Managing Storage for Multithreaded Computations

Multithreading has become a dominant paradigm in general purpose MIMD parallel computation. To execute a multithreaded computation on a parallel computer, a scheduler must order and allocate threads to run on the individual processors. The scheduling algorithm dramatically a ects both the speedup attained and the space used when executing the computation. We consider the problem of scheduling multithreaded computations to achieve linear speedup without using signi cantly more space-per-processor than required for a single-processor execution. We show that for general multithreaded computations, no scheduling algorithm can simultaneously make e cient use of space and time. In particular, we show that there exist multithreaded computations such that any execution schedule X that achieves P -processor execution time TP (X ) T1= , where T1 is the minimum possible serial execution time, must use space at least SP (X ) 1 4( 1) p T1 + S1, where S1 is the space used by an e cient serial execution. For such a computation, even achieving a factor of 2 speedup ( = 2) requires space proportional to the square root of the serial execution time. By restricting ourselves to a class of computations we call strict computations, however, we show that there exist schedulers that can provide both e cient speedup and use of space. Speci cally, we show that for any strict multithreaded computation and any number P of processors, there exists an execution schedule X that achieves time TP (X ) T1=P + T1, where T1 is a lower bound on execution time even for arbitrarily large numbers of processors, and space SP (X ) S1P . We demonstrate such schedules by exhibiting a simple centralized algorithm to compute them. We give a second, somewhat more e cient, algorithm that computes equally good execution schedules; this algorithm is online and should be practical for moderate numbers of processors, but its use of a centralized queue makes it ine cient for large numbers of processors. To demonstrate an algorithm that is e cient even for large machines, we give a randomized, distributed, and online scheduling algorithm that computes an execution schedule X that achieves guaranteed space SP (X ) = O(S1P lg P ) and expected time E [TP (X )] = O(T1=P+T1 lg P ). Though this algorithm uses a lg P factor more space than the centralized algorithm, it can still achieve linear expected speedup | that is E [TP (X )] = O(T1=P ) | provided