Space-Efficient Scheduling of Multithreaded Computations

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. Utilizing a new graph-theoretic model of multithreaded computation, execution efficiency is quantified by three important measures: T1 is the time required for executing the computation on a 1 processor, $T_\infty$ is the time required by an infinite number of processors, and S1 is the space required to execute the computation on a 1 processor. A computation executed on P processors is time-efficient if the time is $O(T_1/P + T_\infty)$, that is, it achieves linear speedup when $P=O(T_1/T_\infty)$, and it is space-efficient if it uses O(S1P) total space, that is, the space per processor is within a constant factor of that required for a 1-processor execution. The first result derived from this model shows that there exist multithreaded computations such that no execution schedule can simultaneously achieve efficient time and efficient space. But by restricting attention to "strict" computations---those in which all arguments to a procedure must be available before the procedure can be invoked---much more positive results are obtainable. Specifically, for any strict multithreaded computation, a simple online algorithm can compute a schedule that is both time-efficient and space-efficient. Unfortunately, because the algorithm uses a global queue, the overhead of computing the schedule can be substantial. This problem is overcome by a decentralized algorithm that can compute and execute a P-processor schedule online in expected time $O(T_1/P + T_\infty\lg P)$ and worst-case space $O(S_1P\lg P)$, including overhead costs.