A decomposition theorem for task systems and bounds for randomized server problems

A lower bound of Ω(√log k/ log log k) is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k + 1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of Ω(log log k) for arbitrary metric spaces [H.J. Karloff, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293-312] and more closely approaches the conjectured lower bound of Ω(log k). For the server problem on k + 1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of Ω(log k/ log log k) is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the "pursuit-evasion game." It is shown that if a metric space script M sign can be decomposed into two spaces script M sign_ℒ and script M sign_{script R sign} such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on script M sign can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.