In the reordering buffer problem, we are given an input sequence of requests for service each of which corresponds to a point in a metric space. The cost of serving the requests heavily depends on the processing order. When serving a request the cost is equal to the distance, in the metric space, between this request and the previously served request. A reordering buffer with storage capacity $k$ can be used to reorder the input sequence in a restricted fashion so as to construct an output sequence with lower service cost. This simple and universal framework is useful for many applications in computer science and economics, e.g., disk scheduling, rendering in computer graphics, or painting shops in car plants.

In this paper, we design online algorithms for the reordering buffer problem where the goal is to minimize the total cost. Our main result is a strategy with a polylogarithmic competitive ratio for general metric spaces. Previous work on the reordering buffer problem only considered very restricted metric spaces. We obtain our result by first developing a deterministic algorithm for weighted trees whose competitive ratio depends on $k$ and the hop-diameter of the tree. Then we show how to improve this competitive ratio to $O(\log^2 k)$ for metric spaces that correspond to hierarchically well-separated trees. Combining this result with the results on the probabilistic approximation of arbitrary metrics by tree metrics due to Fakcharoenphol, Rao, and Talwar, we obtain a randomized strategy for general metric spaces that achieves a competitive ratio of $O(\log^2 k \cdot \log n)$ in expectation against an oblivious adversary. Here $n$ denotes the number of distinct points in the metric space. Note that the length of the input sequence can be much larger than $n$.