Parallel Algorithms for Shared-memory Machines. a Standard Wasteful Implementation

10 technique enables a simple cost-eeective implementation with little eeort. It was used for the rst time to implement a fast optimal parallel hashing algorithm 7]. The hashing algorithm in 7] comprises two parts: the rst part is a randomized geometric decaying algorithm which runs for O(lg lg n) steps. By using the technique of this paper and the O(lg lg n) time load balancing of 6], this part was implemented optimally with expected additive overhead of O(lg lg n lg n) time, which implies an work-optimal implementation that takes O(lg lg n lg n) time, i.e., using p = n= lg lg n lg n processors. The second part runs in O(lg lg n) time using p processors, implying a work-optimal implementation for the entire algorithm that takes O(lg lg n lg n) expected time. Subsequent to a preliminary version of this paper (presented in 7]), other policies of eeective load balancing for other classes of algorithms were introduced, often using amortization arguments similar to the one used here 17, 9, 10, 5] (see also 16]). 9 in an attempt to achieve an o(lg`) overhead. Suppose that t i is set to satisfy t i x i = ` lg`, and let us ignore the extra execution time imposed on the processors due to imbalance. Then, (6) is replaced by t i = ` lg`=x i but (7) and (8) do not change. It is easy to verify that the overhead is still (lg`). Moreover, it is easy to see that the performance of the above policy (when ignoring the extra execution due to imbalance as above) can match the performance of any other policy of load balancing utilization, up to a constant factor. The policy presented in this paper is therefore optimal. 4 Conclusions This paper introduces a simple eeective policy for employing load balancing algorithms for processor scheduling in geometric-decaying algorithms. The resulting implementation is as simple as the standard implementation with the only diierence being a more careful account of when and how often load balancing should be used. The load balancing algorithm is used as a black box, and the suggested policy is thus applicable to many models of parallel computation. We deened the overhead of policies of invoking load balancing to implement Brent's scheduling principle. The overhead can be used to evaluate and compare such policies. It was shown that the overhead for our …