Parallel computation thesis

Abstract: A number of different models of synchronous, unbounded parallel computers have appeared in recent literature. Without exception, running time on these models has been shown to be polynomially related to the classical space complexity measure. The general applicability of this relationship is called “the parallel computation thesis” and strong evidence of its truth is given in this paper by introducing the notion of “conglomerates” - a very large class of parallel machines, including all those which could feasibly be built. Basic parallel machine models are also investigated, in an attempt to pin down the notion of parallel time to within a constant factor. To this end, a universal conglomerate structure is developed with can simulate any other basic model within linear time. This approach also leads to fair estimates of instruction execution times for various parallel models.

Abstract: There are efficient sequential algorithms that use linear programming (LP) for computing maximum weight matchings. Finding a deterministic parallel algorithm for computing maximum weight matchings in complete graphs has been an open problem for some time. Since LP is known to be P-complete, then, by the parallel computation thesis, it is unlikely that there exists an NC algorithm that uses LP to solve the maximum weight matching problem. The authors present an LP-based parallel algorithm for maximum weight matching in a complete weighted graph. The algorithm is designed for the EREW PRAM model of parallel computation, and runs in O(n/sup 3//p+n/sup 2/logn) time for p >

Abstract: It is reasonable to expect parallel machines to be faster than sequential ones. But exactly how much faster do we expect them to be? Various authors have observed that an exponential speedup is possible if sufficiently many processors are available. One such author has claimed (erroneously) that this is a counterexample to the parallel computation thesis. We show that even more startling speedups are possible. In fact, if enough processors are used, any recursive function can be computed in constant time. Although such machines clearly do not obey the parallel computation thesis, we argue that they still provide evidence in favour of it. In contrast, we show that an arbitrary polynomial speedup of sequential machines is possible on a model which satisfies the parallel computation thesis. If, as widely conjectured, P⊈POLYLOGSPACE, then there can be no exponential speedup on such a model.

Abstract: The complexity theory of synchronous parallel computation is studied in this thesis. It is shown that there is a precise relationship between space usage on traditional models of sequential computation and hardware usage on two new models of parallel computation. These two new models of parallelism, called hardware modification machines and aggregates, are also shown to be closely related to existing parallel models, such as combinational circuits; and variants of the new models are used to examine non-uniform and non-deterministic complexity classes. It is also shown that a close relationship exists between the reversal used in a sequential computation and the time used in a corresponding parallel computation. Moreover, this new relationship can be made to hold simultaneously with the hardware/space relationship mentioned above. This simultaneous correspondence between sequential and parallel resource usage can be used as evidence for an extended version of the "parallel computation thesis", which attempts to relate the intuitive concepts of parallel time and hardware to formal sequential resource usage. We study the interrelationships of these two parallel resources, time and hardware, and show that to within polynomial factors, they define the same complexity classes. Finally, we examine some specific simultaneous parallel complexity classes and show them to be the same as previously studied sequential simultaneous classes.