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 = (r₁,r₂,… ,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(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 2^2O(k)-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.

广义k -server 问题是k -server 问题的深远扩展，具有多种应用。这里，每个服务器s _i位于它自己的度量空间M _i中。请求是一个k元组r = ( r ₁ , r ₂ ,… , r _k，它通过将一些服务器s _i移动到点r _i ∈ M _i来服务，目标是最小化通过的总距离服务器。尽管做了很多工作，没有f ( k)-competitive algorithm 以k > 2 服务器的问题而闻名，即使对于统一度量和线路等特殊情况也是如此。

在这里，我们考虑统一度量中的问题，并给出一般k的第一个f ( k ) 竞争算法。特别是，我们分别获得了竞争比为k · 2 ^k和O ( k ³ log k ) 的确定性和随机算法。我们的确定性界限基于多项式方法在在线算法中的新颖应用，并且基本上与众所周知的下界 2 ^k -1 相匹配。我们还给出 2 ^{2 O(k)}- 用于加权统一度量的竞争确定性算法，它也基本上匹配问题的最近双指数下界。