A Primal-Dual Randomized Algorithm for Weighted Paging

We study the weighted version of the classic online paging problem where there is a weight (cost) for fetching each page into the cache. We design a randomized O (log k )-competitive online algorithm for this problem, where k is the cache size. This is the first randomized o ( k )-competitive algorithm and its competitive ratio matches the known lower bound for the problem, up to constant factors. More generally, we design an O (log( k /( k − h + 1)))-competitive online algorithm for the version of the problem where the online algorithm has cache size k and it is compared to an optimal offline solution with cache size h ≤ k . Our solution is based on a two-step approach. We first obtain an O (log k )-competitive fractional algorithm based on an online primal-dual approach. Next, we obtain a randomized algorithm by rounding in an online manner the fractional solution to a probability distribution on the possible cache states. We also give an online primal-dual randomized O (log N )-competitive algorithm for the Metrical Task System problem (MTS) on a weighted star metric on N leaves.