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 s i lies in its own metric space M i . A request is a k -tuple r = ( r 1 , r 2 ,… , r k , which is served by moving some server s i to the point r i ∈ M i , 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 k · 2 k and O ( k 3 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.