A randomized algorithm for two servers in cross polytope spaces

It has been a long-standing open problem to determine the exact randomized competitiveness of the 2-server problem, that is, the minimum competitiveness of any randomized online algorithm for the 2-server problem. For deterministic algorithms the best competitive ratio that can be obtained is 2 and no randomized algorithm is known that improves this ratio for general spaces. For the line, Bartal et al. (1998) [2] give a competitive algorithm, but their algorithm is specific to the geometry of the line.We consider here the 2-server problem over Cross Polytope Spaces M24. We obtain an algorithm with competitive ratio of , and show that this ratio is best possible. This algorithm gives the second non-trivial example of metric spaces with better than2-competitive ratio.The algorithm uses a design technique called the knowledge state technique — a method not specific to M24.