The 3-Server Problem in the plane (extended abstract)

In the k-Server Problem we wish to minimize, in an online fashion, the movement cost of k servers in response to a sequence of requests. The request issued at each step is specified by a point r in a given metric space M. To serve this request, one of the k servers must move to r. (We assume that k ≥ 2.) It is known that if M has at least k + 1 points then no online algorithm for the k-Server Problem in M has competitive ratio smaller than k. The best known upper bound on the competitive ratio in arbitrary metric spaces, by Koutsoupias and Papadimitriou [6], is 2k - 1. There is only a number of special cases for which k-competitive algorithms are known: for k = 2, when M is a tree, or when M has at most k + 2 points. The main result of this paper is that the Work Function Algorithm is 3-competitive for the 3-Server Problem in the Manhattan plane. As a corollary, we obtain a 4.243-competitive algorithm for 3 servers in the Euclidean plane. The best previously known competitive ratio for 3 servers in these spaces was 5.