S1P), where λ is the maximum number of younger brother threads that have the same parent thread and can be blocked during execution. If we assume that λ is a constant, and the space requirements for elder and younger brother threads are the same, then the space required would be bounded by O(S1P). Based on the aligned multithreaded computational model, we show that the aligned multithreaded computational model supports pipelined applications. Furthermore, we propose a multithreaded programming language and show that it can express arbitrary task graph.
Parallel Performance Investigations of an Unstructured Mesh Navier-Stokes Solver