Competitive Algorithms for Generalized k-Server in Uniform Metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server $si$ lies in its own metric space $Mi$. A request is a k-tuple $r = (r1,r2,\dotsc,rk)$ and to serve it, we need to move some server $si$ to the point $ri \in Mi$, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio $O(k 2^k)$ and $O(k³ \log k)$ respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of $2^k-1$. We also give a $2^{{2{O(k)}}$-competitive} deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.