Optimal Prediction for Prefetching in the Worst Case

AMS subject classi cations. 68Q25, 68T05, 68P20, 68N25, 60J20 PII. S0097539794261817Response time delays caused by I/O are a major problem in many systems and database applications. Prefetching and cache replacement methods are attracting renewed attention because of their success in avoiding costly I/Os. Prefetching can be looked upon as a type of online sequential prediction, where the predictions must be accurate as well as made in a computationally e cient way. Unlike other online problems, prefetching cannot admit a competitive analysis, since the optimal o ine prefetcher incurs no cost when it knows the future page requests. Previous analytical work on prefetching [J. Assoc. Comput. Mach., 143 (1996), pp. 771{793] consisted of modeling the user as a probabilistic Markov source. In this paper, we look at the much stronger form of worst-case analysis and derive a randomized algorithm for pure prefetching. We compare our algorithm for every page request sequence with the important class of nite state prefetchers, making no assumptions as to how the sequence of page requests is generated. We prove analytically that the fault rate of our online prefetching algorithm converges almost surely for every page request sequence to the fault rate of the optimal nite state prefetcher for the sequence. This analysis model can be looked upon as a generalization of the com- petitive framework, in that it compares an online algorithm in a worst-case manner over all sequences with a powerful yet nonclairvoyant opponent. We simultaneously achieve the computational goal of implementing our prefetcher in optimal constant expected time per prefetched page using the optimal dynamic discrete random variate generator of Matias, Vitter, and Ni [Proc. 4th Annual SIAM/ACM Symposium on Discrete Algorithms, Austin, TX, January 1993]