Randomized Memoryless Algorithms for the Weighted and the Generalized k -server Problems

Abstract

The weighted k -server problem is a generalization of the k -server problem wherein the cost of moving a server of weight β i through a distance d is β i ⋅ d . On uniform metric spaces, this models caching with caches having different page replacement costs. A memoryless algorithm is an online algorithm whose behavior is independent of the history given the positions of its k servers. In this article, we develop a framework to analyze the competitiveness of randomized memoryless algorithms. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. Using this, we establish tight bounds on the competitive ratio achievable by randomized memoryless algorithms for the weighted k -server problem on uniform metrics. We first prove that there is an α k -competitive memoryless algorithm for this problem, where α k =α k − 1 2 + 3α k − 1 +1; α 1 = 1. We complement this result by proving that no randomized memoryless algorithm can have a competitive ratio less than α k . Finally, we prove that the above bounds also hold for the generalized k -server problem on weighted uniform metrics.