Competitive Algorithms for Generalized k-Server in Uniform Metrics

TL;DR

This paper introduces the first competitive algorithms for the generalized k-server problem in uniform metrics, achieving bounds close to known lower bounds, and extends results to weighted uniform metrics.

Contribution

It provides the first f(k)-competitive algorithms for the generalized k-server problem in uniform metrics, using novel polynomial method techniques.

Findings

Deterministic algorithm with O(k 2^k) competitive ratio

Randomized algorithm with O(k^3 log k) competitive ratio

Weighted uniform metrics algorithm with 2^{2^{O(k)}} competitive ratio

Abstract

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 $r = (r_1,r_2,\dotsc,r_k)$ and to serve it, we need to move some server $s_i$ to the point $r_i \in M_i$, 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 $O(k 2^k)$ 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.